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\begin{document}

 \title{Optimal Control, Optimization and Analytical Mechanics}


 \author{Anthony M. Bloch\thanks{
 Research partially supported by National Science
 Foundation PYI grant DMS--9496221, AFOSR grant 
 F49620-96-1-0100 and a Guggenheim Fellowship.}
 \\Department of Mathematics
 \\ University of Michigan \\ Ann Arbor MI 48109
 \and  Peter E. Crouch\thanks{
 Work supported in part by NSF grant DMS 91011964 and
 NATO grant CRG 910926
 }
 \\ Center for Systems Science
 \\and Engineering
 \\ Arizona State University
 \\Tempe, AZ 85287}



 \date{\small this version: Jan 22, 1998 \\}
 \maketitle

\centerline {\it Dedicated to Roger Brockett on the occasion of
his 60th birthday}

 \begin{abstract}
 In this paper we discuss some of the contributions of Roger Brockett
to problems in optimal control theory, optimization and analytical mechanics.
In particular we consider how ideas in one field have helped to shed
light on the others. We discuss this interaction not only in the work
of Brockett but in our own work and the work of others that has
been inspired by the work of Brockett over the years.

\end{abstract}

% \noindent {\bf Keywords:} 

 \section{Introduction}
In this paper we describe the work of Roger Brockett and other work that
has been inspired by him in the areas of optimal control and analytical
mechanics. We consider particularly the beneficial interaction
between these two areas and describe particular examples that have been
of interest to Roger Brockett, ourselves and other researchers in these
areas.

There is of course a very natural connection between mechanics and 
optimal control through the maximum principle (see Lee and Markus
[1976], %\cite{}
Sussman [1997], Pontryagin et al. [1962])%\cite{}
, which yields from
an optimal control problem a Hamiltonian system. Indeed some of the
history of this connection, going back to Bernoulli's brachystochrone
problem, was beautifully presented in a recent article by Sussmann and 
Willems [1997]. The Lagrangian, or constrained variational, point of view
also turns out to be illuminating.

Before concentrating on optimal control however, let us recall some of
Brockett's early perspectives  on control and mechanics as
described for example in  Brockett [1976]. 
In this paper Brockett sketched a number of connections 
and differences between control theory and mechanics. He made the broad
observation that mechanics is a descriptive science while control theory
is a prescriptive one. He also noted that while much of mechanics
had been put on a sound geometrical footing as described
in the works of Arnold and Avez [1968] and Abraham [1967] for
 example (see also 
Abraham and Marsden [1978] and Arnold [1978])
much that was relevant to control theory, such as Lagrange's equations
with exogenous forces, the Rayleigh dissipation function, nonholonomic
mechanics and quasicoordinates, and stability theory
still resisted such treatment. Interestingly in the years since that
article was written some of these topics have indeed been analyzed
from a geometrical point of view. Lagrange's equations with
forces and the Rayleigh dissipation function are treated in
Bloch, Marsden, Krishnaprasad and Ratiu [1994], [1996] for example
and references therein. In fact a nonlinear version of the Rayleigh
dissipation function in this work is based on the double bracket
equations introduced by Brockett in Brockett [1988] (see also
Bloch, Brockett and Ratiu [1992] and the discussion below).
The right treatment of quasicoordinates and nonholonomic mechanics 
is either Lagrangian reduction as discussed in Marsden and Scheurle [1993]
or the related geometric analysis of nonholonomic systems --
see Bloch and Crouch [1993], [1995a] and Bloch, Krishnaprasad,
Marsden and Murray [1996]. A Lagrangian/Hamiltonian approach to stability is
discussed in Marsden [1992], Bloch, Leonard and Marsden [1997]. 
Brockett's paper treating stability issues in a geometric
setting, Brockett [1983], was very influential. An almost parallel
development was begun by Hermann, see for example, Hermann [1962], 
[1977], [1992].

In his 1976 paper Brockett also made a number of interesting
observations about the structure of mechanical control systems.
He noted for example that the state and control spaces of such a system
had in general the structure of a vector bundle rather than a simple
cross product structure and that one could formulate natural
notions of controlled Hamiltonian and Lagrangian systems. The latter
ideas have been developed in the work of van der Schaft [1982], [1983],
Crouch and van der Schaft [1987], Wang and Krishnaprasad [1992]. These
formulations, and appropriate generalizations, turn out to be important in 
modelling interconnections of mechanical systems, (see Maschke and
van der Schaft [1997]%\cite{}
).

For the purpose of the current paper the notions of Hamiltonian 
or Lagrangian control systems are not so important
however because of the natural mechanical structure arising in any 
optimal control problem as discussed above. However 
reduction, both Lagrangian and Hamiltonian, based on the
use of symmetries and conservation laws as well as ideas
from nonholonomic kinematics and dynamics will play a role, (see for
example Marsden and Ratiu [1994]).

There are several areas of optimal control that have been of particular
interest both to Roger Brockett and the authors of this paper that 
we intend to cover in some detail here. These areas are interconnected
in interesting ways.

One topic of interest is the problem of optimal control on Lie
groups, see for example Brockett [1970], [1973], [1973a], [1976]. Such problems 
give rise, via
the maximum principle, to Hamiltonian systems with symmetries and
thus with a great deal of structure, including the existence of
conservation laws. A useful recent text on mechanical systems
with symmetries is Marsden and Ratiu [1994], while the book of
Jurdjevic [1996] discusses some optimal control on Lie groups
and the control of certain mechanical systems.

Another topic that owes much to the insight of Brockett
( and in particular to the
paper Brockett [1981]) is that of so-called subRiemannian optimal
control problems. These problems concern driftless control systems,
i.e. systems that are locally of the form
$\dot x=\sum_iu_ig_i(x)$, which are controllable, but where the number of 
controls is less than the dimension of the underlying state space.
The corresponding optimal controls are given by a geodesic flow
with respect to a degenerate or subRiemmanian metric. Such
systems have numerous unusual properties including the existence
of a  nonsmooth geodesic ball (see Brockett [1981] and Vershik and
Gershkovich [1988]) and the possible existence of ``rigid extremals'' 
(see for example Sussmann [1992]). Connections
with rigid body mechanics were discussed by the authors in
Bloch, Crouch and Ratiu [1994].

Many of the extremal flows corresponding to
the optimal control problems discussed above
are completely integrable; completely soluble by quadratures and
inversion of functions.
While it is of course interesting to find nonlinear optimal control
problems that are explicitly solvable (see Faybusovich [1988] and
Jurdjevic [1996], Krishnaprasad [1993]), it has also turned out that analysis
of soluble optimal control problems can shed new light on the
classical theory of integrable Hamiltonian systems. Brockett [1994]
pointed out a novel method of obtaining the tridiagonal
Toda lattice equations as the solution to an optimal control problem
and discussed optimal control problems which give rise to geodesic
flows on adjoint orbits of Lie groups. The authors (see Bloch and
Crouch [1996]) analyzed the latter problem
from the point of view of the maximum principle and
showed how such flows could be cast as coupled double bracket equations.  
This approach led us to a novel formulation of the generalized rigid body 
equations on $SO(n)$. 
Together with Brockett (Bloch, Brockett and Crouch [1997]) we 
analyzed the integrability of such flows as well as a related class
of flows on Grassmannians. More recently we have shown that the 
full Toda lattice equations may be written as the solution to an interesting 
optimal control problem, (see Bloch and Crouch [1997]).

\section{%2.
Modeling Variational Problems in Mechanics and Control}\label{S:2}
\subsection{%2.1
Introduction}\label{s:2.1}

In this section we sketch at a more technical level many of the
models that have been used in examining problems in analytical
mechanics and discuss their relation to optimal control problems.
Many of these discussions originate from the work of Brockett [1970]%\cite{ }, 
[1976],%\cite{ }
 but see also Willems [1979]%\cite{}
.

A basic question asked by Brockett was how to take a generic non
linear control system of the form:
\begin{eqnarray}\label{e:2.1}
\dot x&=&f(x,u)\\
y&=& h(x,u)\nonumber
\end{eqnarray}
where $x\in M$, $u\in N$, $y\in P$ are smooth manifolds, $f:M\times N
\to TM$ is a parameterized smooth vector field on $M$, and $h:M\times
N\to P$ is an output mapping, and introduce appropriate additional
structure so that one could systematically study problems in mechanics
with external forces. The output mapping $h$ and manifold $P$ are
important not least in the context of examining the (output) feedback
control $u(t)=k(y(t))$ for mappings $k:P\to N$,  most typically for
this paper in the context of optimal control problems, but also in
the context of modeling the duality between input and output spaces,
and interpreting in this generality such concepts as work and
dissipation. The duality is especially important in modeling
generalizations of mechanical conservative systems. Obvious
candidates for $M$, $P$ and $N$ for mechanical systems are $M=TQ$,
$N=V$, $P=V^*$, where $Q$ models the generalized configuration space,
$V$ is a vector bundle over $Q$ and $V^*$ is the dual bundle to $V$.
Thus if $x=(q,v)\in TQ$, with $v\in T_qQ$, the tangent space to $Q$
at $q\in Q$, the triple $(x,u,y)$ in (\ref{e:2.1}) would have the
form $u\in V_q$, $y\in V^*_q$ where $V_q(V^*_q)$ are the fibers above
$q$ in $V(V^*)$. The cases where $N=TQ$ or $T^*Q$ arise in this
paper, while other choices are also used.

Additional structure needs to be added to the model (\ref{e:2.1}) to
reflect the Hamiltonian or Lagrangian structure, at least for the
uncontrolled system, nominally modeled by choosing $u$ to be the
zero section of the bundle $V$, and also to reflect the manner in
which controls (external forces or torques) interact with the
uncontrolled system. Understanding this additional structure and
appropriate specializations which reflect specific classes of
mechanical systems with external forces has been a topic of much
research in the past two decades, building upon both the existing
control theories and analytical mechanics, but also prompting
further work in both arenas separately.

\subsection{%2.2
Variational Systems Without External Forces}\label{s:2.2}

The basic inspiration for modeling systems in analytical mechanics
is the variational problem:
\begin{equation}\label{e:2.2}
\begin{array}{l}
\displaystyle\min_{q(\cdot)}\displaystyle\int^T_0L(q,\dot q)dt\\
\mbox{subject to: $q$ is a piecewise $C^1$ curve in}\\
{\mathbf{R}}^n\mbox{ satisfying }q(0)=q_0, q(T)=q_T.
\end{array}
\end{equation}
The necessary conditions for this variational problem describe the
equations of motion for many mechanical systems, which are free from
external influence, for appropriate choice of Lagrangian function
$L$. The necessary conditions are the celebrated Lagrangian
equations:
\begin{equation}\label{e:2.3}
\frac d{dt}\frac{\pa L}{\pa\dot q_i}-\frac{\pa L}{\pa q_i}=0,\quad 1
\le i\le n.
\end{equation}

In the case of nondegenerate systems in which the matrix 
\[
\left[\frac{\pa^2L}{\pa\dot q_i\pa\dot q_j}\right]_{1\le i,j\le n}
\]
is everywhere invertible, the Legendre transform ${\L}$ is a
diffeomorphism.
\begin{equation}\label{e:2.4}
{\L}:(q,\dot q)\to(q,p);\quad p_i=\frac{\pa L}{\pa q_i},\quad 1\le i\le
n,
\end{equation}
and maps the Lagrangian system (\ref{e:2.3}) into the Hamiltonian
system
\begin{equation}\label{e:2.5}
\dot q_i=\frac{\pa H}{\pa p_i},\quad \dot p_i=-\frac{\pa H}{\pa q_i},
\quad 1\le i\le n,
\end{equation}
where
\begin{equation}\label{e:2.6}
H(q,p)=\sum^n_{i=1}\frac{\pa L}{\pa\dot q_i}\dot q_i-L\bigg|_{(q,\dot
q)={\L}^{-1}(q,p)}.
\end{equation}
The classical Hamiltonian system (\ref{e:2.5}) is abstracted very
simply in terms of a cotangent bundle $T^*Q$ and a natural
symplectic form $\omega$ on $T^*Q$. In local coordinates for $T^*Q$,
$(q_1,\ldots,q_n,p_1,\ldots,p_n)$ we have
\begin{equation}\label{e:2.7}
\omega=\sum^n_{i=1}dq_i\wedge dp_i.
\end{equation}
In general, given a function $H$ on $T^*Q$, the Hamiltonian function,
the corresponding Hamiltonian vector field on $T^*Q$ is denoted by
$X_H$, and defined by
\[
\omega(X_H,Z)=dH(Z),\quad Z\in\G(TT^*Q),
\]
where $\G(V)$ denotes the space of sections of a vector bundle $V$.
The abstract Hamiltonian system on $T^*Q$ is now given by $X_H$ as
\[
\dot x=X_H(x),\qquad x\in T^*Q.
\]

We define two projections
\[
\pi_1:T^*Q\to Q,\qquad \pi_2:TQ\to Q.
\]
In particular, we may define a one form $\th\in\G(T^*T^*Q)$. If
$(q,p)\in T^*Q$  then we set 
\[
\th_{(q,p)}(Z)=p(\pi_{1*}Z),\quad Z\in\G(TT^*Q).
\]
The natural symplectic two form on $T^*Q$ is now defined as $\omega=-d
\th$. Clearly, $\omega$ is closed and nondegenerate.

We may now abstract the Lagrangian system (\ref{e:2.3}) by
introducing a function $L$ on $TQ$ and defining the fiber derivative
of $L$ at $(q,\dot q)\in TQ$
\begin{equation}\label{e:2.8}
L_F(q,\dot q)(v)=\lim_{h\to 0}\frac1h(L(q,\dot q+hv)-L(q,\dot q))
\end{equation}
where $v\in T_qQ$. Thus $L_F:TQ\to T^*Q$. We assume that $L_F$ is in
fact a vector bundle isomorphism. We define another function $H_L$ on
$TQ$.
\begin{equation}\label{e:2.9}
H_L(q,\dot q)=L_F(q,\dot q)(\dot q)-L(q,\dot q),
\end{equation}
so that
\begin{equation}\label{e:2.10}
H=H_L\circ L^{-1}_F
\end{equation}
generalizes the Hamiltonian function (\ref{e:2.6}) to a function on
$T^*Q$.

We say that a vector field $X\in\G(TTQ)$ is second order if
\[
\pi_{2*}X_{(q,\dot q)}=\dot q,\qquad \dot q\in T_qQ.
\]
Setting $\Omega_L=L^*_F\omega$, we obtain a symplectic form on $TQ$,
since we assumed that $L_F$ is a vector bundle isomorphism. We may
now define the Lagrangian vector field $X_L\in\G(TTQ)$, corresponding
to the Hamiltonian (\ref{e:2.9}) as the second order vector field
satisfying
\begin{equation}\label{e:2.11}
\Omega_L(X_L,Z)=dH_L(Z),\qquad Z\in\G(TTQ).
\end{equation}
It is easily verified that $X_L$ is related to the Hamiltonian vector
field $X_H$, with Hamiltonian $H$ defined in (\ref{e:2.10}), by the
relation
\[
L_{F*}X_L=X_H.
\]
The system of Lagrangian equations (\ref{e:2.3}) may now be
abstracted by the dynamical system on $TQ$ given by
\begin{equation}\label{e:2.12}
\dot x=X_L(x),\qquad x\in TQ.
\end{equation}
A vector field $Z$ on $TQ$ is said to be vertical if $\pi_{2*}Z=0$.
Using local coordinates one easily shows that (\ref{e:2.11}) is
satisfied identically for all vertical vector fields and any second
order vector field $X_L$.

Corresponding to the variational problem (\ref{e:2.2}), we may pose
the more abstract problem on the manifold $Q$ and defined by a
function $L$ on $TQ$
\begin{equation}\label{e:2.13}
\begin{array}{l}
\displaystyle\min_{q(\cdot)}\displaystyle\int^T_0L(q,\dot q)dt\\
\mbox{subject to: $q$ is a piecewise $C^1$ curve in $Q$}\\
\mbox{satisfying, }q(0)=q_0, q(T)=q_T.
\end{array}
\end{equation}
Variational calculus can be extended to manifolds in a
straightforward manner. A variation of a trajectory $q(\cdot)$
satisfying the conditions in (\ref{e:2.13}) is a piecewise $C^1$
mapping
\[
(t,\e)\to q(t,\e),\qquad 0\le t\le T,\qquad \e\in(-\d,\d)\subset
\mathbf{R},
\]
satisfying
\begin{enumerate}
\item[(i)] $q(t,0)=q(t)$, $t\in[0,T]$
\item[(ii)] $q(0,\e)=q_0$, $q(T,\e)=q_T$.
\end{enumerate}
Setting $\d q(t)=(\pa/\pa\e)q(t,\e)|_{\e=0}\in T_{q(t)}Q$, we have
\begin{equation}\label{e:2.14a}
\d q(0)=\d q(T)=0.
\end{equation}

If $\bar q(t)=(q(t),\dot q(t))$, $\dot q(t)\in T_{q(t)}Q$ we have
\[
\d{\bar q}(t)=\bigg(\d q(t),\frac d{dt}\d q(t)\bigg)\in T_{\bar
q(t)}TQ.
\]
One can show using local coordinates that 
\begin{equation}\label{e:2.14b}
\d\int^T_0L(q,\dot q)dt=\int^T_0(\Omega_L(\dot{\bar q},\d{\bar
q})-dH_L(\d{\bar q}))dt.
\end{equation}
It easily follows that necessary conditions for problem 
(\ref{e:2.13}) are indeed satisfied by the flow of the system 
(\ref{e:2.12}).

In problem (\ref{e:2.13}), if we assume that there are additional 
holonomic constraints of the form $\phi_i(q)=0$, $1\le i\le m$, then
necessary conditions may be generated by considering an augmented
variational problem with Lagrange multipliers
\begin{equation}\label{e:2:15}
\begin{array}{l}
\displaystyle\min_{q(\cdot)}\displaystyle\int^T_0\bigg(L(q,\dot q)+\sum^m_{i=1}
\la_i\phi_i(q)
\bigg)dt\\
\mbox{subject to: $q$ is a piecewise $C^1$ curve in $Q$}\\
\mbox{satisfying $\phi_i(q)=0$; $1\le i\le m$ and $q(0)=q_0$}\\
q(T)=q_T,\quad \phi_i(q_0)=\phi_i(q_T)=0,\quad 1\le i\le m.
\end{array}
\end{equation}
Necessary conditions may therefore be written as
\begin{equation}\label{e:2.16}
\begin{array}{c}
i_{\dot{\bar q}}\Omega_L-dH_L+\displaystyle\sum^m_{i=1}\la_i\pi^*_2d\phi_i
\equiv0\\
\phi_i(q)=0,\quad 1\le i\le m.
\end{array}
\end{equation}
Suppose in addition that
\[
Q_c=\{q\in Q;\phi_i(q)=0\}\subset Q
\]
is a regular smooth submanifold of $Q$. If $i:Q_c\to Q$ is the inclusion
mapping then $\bar i=i_*:TQ_c\to TQ$ and $\bar i^*\pi^*_2d\phi_i=d(\phi_i
\circ\pi_2\circ\bar i)\equiv0$. It follows that the necessary
conditions (\ref{e:2.16}) are satisfied by simply pulling back via $
\bar i$ to $Q_c$ to obtain a Lagrangian system on $Q_c$ with Lagrangian
$L\circ\bar i$. Thus, as long as the holonomic constraints are
sufficiently regular, we may remove the constraints by restricting to
the space defined by the constraints.

In problem (\ref{e:2.13}) more general constraints of the form $
\phi(q,\dot q)=0$ cannot be dealt with using the above approach.
These nonholonomic constraints will be considered in the next
section.

There is another mechanism by which one may characterize Hamiltonian
systems. Suppose that $M$ is a manifold which admits a nondegenerate
two form $\Omega$. A distribution $D$ on a submanifold $N\subset M$ is said to 
be isotropic (co-isotropic) if
\[
D^{\perp}_x=\{Y_x\in T_xM;\Omega(D_x,Y_x)=0\},\quad x\in{N},
\]
satisfies $D^{\perp}_x\subset D_x$ ($D^{\perp}_x\supset D_x$) for all
$x\in{N}\subset M$. A distribution $D$ is said to be
Lagrangian if $D$ is both isotropic and co-isotropic. In this case
$\mbox{Dim}D_x=1/2\mbox{Dim}M$. A submanifold $N\subset M$, with
inclusion map $i:N\to M$, is said to be Lagrangian when $\Omega$ is a
symplectic form and $i_*TN$ is Lagrangian distribution on $N\subset
M$. In the case $M=T^*Q$, there is a natural symplectic form $
\omega$, and an induced symplectic form $\dot\omega$ on $TT^*Q$ given
in local coordinates by
\[
\dot\omega=\sum_id{\dot q_i}\wedge d{p_i}+\sum_id{q_i}\wedge d{
\dot p_i}.
\]
If $X_H$ is a Hamiltonian vector field on $T^*Q$, then set
\[
N=\{(x,X_H(x);x\in T^*Q\}\subset TT^*Q.
\]
It turns out that $N$ is a Lagrangian submanifold of $TT^*Q$, (with
symplectic form $\dot\omega$), and every Lagrangian submanifold of
$TT^*Q$ is locally parameterized by a Hamiltonian vector field on
$T^*Q$. Thus Hamiltonian vector fields on $T^*Q$ are equivalently
defined by the Lagrangian submanifolds of $TT^*Q$ which are
parameterized by a single Hamiltonian function on $T^*Q$. See Van der
Schaft [1982]%\cite{ }
, [1983] %\cite{ }
 for details.

\subsection{%2.3 
Mechanical systems with external forces}\label{s:2.3}

Most mechanical systems interact with the environment through
(generalized) forces. The basic mechanisms which describe these 
interactions are Newton's Laws and the Lagrange D'Alembert principle.
Since these (external) forces are modeled by the controls in the
general model (\ref{e:2.1}), these principles are employed to give
important information as to how the controls interact with the
uncontrolled system in the model (\ref{e:2.1}).

\paragraph{Newton's Laws}
To describe a version of Newton's second law consistent with our
abstract configuration space modelled by the manifold $Q$, we
introduce a bundle isomorphism
\begin{equation}\label{e:2.17}
\hat J:TQ\to T^*Q
\end{equation}
where $\hat J_q:T_qQ\to T^*_qQ$ defines the relationship between the
phase velocity $\dot q\in T_qQ$ and the momentum $p\in T^*_qQ$;
\begin{equation}\label{e:2.18}
p=\hat J_q(\dot q).
\end{equation}

We first need a notion of uniform motion given by Newton's First Law,
which we model as the flow on $TQ$ induced by a second order vector
field $X_0$ on $TQ$. We do not say more about $X_0$ until we supply
further structure. Now $\hat J_*:TTQ\to TT^*Q$ is also a bundle
isomorphism so $\hat J_*X_0$ defines a vector field on $T^*Q$.

Our abstract notion of a force field will be a time varying one form
$F(t)$ on $Q$. Thus $F(t)\in\G(T^*Q)$ for each $t$.

If $Z\in\G(TQ)$ then we may lift $Z$ to a function $Z^H$ on $T^*Q$ by
setting
\[
Z^H(q,p)=p(Z_q),\quad p\in T^*_qQ.
\]
If $\eta\in\G(T^*Q)$ then we define the vertical lift of $\eta$,
denoted $\dot\eta\in\G(TT^*Q)$ by setting
\[
\dot\eta(Z^H)=\eta(Z),\quad Z\in\G(TQ).
\]
Locally, if $\eta=\sum_i\a_idq_i$, then $\dot\eta=\sum_i\a_i(\pa/\pa
p_i)$.

Our definition of a Newtonian vector field $X_F\in\G(TT^*Q)$,
corresponding to an external force field $F$ is given by
\begin{equation}\label{e:2.19}
X_F=\hat J_*X_0+\dot F.
\end{equation}
Since $X_0$ is second order, in local coordinates $(q,v)$ for $TQ$ we
may write it in the form:
\[
\dot q=v,\qquad \dot v=a_v(q,v).
\]
The dynamical system $\dot x=X_F(x)$, $x\in T^*Q$ can therefore be
expressed in local coordinates $(q,p)$ for $T^*Q$ in the form
\begin{eqnarray}\label{e:2.20}
\dot p&=&a_p(q,p)+F\\
\dot q&=&\hat J^{-1}_q(p)
\nonumber
\end{eqnarray}
where $a_p(q,p)=\frac d{dt}\hat J_q(v)$, $\dot v=a_v(q,v)$, $\dot
q=v=\hat J^{-1}_q(p)$. The existence of a drift term when we set $F
\equiv0$ relates to the fact that Newton's second law is cast in
terms of inertial axes. In the present situation this may be
expressed as a set of axes in a fixed space $T^*_aQ$. To express a 
Newtonian flow corresponding to a vector field $X_F$ in such a frame,
we need the additional structure on $Q$ that will be introduced later
in this section.

However, the representation (\ref{e:2.20}) is useful in distinguishing
two important components of the Newtonian system on $T^*Q$. The
kinematics is the system on $Q$ defined locally by
\[
\dot q=\hat J^{-1}_q(p)
\]
which describes the evolution of $q$ with $p\in T_qM$ as an input.
The dynamics is the system on $T^*Q$ defined locally by
\[
\dot p=a_p(q,p)+F
\]
which describes the evolution of $p$ with $F\in T^*_qM$ as an input.
Note that, in the context of model (\ref{e:2.1}), our input space in
each of the examples above is indeed a vector bundle.

\paragraph{Lagrange D'Alembert}
The Lagrange D'Alembert principle gives an alternate means of
describing motion subject to an external force field $F$. Given a
Lagrangian function $L$, the motion is governed by solutions of the
variational system:
\begin{equation}\label{e:2.21}
\begin{array}{l}
\displaystyle\d\int^T_0L(q,\dot q)dt+\int^T_0F(\d q)dt=0,\quad q\in Q\\
\mbox{subject to: $q$ is piecewise $C^1$ curve}\\
\mbox{satisfying }q(0)=q_0, q(T)=q_T.
\end{array}
\end{equation}
>From (\ref{e:2.14b}) it is clear that solutions of this variational
system are flows of the vector field $X_L$ on $TQ$ satisfying
\begin{equation}\label{e:2.22}
i_{X_L}\Omega_L-dH_L+\pi^*_2F(t)\equiv0.
\end{equation}
As in the previous section the formulation (\ref{e:2.22}) includes
holonomic constraints. To incorporate nonholonomic constraints
additional ``forces'' must be included to ensure that the constraints
are satisfied. We only consider non\-holonomic constraints which are
linear in the velocity for simplicity. Thus we model the
nonholonomic constraints by a collection of $m$, everywhere
independent, one forms $\omega_i$, $1\le i\le m$, on $Q$. The
constraints take the form
\begin{equation}\label{e:2.23}
\omega_i(q)(\dot q)=0,\qquad 1\le i\le m.
\end{equation}

\paragraph{Classical Nonholonomic Systems}
The classical nonholonomic system corresponding to the variational
problem (\ref{e:2.13}) and constraints (\ref{e:2.23}) is determined
by the Lagrange D'Alembert principle as
\begin{equation}\label{e:2.24}
\begin{array}{l}
\d\displaystyle\int^T_0L(q,\dot q)dt+\int^T_0\displaystyle\sum^m_{i=1}\mu_i
\omega_i(\d q)dt+
\int^T_0F(t)(\d q)dt=0\\
\mbox{subject to: $q$ is piecewise $C^1$ curve satisfying}\\
\mbox{the constraints (\ref{e:2.23}) and $q(0)=q_0$,
$q(T)=q_T$.}
\end{array}
\end{equation}

The corresponding vector field $X_L$ on $TQ$ defining the flow
satisfying (\ref{e:2.24}) is determined from
\begin{equation}\label{e:2.25}
\begin{array}{c}
i_{X_L}\Omega_L-dH_L+\pi^*_2\bigg(\displaystyle\sum^m_{i=1}\mu_i\omega_i+F(t)
\bigg)\equiv0\\
\omega_i(\dot q)\equiv0,\quad 1\le i\le m.
\end{array}
\end{equation}

\paragraph{Constrained Variational Mechanics}
The constraints (\ref{e:2.23}) can instead be treated as classical
constraints in the problem (\ref{e:2.13}). The normal extremals may
be determined using Lagrange multipliers, and the resulting Lagrangian
system (sometimes referred to as a Vakonomic system, Arnold [1988]%\cite{ }
)
is derived from the variational system:
\begin{equation}\label{e:2.26}
\begin{array}{c}
\d\displaystyle\int^T_0\left(L(q,\dot q)+\sum^m_{i=1}\la_i\omega_i(\dot q)
\right)dt+\int^T_0F(t)(\d q)dt=0\\
q(0)=q_0,q(T)=q_T,\qquad 1\le i\le m,\omega_i(q)=0.
\end{array}
\end{equation}
Now for a variation $(t,\e)\mapsto q(t,\e)$ of a curve $q$, we have
\begin{equation}\label{e:2.27}
d\omega_i\left(\frac{\pa q}{\pa t},\frac{\pa q}{\pa\e}\right)=\frac{
\pa}{\pa t}\omega_i\left(\frac{\pa q}{\pa\e}\right)-\frac{\pa}{\pa\e}
\omega_i\left(\frac{\pa q}{\pa t}\right).
\end{equation}
Thus the Lagrangian flow satisfying (\ref{e:2.26}) is determined by
the vector field $X_L$ on $TQ$ satisfying 
\begin{equation}\label{e:2.28}
\begin{array}{c}
i_{X_L}\Omega_L-dH_L-\pi^*_2\left(\displaystyle\sum^m_{i=1}\dot\la_i\omega_i+
\la_i
i_{\dot q}d\omega_i\right)+\pi^*_2F(t)\equiv0\\
1\le i\le m,\qquad \omega_i(\dot q)\equiv0.
\end{array}
\end{equation}

It follows that the nonholonomic system (\ref{e:2.25}) is obtained
from (\ref{e:2.28}) by setting $\dot\la_i=-\mu_i$ and $\la_i=0$. If the
one form $\omega_i$ is exact, $\omega_i=d\phi_i$, then clearly in
both (\ref{e:2.25}) and (\ref{e:2.28}) the corresponding nonholonomic
constraint may be treated as a holonomic constraint.

\paragraph{Hamiltonian Control Systems}
One can take an alternate point of view of defining Hamiltonian (or
Lagrangian) systems with extremal forces, or controls, by generalizing
the definition of a Hamiltonian vector field as a parameterization of
a Lagrangian submanifold, as discussed in section~\ref{s:2.2}. If in system 
(\ref{e:2.1}), $N=P={\mathbf{R}}^k$, set
\[
\omega^e=\sum^k_{i=1}du_i\wedge dy_i
\]
as a symplectic form on ${\mathbf{R}}^{2k}$, parameterized by
coordinates $(u_1,\ldots,u_k,y_1,\allowbreak\ldots,y_k)$. Assume that $M$ is
symplectic with symplectic form $\omega$, then $TM\oplus{\mathbf{R}}^{2k}$
is a symplectic manifold with symplectic form $\dot\omega+\omega^e$.
If $H(x,u)$ is a function on $M\oplus{\mathbf{R}}^k\subset TM\oplus
{\mathbf{R}}^{2k}$, we may consider the following system generated by
$H$
\begin{eqnarray}\label{e:2.29}
\dot x&=&X_{H(x,u)}(x)\\
y&=&\frac{\pa H}{\pa u}(x,u)\nonumber
\end{eqnarray}
where $X_H$ is the Hamiltonian vector field on $M$ (with symplectic
form $\omega$). System (\ref{e:2.29}) defines a Lagrangian
submanifold $N$ of $TM\oplus{\mathbf{R}}^{2k}$ with symplectic form $
\dot\omega+\omega^e$, $N=\{(x,u),(X_H,(\pa H/\pa u))\}$. It
follows that one may define a Hamiltonian control system 
(\ref{e:2.1}), with $N=P={\mathbf{R}}^k$, as globally parameterized
Lagrangian submanifolds of $TM\oplus{\mathbf{R}}^{2k}$ with symplectic form $ 
\dot\omega+\omega^e$. See Van der Schaft [1982] [1983] %\cite{ } \cite{ }
for a detailed discussion.

If $H$ is affine in the coordinates $u_i$, the vector field $X_H$ is
also affine in $u_i$, and so one can directly compare $X_H$ with
$X_L$ (and $L_F*X_L=X_H$) in (\ref{e:2.22}), where $M=TQ$ and $F(t)$
has the form
\begin{equation}\label{e:2.30}
F(t)=\sum^k_{i=1}\eta_iu_i(t),\qquad \eta_i\in\G(T^*Q).
\end{equation}
Unfortunately, the restrictions imposed by the structure 
(\ref{e:2.29}) do not make for direct comparisons, except in some
special cases. However, the model (\ref{e:2.29}) has some added
attractions when studying the input-output properties of systems 
(\ref{e:2.1}).

In the work of Brockett and Rahimi [1972]%\cite{ }
, it was shown that one
could characterize initialized, linear, time invariant, Hamiltonian
systems (\ref{e:2.29}) with input-output maps
\begin{equation}\label{e:2.31}
y(t)=\int^t_0W(t-\s)u(\s)d\s
\end{equation}
by the additional symmetry property of the impulse response
\begin{equation}\label{e:2.32}
W(t)=-W(-t)^T.
\end{equation}

This result was generalized to nonlinear systems (\ref{e:2.29}) by
Crouch and Van der Schaft [1987]%\cite{ }
, using a variational property.
Specifically, if $(u,y)$ is the input response of a system 
(\ref{e:2.1}), with $N=V={\mathbf{R}}^k$, $P=V^*={\mathbf{R}}^k$, and
defined on the infinite interval $(-\infty,\infty)$, then we define
an admissible variation of $(u,y)$ to be essentially a mapping
\[
(t,\e)\longmapsto(u(t,\e),y(t,\e)),(t,\e)\in(-\infty,\infty)\times(-
\d,\d)
\]
satisfying
\begin{enumerate}
\item[(i)] $(u(t,0),y(t,0))=(u(t),y(t))$
\item[(ii)] $(u(t,\e),y(t,\e))$ is an input-output response of a
system (\ref{e:2.1}) for each $\e\in(-\d,\d)$
\item[(iii)] $(\d u(t),\d y(t))=(\pa/\pa\e)|_{\e=0}(u(t,\e),y(t,\e))$ 
has compact support along $(u,y)$.
\end{enumerate}

We may define a manifold ${\M}_V$ of maps consisting of all
input-output behaviors $(u,y)$ of systems (1), corresponding to the
same input-output space $N=V={\mathbf{R}}^k$, $P=V^*={\mathbf{R}}^{k^*}$.
The set of all admissible variations of $(u,y)\in{\M}_V$ defines a
set of variational fields $(\d u,\d y)$ along $(u,y)$, which may be
interpreted as the tangent space to ${\M}_V$ at $(u,y)$. There is a
skew form on this tangent space defined by 
\begin{equation}\label{e:2.33}
\omega_{(u,y)}((\d_1u,\d_1y),(\d_2u,\d_2y))=\int^\infty_{-\infty}(
\d_2y(\d_1u)-\d_1y(\d_2u))dt.
\end{equation}

The notion of isotropic, co-isotropic and Lagrangian submanifolds can
be defined as for finite dimensional manifolds. It is shown in 
Crouch and Van der Schaft [1987] %\cite{ }
 that the set of input-output behaviors for Hamiltonian
systems (\ref{e:2.29}) form a Lagrangian submanifold of the manifold
${\M}_V$.

It is interesting to note that this variational characterization of
Hamiltonian systems (\ref{e:2.28}) is related to the classical inverse
problem in mechanics, Santilli [1978]%\cite{ }
. In its most basic version,
this problem considers an implicit system of second order equations
\[
R_i(q,\dot q,\ddot q)=0,\quad 1\le i\le n,\quad q\in{\mathbf{R}}^n
\]
where $[\pa^2R_k/\pa\ddot q_i\pa\ddot q_j]_{1\le i,j\le n}$ is an
invertible matrix for all values of $(q,\dot q,\ddot q)$, and asks
when this system is equivalent to a system of equations (\ref{e:2.3})
for some choice of Lagrangian function $L$? This problem is in fact a
special case of the problem of determining when there exists a
function $H(x,u)$ such that system (\ref{e:2.1}) coincides with the
Hamiltonian system (\ref{e:2.29}). This problem is answered in detail
in Crouch and Van der Schaft [1978]%\cite{ }
. Clearly the conditions that are developed in this work ensure 
that the symplectic form (\ref{e:2.33}) vanishes and, for linear
systems, they are equivalent to the condition (\ref{e:2.32}) of Brockett
and Rahimi [1972]%\cite{ }
.

\subsection{%2.4
Relation to optimal control}\label{s:2.4}

\paragraph{Pontryagin Maximum Principle}
For control systems, optimal control problems have been typically
cast in a different setting from the classical variational problems, 
more closely associated with mechanics and discussed in the previous
section. The basic difference lies in the way in which the
trajectories are formulated; in the optimal control setting the
trajectories are ``parameterized'' by the controlled vector field, while
in the traditional variational setting trajectories are simply
``constrained.'' The other basic difference is that the necessary
conditions for extremals in the optimal control setting are typically
expressed using a Hamiltonian formulation using the Pontryagin
maximum principle, rather than the Lagrangian setting as discussed in
the previous section (see Pontryagin et al. [1962]%\cite{}
, Lee and Markus [1976]%\cite{}
, and Sussmann [1977]%\cite{}
). We state a typical optimal control problem:
\begin{equation}\label{e:2.34}
\begin{array}{l}
\qquad\qquad\qquad\displaystyle\min_{u(\cdot)}\int^T_0h_0(x,u)dt\\
\mbox{subject to:\ (i) } \dot x=f(x,u),\quad x\in M^n,\quad u\in\Omega
\subset{\mathbf{R}}^k,\\
\qquad\qquad\quad\mbox{(ii) } x(0)=x_0,x(T)=x_T
\end{array}
\end{equation}
where $f$ and $h_0\ge0$ are smooth and $\Omega$ is a closed subset of
${\mathbf{R}}^k$. To state necessary conditions dictated by the
Pontryagin maximum principle, we introduce a parameterized
Hamiltonian function on $T^*M$
\begin{equation}\label{e:2.35}
\hat H(x,p,u)=p(f(x,u))-p_0h_0(x,u)
\end{equation}
where $p_0\ge0$ a fixed positive constant, and $p\in T^*M$. We denote
by $t\mapsto u^*(t)$ a curve which satisfies the following
relationship along a trajectory $t\mapsto(x(t),p(t))$ in $T^*M$:
\begin{equation}\label{e:2.36}
H^*(x(t),p(t),t)\stackrel{\Delta}{=}\hat H(x(t),p(t),u^*(t))=\max_{u\in
\Omega}\hat H(x(t),p(t),u).
\end{equation}
The time varying Hamiltonian function $H^*$ defines a time varying
Hamiltonian vector field $X_{H^*}$ on $T^*M$, with respect to the
canonical symplectic structure on $T^*M$. One statement of
Pontryagin's maximum principle gives necessary conditions for
extremals of the problem (\ref{e:2.34}) as follows: an extremal trajectory $t
\mapsto x(t)$ of the problem (\ref{e:2.34}), is the projection onto
$M$ of a trajectory of the flow of the vector field $X_{H^*}$, which
satisfies the boundary condition (\ref{e:2.34})(ii), and for which $t
\mapsto(p(t),p_0)$ is not identically zero on $[0,T]$. 
The extremal is called normal when $p_0\ne0$, (in which case we set
$p_0=1$). When $p_0=0$ we call the extremal abnormal, corresponding
to the case where the extremal is determined by constraints alone.

If the extremal control function $u^*$ is not determined by the
system (\ref{e:2.36}) along the extremal trajectory, then the
extremal is said to be singular, in which case further necessary
conditions are needed to determine $u^*$. In this paper we are only
interested in nonsingular situations in which $\Omega={\mathbf{R}}^k$,
(or indeed more general vector bundles). We also suppose that the
data is sufficiently regular so that $u^*$ is determined uniquely
from the condition
\begin{equation}\label{e:2.37}
0\equiv\frac{\pa\hat H}{\pa u}(x(t),p(t),u^*(t)),\quad t\in[0,T].
\end{equation}
It follows that there exists a function $k$ such that
$u^*(t)=k(x(t),p(t))$. We then set
\begin{equation}\label{e:2.38}
H(x,p)\stackrel{\Delta}{=}\hat H(x,p,k(x,p)).
\end{equation}
It follows that along an extremal
\begin{equation}\label{e:2.39}
H(x(t),p(t))=H^*(x(t),p(t),t).
\end{equation}
Given the assumed regularity, and condition (\ref{e:2.37}), the
extremal flow is also a flow of the time invariant Hamiltonian
function $H$. Further, assuming that this function $H$ is
sufficiently regular, the extremal will also be governed by a
Lagrangian system. However, it is not immediately obvious what
variational problem would result in this Lagrangian system. This
issue was investigated in depth in Bloch and Crouch [1994]%\cite{ }
,
employing the classical results (Rund [1966]%\cite{ }
, Bliss [1930]%\cite{ }
),
which relate classical constrained variational problems to
Hamiltonian flows, although not optimal control problems. We briefly
outline the simplest relationship as discussed in Bloch and Crouch 
[1994]%\cite{ }
.

\paragraph{Constrained Variational Problems}
We consider the following variational system arising from a typical
constrained variational problem:
\begin{eqnarray}\label{e:2.40}
\d\displaystyle\int^T_0\left(L(q,\dot q)+\sum^m_{i=1}\la_i\phi_i(q,\dot q)
\right)dt&=&0,\quad
q(0)=q_0,\quad q(T)=q_T,\\
\phi_i(q,\dot q)&=&0,\quad 1\le i\le m,\quad q\in {\mathbf{R}}^n.
\nonumber
\end{eqnarray}
Setting $\La(q,\dot q,\la)=L(q,\dot q)+\sum^m_{i=1}\la_i\phi_i(q,\dot
q)$, and denoting by $\Phi(q,\dot q)$ the vector of constraints, the
usual Euler-Lagrange system for (\ref{e:2.40}) becomes
\begin{equation}\label{e:2.41}
\frac d{dt}\frac{d\La}{\pa\dot q}-\frac{\pa\La}{\pa q}=0,\qquad 
\Phi(q,\dot q)=0.
\end{equation}
Consider the system of $n+m$ implicit equations for $\la$ and $\dot
q$
\begin{equation}\label{e:2.42}
\frac{\pa\La}{\pa\dot q}(q,\dot q,\la)-p^T=0,\qquad \Phi(q,\dot q)=0,
\end{equation}
and suppose that these have a unique smooth solution
\begin{equation}\label{e:2.43}
\la=\psi(q,p),\qquad\dot q=\phi(q,p).
\end{equation}
This can be guaranteed by assuming that the Jacobian matrix
\begin{equation}\label{e:2.44}
\left[\begin{array}{cc}
\displaystyle\frac{\pa^2\La}{\pa\dot q^2}&\displaystyle\frac{\pa\Phi^T}{\pa\dot 
q}\\
\\
\displaystyle\frac{\pa\Phi}{\pa\dot q}&0\end{array}\right]
\end{equation}
has full rank everywhere. Clearly the system (\ref{e:2.42})
generalizes the Legendre transformation, and one can define a
Hamiltonian function as in the traditional case by setting
\begin{equation}\label{e:2.45}
H(q,p)\stackrel{\Delta}{=} p^T\phi(q,p)-L(q,\phi(q,p)).
\end{equation}
The resulting Hamiltonian formulation of the extremal flow of the
system (\ref{e:2.40}) is classical, Rund [1966]%\cite{ }
, Bliss [1930]%\cite{ }
.
The following result is obtained in Bloch and Crouch [1994]%\cite{ }
.

%\begin{theorem}
\noindent
{\bf Theorem }{\em 
If $M={\mathbf{R}}^n$, $k=n-m$, $\Omega={\mathbf{R}}^k$, then the optimal
control problem (\ref{e:2.34}) and the variational system 
(\ref{e:2.40}) have equivalent (normal) extremal flows in case
\begin{enumerate}
\item[(i)] the Jacobian (\ref{e:2.44}) has full rank everywhere
\item[(ii)] $\Phi(x,\dot x)=0\Leftrightarrow\exists u\in{\mathbf{R}}^k$
such that $\dot x=f(x,u)$
\item[(iii)] $L(x,f(x,u))=h_0(x,u)$.
\end{enumerate}
In this case the Hamiltonian functions $H$ defined in equations 
(\ref{e:2.38}) and (\ref{e:2.45}) coincide. The equivalence of the
extremal flows is determined through the mapping (\ref{e:2.43}), $(q,p)\mapsto(
q,\dot q,
\la)$.
}
%\end{theorem}

\vspace{2ex}
It is clear that the condition (ii) expresses the relation between
the constraints and parameterization of the system trajectories,
while (ii) and (iii) together relate the two sets of data in the
problem formulations. However, the Euler-Lagrange system 
(\ref{e:2.41}), has $2n+m$ degrees of freedom, $\dot q(0),q(0),
\la(0)$ whereas the Hamiltonian formulation via the optimal control
problem (\ref{e:2.34}), has only $2n$ degrees of freedom,
$q(0),p(0)$. The paper by Bloch and Crouch [1994] %\cite{ } 
demonstrates how
the system (\ref{e:2.41}) may indeed always be reduced to a dynamical
system with $2n$ degrees of freedom, assuming that condition (i)
above holds. The resulting systems are strictly equivalent, and
related by a diffeomorphism of phase spaces.

\paragraph{Mechanics and Optimal Control Theory}
The type of relationship discussed above between optimal control
problems and constrained variational problems provides the basis for
the interesting interaction between mechanics and optimal control
theory. The types of interaction we explore in this paper include the
following.

\begin{enumerate}
\item[2.47 i)] Certain optimal control problems give rise to extremal
flows which occur naturally in mechanics as Lagrangian or Hamiltonian
flows. Their formulation as a control problem brings new insights to
the mechanics problem, and vice versa.

\item[2.47 ii)] Optimal control problems often give rise to extremal
flows which are equivalent to constrained variational problems, where
the constraints are nonholonomic in nature, but included in the
variational problem through the Lagrange multiplier technique. Thus
optimal control problems typically yield constrained variational
mechanics (\ref{e:2.28}), rather than classical nonholonomic systems 
(\ref{e:2.25}).

\item[2.47 iii)] Some natural optimal control problems are equivalent
to higher order (constrained) variational problems, which typically
do not occur in classical mechanics, but can be no less interesting
physically.

\item[2.47 iv)] Some optimal control problems give rise to interesting
variants of classical variational problems. For example, certain 
parameterizations of system trajectories which are natural in a 
control setting, yield highly singular situations when viewed as
constrained variational problems.
\end{enumerate}

\paragraph{Model Problem}
We now consider a specific class of optimal control problems made
famous by the initial work of Brockett [1981]%\cite{ }
. We consider control
systems of the form:
\refstepcounter{equation}
\begin{equation}\label{e:2.47}
\dot x=\sum^k_{i=1}X_iu_i,\quad x\in M^n,\quad u\in\Omega\subset
{\mathbf{R}}^k
\end{equation}
where $\Omega$ contains an open subset which contains the origin, and
the vector fields $F=\{X_1,\ldots,X_k\}$ are complete. Setting
\[
D_F(x)=\left\{X_x(x); X=\sum^k_{i=1}\alpha_iX_i,\alpha_i\in
\mathbf{R}\right\},
\]
then $x\mapsto D_F(x)$ defines a distribution on $M$, whose
involutive closure is just the distribution $D_L$ defined by the Lie
algebra $L$ of vector fields generated by $F$. (We assume that all
vector fields in $L$ are also complete). We say that $D_L$ has the
spanning property if $D_L(x)=T_xM$ for all $x\in M$. System 
(\ref{e:2.47}) is said to have the controllability property, if
whenever $x_0$, $x_T$ are two points on $M$, there exists a piecewise
constant control $u(\cdot)$ on $[0,T]$ such that the corresponding
solution $x(\cdot)$ of (\ref{e:2.47}) satisfies $x(0)=x_0$,
$x(T)=x_T$. It is a fundamental fact of control theory (see Sussmann 
[1973]%\cite{ }
) that if $D_L$ has the spanning property then (\ref{e:2.47})
has the controllability property. (In the analytic case this
condition is also necessary for controllability.)

Thus in case $D_L$ has the spanning property the following optimal
control problem is well posed:
\begin{equation}\label{e:2.48}
\begin{array}{c}
\displaystyle\min_{u(\cdot)}\displaystyle\int^T_0\frac12\sum^k_{i=1}u^2_i(t)dt\\
\mbox{subject to: dynamics (\ref{e:2.47}) and $x(0)=x_0$, $x(T)=x_T$}.
\end{array}
\end{equation}


To view this as a constrained variational problem we make some
additional regularity assumptions, which are not necessary, but even
when they hold produce a very rich class of problems.

\vspace{2ex}
\noindent
{\bf Assumption}

\begin{enumerate}
\item[(i)] For the problem defined by (\ref{e:2.47}) and (\ref{e:2.48}) the 
distribution $D_L$
satisfies the spanning property.
\item[(ii)] The dimension of $D_F$ is constant on $M$ and equal to
$k$. (Thus the vector fields $X_1\cdots X_k$ are everywhere
independent).
\item[(iii)] There exist exactly $n-k=m$ one forms on $M$ $\omega_1
\cdots \omega_m$, such that the co-distribution
\[
D^\perp_F(x)=\{\bar\omega\in T^*_xM;\quad \bar\omega D_F(x)=0\}
\]
is spanned by $\omega_1,\ldots,\omega_m$ everywhere. (This condition
implies that $M$ is parallelizable.)
\end{enumerate}

Since $D_F$ has constant dimension on $M$ we may define a norm on
each subspace $D_F(x)$; if $X\in D_F(x)$, and $X=\sum^k_{i=1}
\alpha_iX_i(x)$ then
\[
|X|=\sum^k_{i=1}\alpha^2_i.
\]
This norm defines an inner product on $D_F(x)$, denoted $\langle
\cdot,\cdot\rangle_x$, which can be extended to a metric on $M$. The
optimal control problem (\ref{e:2.48}) is now equivalent to the 
following constrained variational problem when the assumptions (i),
(ii), (iii) hold:
\begin{eqnarray}%(2.49, 2.50)
&&\qquad\qquad\qquad\qquad\min_{x(\cdot)}\frac12\int^T_0\langle\dot x,\dot
x\rangle_xdt\\
&&\mbox{subject to: $x(\cdot)$ is a piecewise $C^1$ curve in $M$ such that}
\\
&&x(0)=x_0,\quad x(T)=x_T\quad\mbox{and}\quad\omega_i(x)(\dot x)=0,
\quad 1\le i\le m.\nonumber
\end{eqnarray}
Problem (2.50) is often referred to as the sub-Riemannian geodesic
problem, to distinguish it from the Riemannian geodesic problem, in
which the constraints (2.51) are absent. These problems were studied
by Griffiths [1983] %\cite{ }
 from the constrained variational viewpoint,
before they gained much wider recognition, from the
optimal control viewpoint by Brockett [1981]%\cite{ }
. This ``singular''
geodesic problem is an example of the situation described in (2.47 iv).
It is the case however that in the sub-Riemannian geodesic problem,
abnormal extremals play an important role, see Sussmann [1992]%\cite{ }
. See also work by Hermann [1962]%\cite{}
, and Strichartz [1983]%\cite{}
, [1989]%\cite{}
.

We note that the solution of the model problem above, for normal
extremals, involves adding the constraints (2.51) to the Lagrangian
by Lagrange multipliers, resulting in necessary conditions in the form of 
constrained variational mechanics (\ref{e:2.28}) with $L(q,\dot q)=1/2\langle
\dot q,
\dot q\rangle_q$. Perhaps unfortunately, in some of the Russian literature,
Vershik and Gershkovich [1988]%\cite{ }
, for example, distributions $D_F$
for which $D_L$ has the spanning property, are said to be completely
nonholonomic, and so the system (\ref{e:2.47}) is sometimes said to
be a nonholonomic control system. Clearly, there is another system
which has the right to be called the non-holonomic control system
generated by the Lagrangian $L=1/2\langle\dot q,\dot q\rangle_q$ and
constraints (2.51); namely the system (\ref{e:2.25}), in which the
controls are determined by the external force field $F$ given in the
form (\ref{e:2.30}). This latter system is clearly ``dynamic'' in nature,
whereas the former system is clearly ``kinematic'' in nature in the
sense of the discussion in section~\ref{s:2.3}. In the presence of
the constraints (2.51), the kinematic system (\ref{e:2.47})
parameterizes the admissible velocities by the controls.

However, the term completely nonholonomic distribution is also well
deserved as we briefly demonstrate. Suppose that one of the forms $
\omega_k$ is exact, $\omega_k=dP$, for some function $P$ on $M$. By
the assumption (iii) $dP(x)\ne0$ for all $x$ in $M$, and
\[
0=\omega_k([X_i,X_j])=dP([X_i,X_j])=X_i(X_j(P))-X_j(X_i(P))
\]
since $dP(X_i)\equiv dP(X_j)\equiv0$. It follows that the
distribution $D_L$ is annihilated by $\omega_k$ everywhere on $M$,
and so cannot have the spanning property. Hence the assumption (i)
implies that none of the forms $\omega_k$, $1\le k\le m$, are exact,
and so none of the constraints may be absorbed as holonomic
constraints.

The singular nature of the sub-Riemannian geodesic problem is
manifested in many ways, such as the existence of distinct abnormal
extremals, and the singular nature of the sub-Riemannian geodesic
ball, as first investigated by Brockett [1981]%\cite{ }
. Defining a metric on
$M$ by setting
\begin{eqnarray*}
d(x_0,x_T)&=&\min_{x(\cdot)}\int^T_0|\dot x|dt,\dot x\in
D_F(x),x(0)=x_0,x(T)=x_T\\
B^F_{\e}(x_0)&=&\{\bar x\in M;d(\bar x,x_0)\le\e\}
\end{eqnarray*}
then the sub-Riemannian geodesic ball $S^F_{\e}(x_0)$ is simply the
boundary of $B^F_\e(x_0)$. In Brockett [1981] %\cite{ }
 a particularly simple
example of the sub-Riemannian geodesic problem was considered, which
expressed as a control problem becomes:
\[
\begin{array}{l}
\displaystyle\min_{u(\cdot)}\int^T_0\frac12(u^2_1+u^2_2)dt,\\
\mbox{subject to: }\dot x_1=u_1\\
\hphantom{\mbox{subject to: }}\dot x_2=u_2\\
\hphantom{\mbox{subject to: }}\dot x_3=x_1u_2-x_2u_1\\
x(0)=x_0,\quad x(T)=x_T.
\end{array}
\]
In this case $F=\{(\pa/\pa x_1)-x_2(\pa/\pa x_3),(\pa/\pa x_2)+x_1(\pa/\pa x_3)
\}$, while $D^\perp_F$ is spanned by $\omega=x_1dx_2-x_2dx_1-dx_3$.
The Lie algebra $L$ is simply the Heisenberg Lie algebra. $S^F_{
\e}(0)$ has a singularity along the $x_3$ axis. This class of problem
continues to invoke a great deal of interest, especially in the areas
of establishing when abnormal extremals are optimal and obtaining a 
precise description of $S^D_{\e}(x_0)$.

\subsection{%2.5
Reduction}\label{s:2.5}

Much of the interest in mechanics and optimal control concerns
situations where the various systems discussed in the previous sections have
explicit solutions. This can be related to the rich theory
surrounding integrability
for dynamical systems occurring in analytical mechanics, and 
reduction techniques, where flows on higher
order tangent bundles are reduced to flows evolving on lower order
bundles. Typically this ``reduction'' is made possible through the
existence of ``symmetries,'' following Noether's theorem, %\cite{}
see Marsden and Ratiu [1994]%\cite{}
. Since many of these reduction
techniques revolve around Lagrangian or Hamiltonian systems, one can
expect that similar techniques are applicable in optimal control
problems, illustrating the interaction (2.47 i). Indeed in the control
theory literature, a notable example of this technique is
demonstrated in Faybusovich [1988]%\cite{ }
, Jurdjevic [1996] %\cite{ }
 and
Krishnaprasad [1993]%\cite{ }
. Most of these efforts do indeed date back to
the work of Brockett dealing with optimal control problems on Lie
groups, Brockett [1970]%\cite{ }
, [1973] %\cite{ }
and [1973a]. However, for simplicity we concentrate our
attention here on the reduction process applied to the variational
problems (\ref{e:2.13}).

The reduction process for mechanics problems is often achieved by
decoupling the dynamics and kinematics, and in some way reducing the
dynamics to a system on a  lower dimension space. For general
problems reduction is not possible, but for systems in which the data
exhibits an invariance property reduction can be possible. Assuming
that a problem is posed on a configuration manifold $Q$ with phase
space $TQ$ or $T^*Q$ a common invariance property occurs when a Lie
group $G$ acts on $Q$ and leaves the system data invariant. If $G$
acts transitively on $Q$, then one can hope to reduce the dynamics to
a flow on the Lie algebra $\mathcal{G}$, or its dual $\mathcal{G}^*$, and 
reduce the
kinematics to a flow on $G$, driven by the dynamic flow. For
systems with nonholonomic constraints, either classical
non-holonomic systems or constrained variational mechanics corresponding to 
optimal control problems, similar reduction techniques have been, and are
currently being, developed, see for example Marsden and Scheurle
[1993]%\cite{ }
, and Bloch, Marsden, Krishnaprasad and Murray [1996]%\cite{ }
, Koon and Marden [19971,b] and Zenkov, Bloch and Marsden [1997].

\paragraph{Group Invariance}
We first consider the case where $Q$ is a Lie group ($=G$) in the
context of the variational problem (\ref{e:2.13}). 

\vskip12pt
\noindent
{\bf Assumption}
There exists a function $l$ on the Lie algebra $\mathcal{G}$ of $G$ such that
\begin{equation}\label{e:2.51}
L(q,\dot q)=L(e,R^{-1}_{q*}\dot q)=l(R^{-1}_{q*} \dot q)
\end{equation}
where $R_q$ is right multiplication on $G$ by $q\in G$, and $\dot q
\in T_qG$ so that $R^{-1}_{q*}\dot q\in T_eG\simeq\mathcal{G}$. (A
similar theory can be established for left invariance by $L_q$, the 
left multiplication operator on $G$.) \qed

\vskip12pt
We may find a basis for $
\mathcal{G}^*$, denoted $\eta_1,\ldots,\eta_n$, and introduce a
function on $\mathbf{R}^n$ which satisfies
\[
L(q,\dot q)=\hat l(\eta_1(R^{-1}_{q*}\dot q),\ldots,
\eta_n(R^{-1}_{q*}\dot q)).
\]
It follows that $R^{-1*}_q\eta_k=\omega_k$, $1\le k\le n$, are right
invariant one forms on $G$, and we also have $L(q,\dot q)=\hat l(
\omega_1(\dot q),\ldots,\omega_n(\dot q))$. We may now compute under
the assumption (\ref{e:2.51}),
\[
\d\int^T_0L(q,\dot q)dt=\int^T_0\sum_k\frac{\pa\hat l}{\pa\eta_k}\d(
\omega_k(\dot q))dt.
\]
Employing the identity (\ref{e:2.27}) we may rewrite this in the form
\begin{equation}\label{e:2.52}
\d\int^T_0L(q,\dot q)dt=-\int^1_0\sum_k\left(\frac d{dt}\left(\frac{
\pa\hat l}{\pa\eta_k}\right)\omega_k(\d{q_t})+\frac{\pa\hat l}{\pa
\eta_k}d\omega_k(\dot q,\d{q_t})\right)dt.
\end{equation}
Setting $\d{q_t}=R_{q*}W_t$, $\dot q_t=R_{q*}X_t$, for time varying
elements, $W_t,X_t\in\mathcal{G}$, we obtain, from the right
invariance of the forms $\omega_k$, $\omega_k(\d{q_t})=\eta_k(W_t)$,
and
\[
d\omega_k(\dot q_t,\d q_t)=-\eta_k([X_t,W_t]).
\]
Thus the expression (\ref{e:2.52}) yields the following necessary
conditions as a flow on $\mathcal{G}$:
\[
\sum_k\frac d{dt}\left(\frac{\pa\hat l}{\pa\eta_k}\right)
\eta_k+ad^*X_t\left(\frac{\pa\hat l}{\pa\eta_k}\right)\eta_k=0
\]
where
\begin{equation}\label{e:2.53}
ad^*X(\eta)(Y)=-\eta([X,Y]),\quad \eta\in\mathcal{G}^*,X,Y\in
\mathcal{G}.
\end{equation}
Finally, this may be written in the form known as the Euler-Poincare
equations (see Marsden and Scheurle [1993]):
\begin{equation}\label{e:2.54}
\frac d{dt}\frac{\pa l}{\pa X_t}+ad^*X_t\left(\frac{\pa l}{\pa
X_t}\right)=0.
\end{equation}

Thus, under the symmetry condition (\ref{e:2.51}), the extremal flow
on $TG$ for the variational problem (\ref{e:2.13}), is reduced to the
flow (\ref{e:2.54}) on $\mathcal{G}$ together with the kinematic flow
\begin{equation}\label{e:2.55}
\dot q_t=R_{q_t^*}X_t.
\end{equation}

We now consider this result in the context of external forces,
governed by the Lagrange D'Alembert principle (\ref{e:2.21}). By
combining the equations (\ref{e:2.52}) and (\ref{e:2.21}) we obtain
the system:
\begin{equation}\label{e:2.56}
\frac d{dt}\left(\frac{\pa l}{\pa X_t}\right)+ad^*X_t\left(\frac{\pa
l}{\pa X_t}\right)=R^*_{q_t}F(t).
\end{equation}

To obtain a version of this equation consistent with Newton's second
law, as discussed in section~(\ref{s:2.3}), we introduce the momentum
\begin{equation}\label{e:2.57}
p_t=L_F(q_t,\dot q_t)\in T^*_{q_t}G
\end{equation}
which, by the assumption (\ref{e:2.51}) and the definition of $X_t$,
we may express as
\begin{equation}\label{e:2.58}
R_{q_t}^*p_t=\frac{\pa l}{\pa X_t}.
\end{equation}
We also define the momentum, as measured in the fixed set of axes
$T^*_eG$,
\begin{equation}\label{e:2.59}
P_t=L^*_{q_t}p_t=L^*_{q_t}R^*_{q^{-1}_t}\frac{\pa l}{\pa X_t}\in
T^*_eG.
\end{equation}
Now for a Lie group $G$ the operator $Adh$ is the derivative of the mapping
$g\mapsto h^{-1}gh$ at the identity in $G$, for each $h\in G$, 
which is also a Lie algebra isomorphism.
Thus $Adq_t=L_{q^{-1}_t*}R_{q^*_t}$, and from (\ref{e:2.55}) we
obtain
\begin{equation}\label{e:2.60}
\frac d{dt}Ad{q_t}=Ad{q_t}ad X_t.
\end{equation}

We introduce the coadjoint action of $G$ on $\mathcal{G}^*$
corresponding to the infinitesimal action (\ref{e:2.53}):
\begin{equation}\label{e:2.61}
Ad^*g(\eta)(X)=-\eta(Adg(X)),\eta\in{\mathcal{G}}^*,X\in{\mathcal{G}}, g
\in G.
\end{equation}

It follows from (\ref{e:2.60}) that
\begin{equation}\label{e:2.62}
\frac d{dt}Ad^*{q_t}=-ad^*X_tAd^*{q_t},
\end{equation}
and moreover
\[
\frac d{dt}(Ad^*q_t)^{-1}=(Ad^*{q_t})^{-1}ad^*X_t.
\]

Now from (\ref{e:2.59}) we have
\begin{equation}\label{e:2.63}
P_t=(Ad^*q_t)^{-1}\frac{\pa l}{\pa X_t}.
\end{equation}
It follows from (\ref{e:2.55}) and (\ref{e:2.56}) that 
\begin{equation}\label{e:2.64}
\frac{dP_t}{dt}=L^*_{q_t}F(t),\qquad \dot q_t=R_{q_t*}X_t.
\end{equation}

Now equations (\ref{e:2.64}) represent a version of Newton's second
law corresponding to the Lagrange D'Alembert principle (\ref{e:2.21}), with the
symmetry assumption (\ref{e:2.51}). In particular, body fixed frames
are represented by a right invariant basis for $T_{q_t}Q$, while an
inertial frame is represented by a fixed basis for $T_eQ$. When the
external force field $F(t)$ vanishes identically, the momentum $P_t$,
in the inertial frame, is constant.

\paragraph{Invariance by Isometry}
In this section we will consider another symmetry of the Lagrangian,
but this time based on a Riemannian structure. We suppose in this
case that $Q$ is a Riemannian manifold under an ambient metric,
denoted $(\cdot,\cdot)$, or $g$, and a mechanical metric denoted $
\langle\cdot,\cdot\rangle$. The mechanical metric is determined from
the ambient metric by a symmetric
bundle isomorphism (relative to the ambient metric)
\begin{equation}\label{e:2.65}
J:TQ\to TQ,\qquad J_q:T_qQ\to T_qQ.
\end{equation}
Thus
\[
\langle X,Y\rangle=(JX,Y)=(X,JY),\quad X,Y\in\G(TQ).
\]
We also refer to $\langle\cdot,\cdot\rangle $ by $g^J$. Physically
$J$ represents an appropriate inertia tensor. Clearly $J$ is related
to the bundle isomorphism $\hat J$ introduced in section~\ref{s:2.3}
by
\begin{equation}\label{e:2.66}
\hat J(X)(Y)=(JX,Y)=\langle X,Y\rangle,\quad X,Y\in\G(TQ).
\end{equation}

Many mechanical systems are defined by Lagrangians of the form:
\begin{equation}\label{e:2.67}
L=\frac12\langle\dot q,\dot q\rangle-V(q)=\frac12(\dot q,J_q(\dot
q))-V(q),
\end{equation}
representing the contributions of kinetic and potential energy.

Corresponding to the metric $g^J$ is a unique (symmetric) Riemannian
connection $\nabla^J$ satisfying
\begin{enumerate}
\item[(i)] $\nabla^J_{fX}hY=f(h\nabla^J_XY+YX(h))$,
\item[(ii)] $\nabla^J_XY-\nabla^J_YX-[X,Y]\stackrel{\Delta}{=}T(X,Y)=0$.
\item[(iii)] $(\nabla^J_Zg^J)(X,Y)=Z(g^J(X,Y))-g^J(
\nabla^J_ZX,Y)-g^J(X,\nabla^J_ZY)=0$, for all $f,h\in C^\infty(Q)$,
$X,Y,Z\in\G(TQ)$.
\end{enumerate}
Corresponding to $\nabla^J$ is a covariant derivative along a curve
$t\mapsto q(t)$ in $Q$, of a curve $t\mapsto Z_t\in T_{q(t)}Q$, 
\[
\frac{D^JZ_t}{\pa t}=(\nabla^J_{\dot q}Z)(q(t)),
\]
where $Z\in\G(TQ)$ extends $Z_t$ in neighborhood of $q(\cdot)$;
$Z(q(t))=Z_t$. $\nabla^J$ extends to $\G(T^*Q)$ by setting
\begin{equation}\label{e:2.68}
(\nabla^J_X\omega)(Y)=X(\omega(Y))-\omega(\nabla^J_XY),\quad X,Y\in
\G(TQ),\quad \omega\in\G(T^*Q).
\end{equation}

Given $\omega\in\G(T^*Q)$, there exists a unique vector field $X^
\omega\in\G(TQ)$ such that 
\[
\omega(Y)=g^J(X^\omega,Y),\qquad Y\in\G(TQ).
\]
We have $(\nabla^J_Z\omega)(Y)=g^J(\nabla^J_ZX^\omega,Y)$. If $t
\mapsto \omega_t\in T^*_{q_t}Q$ is a curve in $T^*Q$, along the curve
$q(\cdot)$ in $Q$, we have from (\ref{e:2.68}),
\begin{equation}\label{e:2.69}
\frac d{dt}\omega_t(Z_t)=\left(\frac{D^J\omega_t}{\pa t}\right)(Z_t)+
\omega_t\left(\frac{D^JZ_t}{\pa t}\right).
\end{equation}

Parallel transport along a curve $q(\cdot)$ in $Q$, with $q(0)=a$, is
determined by the equations
\begin{equation}\label{e:2.70}
\frac{D^JW_t}{\pa t}=0,\qquad W_t\in T_{q_t}Q.
\end{equation}
We set $P^t_aW_0=W_t$, where $P^t_a$ is the parallel transport
operator $P^t_a:T_aQ\to T_{q_t}Q$. $P^t_a$ is an isometry of tangent
spaces $T_aQ$ and $T_{q_t}Q$ under the metric $g^J$. It is easy to
prove that, if $t\mapsto Z_t\in T_{q_t}Q$ is a curve in $TQ$ along
$q(\cdot)$, then
\begin{equation}\label{e:2.71}
\frac{D^JZ_t}{\pa t}\bigg|_{t=0}=\frac d{dt}(P^t_a)^{-1}Z_t
\bigg|_{t=0}.
\end{equation}

We now introduce our second symmetry assumption on the Lagrangian in
the variational problem (\ref{e:2.21}).

\vskip12pt
\noindent
{\bf Assumption}
\begin{equation}\label{e:2.72}
L(q_t,\dot q_t)=L(a,(P^t_a)^{-1}\dot q_t)
\end{equation}
along all curves $t\mapsto q_t$ in $Q$, with $q_0=a$.\qed
\vskip12pt

Computing under the assumption (\ref{e:2.72}) we obtain
\begin{eqnarray*}
\d\int^1_0L(q,\dot q)dt&=&\frac d{d\e}\bigg|_{\e=0}\int^1_0L(q_{
\e}(t),q_{\e}(t))dt\\
&=&\frac d{d\e}\bigg|_{\e=0}\int^1_0L(q(t),(P^{\e}_{q(t)})^{-1}\dot
q_{\e}(t))dt
\end{eqnarray*}
where $P^{\e}_{q(t)}$ is the parallel transport operator along the
curve $\e\mapsto q(t,\e)=q_{\e}(t)$, and $q(\cdot,\cdot)$ is a
variation of the curve $q(\cdot)$. Thus, using the identity 
(\ref{e:2.71}) we obtain
\begin{eqnarray}\label{e:2.73}
\d\int^1_0L(q,\dot q)dt&=&\int^1_0L_F(q,\dot q)\frac{D^J\dot q_{\e}}{
\pa\e}\bigg|_{\e=0}dt\\
&=&\int^1_0L_F(q,\dot q)\frac{D^J}{\pa t}\d q(t)dt\nonumber\\
&=&-\int^1_0\frac{D^J}{\pa t}L_F(q,\dot q)\d q(t)dt.\nonumber
\end{eqnarray}

Thus extremals of the problem (\ref{e:2.21}), where the Lagrangian
satisfies assumption (\ref{e:2.72}), are governed by 
\begin{equation}\label{e:2.74}
\frac{D^J}{\pa t}L_F(q,\dot q)=F(t)\bigg(=\frac{D^J}{\pa t}p_t\bigg).
\end{equation}

Lagrangians which satisfy assumption (\ref{e:2.72}) are functions of the 
mechanical metric since $P^t_a$ is an isometry,
\[
\langle\dot q,\dot q\rangle_q=\langle(P^t_a)^{-1}\dot q,(P^t_a)^{-1}
\dot q\rangle_a.
\]
Specifically, when the Lagrangian $L$ is given by the expression 
(\ref{e:2.67}), we have from (\ref{e:2.57}) $p=L_F(q,\dot q)=
\langle\dot q,\cdot\rangle=\hat J_q(\dot q)$, and from (\ref{e:2.73})
\begin{equation}\label{e:2.75}
\frac{D^Jp_t}{\pa t}+dV(q_t)=F(t),\qquad \dot q_t=\hat
J^{-1}_{q_t}(p_t).
\end{equation}
Thus, once more, a symmetry assumption enables us to reduce the
Lagrangian flow defined by a second order vector field in $\G(TTQ)$
satisfying equation (\ref{e:2.22}), to a system of equations in $T^*Q$. In the
case that the potential $V$ is constant and $F\equiv0$, then the
motion (\ref{e:2.75}) reduces to
\[
\frac{D^{J^2}q_t}{\pa t^2}=0.
\]
Thus the uniform flow in the sense of the discussion on Newton's
first law is simply the geodesic flow, as is commonly understood. 
Equations (\ref{e:2.75}) yield a specific case of the Newton system
of equations (\ref{e:2.20}).
However, to give a suitable interpretation to Newton's second law, we
define a momentum in the fixed space $T^*_aQ$,
\begin{equation}\label{e:2.76}
P_t=(P^t_a)^*p_t\in T_aQ
\end{equation}
along the curve $t\mapsto q_t$ in $Q$, with $q_0=a$. Now for any $Z
\in T_aQ$ we may compute
\[
\frac d{dt}(P^t_a)^*p_t(Z)=\frac d{dt}p_tP^t_aZ=\frac{D^Jp_t}{\pa
t}(P^t_aZ),
\]
since $P^t_aZ$ is indeed parallel along $q(\cdot)$. Thus from 
(\ref{e:2.57}) and (\ref{e:2.74}) we obtain the following
representation of the extremals for the problem (\ref{e:2.21}), with
a Lagrangian satisfying the symmetry assumption (\ref{e:2.72})
\begin{equation}\label{e:2.77}
\frac d{dt}P_t=P^{t*}_aF(t),\qquad \dot q=\hat
J^{-1}_q((P^t_a)^{-1*}P_t).
\end{equation}

In this setting we again formulate Newton's second law for the
problem, so that in the absence of external forces, $F\equiv0$, we
obtain uniform motion $(d/dt)P_t=0$.

It is interesting to write down the classical nonholonomic motion
corresponding to a Lagrangian $L$, which satisfies the assumption 
(\ref{e:2.72}), and the nonholonomic constraints (\ref{e:2.23}). From
(\ref{e:2.73}) and (\ref{e:2.24}) we directly obtain
\begin{equation}\label{e:2.78}
\frac{D^J}{\pa t}L_F(q,\dot q)=\sum^m_{L=1}\mu_i\omega_i+F(t); \quad 
\omega_i(\dot q)=0,\quad 1\le i\le m.
\end{equation}
Similarly, constrained variational mechanics under the same conditions is 
obtained from (\ref{e:2.26}), (\ref{e:2.28}), and (\ref{e:2.73}), and takes
the form
\begin{equation}\label{e:2.79}
\begin{array}{c}
\displaystyle\frac{D^J}{\pa t}L_F(q,\dot q)+\sum^m_{i=1}\dot\la_i\omega_i+
\la_ii_{\dot q} d\omega_i=F(t),\\
\omega_i(\dot q)=0,\qquad 1\le i\le m.
\end{array}
\end{equation}
Systems (\ref{e:2.78}) may be written directly as a system on $TQ$
when $L(q,\dot q)=1/2\langle\dot q,\dot q\rangle$ and $\omega_i(Z)=
\langle X^\omega_i,Z\rangle$, $1\le i\le m$, in the following way.
\begin{equation}\label{e:2.80}
\begin{array}{c}
\displaystyle\frac{D^{J^2}q}{\pa t^2}=\sum^m_{i=1}\mu_iX^{\omega}_i+\hat
J^{-1}_q(F(t)),\\
\langle X^\omega_i,\dot q\rangle=0,\quad 1\le i\le m.
\end{array}
\end{equation}
Bloch and Crouch [1993]%\cite{ }
, [1995a] %\cite{ }
 have studied
nonholonomic systems in this precise form. (For another view
see Bloch, Krishnaprasad, Marsden and Murray [1996].)

To express constrained variational mechanics in a similar form, we need an
expression for $i_{\dot q}d\omega$. We have
\[
d\omega_i(X,Y)=(\nabla^J_X\omega_i)(Y)-(\nabla^J_Y\omega_i)(X),\quad
X,Y\in\G(TQ).
\]
Thus,
\[
(i_{\dot q}d\omega_i)(X)=\langle\frac{D^JX^{\omega}_i}{\pa t},X
\rangle-\hat J_q(\dot q)(\nabla^J_XX^\omega_i).
\]
Now $X_q\mapsto\hat J_q(\dot q)(\nabla^J_{X_q}X^\omega_i)$ defines a
one form denoted $\hat J_q(\dot q)(\nabla^JX^\omega_i)$. Thus we have
\[
\hat J^{-1}_q(i_{\dot q}d\omega_i)=\frac{D^JX^\omega_i}{\pa t}-\hat J^{-1}_q(
\hat J_q(\dot q)(\nabla ^JX^\omega_i)).
\]
It follows that the constrained variational mechanics corresponding to the 
Lagrangian
$1/2\langle\dot q,\dot q\rangle$ and constraints (\ref{e:2.23}) is
described by
\begin{equation}\label{e:2.81}
\begin{array}{c}
\displaystyle\frac{D^{J^2}q}{\pa t^2}+\sum^m_{i=1}\dot\la_iX^\omega_i+\sum^m
_{i=1}
\la_i\left(\frac{D^JX^\omega_i}{\pa t}-\hat J^{-1}_q(\hat J_q(\dot
q)(\nabla^JX^\omega_i))\right)=0,\\
\langle X^\omega_i,\dot q\rangle=0,\quad 1\le i\le m.
\end{array}
\end{equation}
The systems (\ref{e:2.80}) and 
(\ref{e:2.81}) were contrasted in Bloch and Crouch [1993]%\cite{ }
.

\paragraph{Two Optimal Control Problems}
Often reduced mechanical systems are conveniently described in terms
of Poisson structures. A Poisson structure on a manifold $M$ is a
bilinear operation $\{\cdot,\cdot\}$  on $C^\infty(M)$ such that $(C^
\infty(M),\{\cdot,\cdot\})$ is a Lie algebra and $\{FG,H\}=\{F,H\}G+F
\{G,H\}$ for all $F,G,H\in C^\infty(M)$. For $H\in C^\infty(M)$ the
Hamiltonian vector field defined by $H$ and the Poisson structure is
determined from the identity
\begin{equation}\label{e:2.82}
X_H(G)=\{G,H\},\qquad G\in C^\infty(M).
\end{equation}
Locally, if $M$ is represented by coordinates $\{x_1,\ldots,x_m\}$
then
\[
\{G,H\}_x=P_{ij}(x)\frac{\pa G(x)}{\pa x_i}\frac{\pa H(x)}{\pa x_j}
\]
where, for each $x\in M$, $[P_{ij}]_{1\le i,j\le m}$ is a skew matrix
which is not necessarily nondegenerate, and the dynamical system
\[
\dot x=X_H(x),
\]
has the form
\[
\dot x_i=P_{ij}\frac{\pa H}{\pa x_j}.
\]
In the case that $P$ is an invertible matrix for each $x\in M$, then
$P$ is the inverse of a corresponding symplectic structure.

There are two cases of special interest where the configuration space
$Q$ is a Lie group $G$, or a homogeneous space $G/H$, where $H$ is a
closed Lie subgroup of $G$. These two cases occur when the
homogeneous spaces are realized as orbits of a point in the Lie
algebra $\mathcal{G}$ or dual Lie algebra $\mathcal{G}^*$, under the
adjoint or coadjoint actions of $G$, respectively. Since
\[
Ad\exp_GX=\exp_{Gl(\mathcal{G})}adX,\quad X\in\mathcal{G},
\]
tangent vectors to the adjoint orbit of $Y\in\mathcal{G}$ have the
form $ad X(Y)$, while tangent vectors to the co-adjoint orbit of $
\eta\in\mathcal{G}^*$ have the form $ad^*X(\eta)$.

Unlike adjoint orbits, co-adjoint orbits have a natural symplectic
structure, the Kirillov-Kostant-Souriau form $\omega$, which at $\mu
\in\mathcal{G}^*$ is given by
\[
\omega(\mu)(ad^*X(\mu),ad^*Y(\mu))=\mu([X,Y]),\quad X,Y\in\mathcal{G}.
\]
There is a special Poisson structure on $\mathcal{G}^*$. If $F$ and
$H$ are functions on $\mathcal{G}^*$, $dF/d\mu$, $dH/d\mu$ may be
identified as elements of $\mathcal{G}$. The Lie-Poisson bracket is
defined by
\[
\{F,H\}_{\mu}=-\mu\left(\left[\frac{dF(\mu)}{d\mu},\frac{dH(\mu)}{d
\mu}\right]\right).
\]
Thus, if $X_H$ is the Hamiltonian vector field determined by $H$, we
have
\[
X_H(\mu)=-ad^*_{\frac{dH}{d\mu}}\mu.
\]
In particular, the Lie-Poisson equations for the Hamiltonian $H$ are
given by
\begin{equation}\label{e:2.83}
\dot\mu_t=-ad^*_{\frac{dH}{d\mu}}\mu_t,
\end{equation}
and restrict to a Hamiltonian system on coadjoint orbits.

A specific reduction process is concerned with the case where $Q=G$
and employs the moment map
\begin{equation}\label{e:2.84}
\phi:T^*G\to{\mathcal{G}}^*,\phi(p,q)=R^*_qp,\quad p\in T^*_qG.
\end{equation}
(See Marsden Ratiu [1994] %\cite{ }
for details). One can easily check that
\begin{equation}\label{e:2.85}
\phi(L^*_{h^{-1}}p,L_hq)=-Ad^*h(R^*_qp),\quad h\in G,\quad p\in
T^*_qG,
\end{equation}
and so $\phi$ is equivariant for the action of $G$ on $T^*G$,
\[
(p,q)\longmapsto(L^*_{h^{-1}}p,L_hq)\stackrel{\Delta}{=}h\cdot(p,q),
\quad h\in G,\quad p\in T^*_qG,
\]
and the coadjoint action of $G$ on $\mathcal{G}^*$. In particular,
\[
\phi(h\cdot(\eta,e))=-Ad^*h(\eta).
\]

Consider the optimal control problem on a Lie group $G$,
\begin{equation}\label{e:2.86}
\begin{array}{l}
\displaystyle\min_u\int^T_0h_0(u)dt;\\
\mbox{subject to: }\dot q_t=\sum^k_{i=1}R_{q_t*}X_iu_i,q\in G^n,q_0
\mbox{ and }\\
q_T\mbox{ fixed, where }X_1,\ldots,X_n\mbox{ is a basis}\\
\mbox{for the Lie algebra $\mathcal{G}$ of $G$}.
\end{array}
\end{equation}

In the case $k<n$, the problem is well posed as long as the Lie
algebra generated by $X_1,\ldots,X_k$ is equal to $\mathcal{G}$. We
set $X_t=\sum^k_{i=1}X_iu_i(t)\in\mathcal{G}$ and assume that there
are functions $L$ on $TG$ and $l$ on $\mathcal{G}$ such that 
\begin{equation}\label{e:2.87}
L(q,\dot q)=l(R_{q^{-1}}*\dot q)=l(X_t)=h_0(u).
\end{equation}

Thus in case $m=n$, and (\ref{e:2.87}) holds, problem (\ref{e:2.86})
is equivalent to the problem (\ref{e:2.13}) where assumption 
(\ref{e:2.51}) holds. In particular, the extremals are governed by
the Euler-Poincare equations (\ref{e:2.54}). However, we may also apply
the Pontryagin maximum principle to the formulation (\ref{e:2.86}).
Thus we define the Hamiltonian
\begin{eqnarray*}
\hat H(q,p,u)&=&p_t\sum^n_{i=1}R_{q^*_t}X_iu_i-h_0(u)\\
&=&R^*_{q_t}p_tX_t-l(X_t).
\end{eqnarray*}

Setting $\mu_t=R^*_{q_t}p_t\in\mathcal{G}^*$, $\hat H(q,p,u)=
\mu_t(X_t)-l(X_t)$. It follows that the optimal control is determined
through
\begin{equation}\label{e:2.88}
\mu_t=\frac{\pa l}{\pa X_t},
\end{equation}
and that
\[
\bar H(q_t,p_t)\stackrel{\Delta}{=}H(\mu_t)=\mu_t(X_t)-l(X_t)\Big|_{\mu_t=\pa 
l/\pa X_t}.
\]
Clearly $dH/d\mu(\mu_t)=X_t$, and so the Euler-Poincare equations 
(\ref{e:2.54}) are mapped by the Legendre transform (\ref{e:2.88})
onto the Lie Poisson equations (\ref{e:2.83}). It follows that the
momentum map $\phi:T^*G\to\mathcal{G}^*$ maps the extremal flow of
the optimal control problem (\ref{e:2.86}) onto the Lie Poisson
equations, with $H\circ\phi=\bar H$.

We briefly consider another optimal control problem which is equivalent to a
higher order variational problem and is an example of (2.47 iii), in
the connection between mechanics and optimal control theory. We pose this 
problem in the context of a Riemannian setting.
\begin{equation}\label{e:2.89}
\begin{array}{l}
\displaystyle\min_{F(\cdot)}\int^T_0\frac12\|F(t)\|^2_{q_t}dt\\
\displaystyle\mbox{subject to: }\frac{D^Jp_t}{\pa t}=F(t),\quad q(0)=q_0,\quad
\dot q(0)=\dot q_0\\
q(T)=q_T,\quad\dot q(T)=\dot q_T,\quad p_t=\hat J_q(\dot q),\quad q
\in(M,g^J).
\end{array}
\end{equation}

In the above, $\|F\|_q$ is the norm of $F$ as a linear functional on
$T_{q}M$, using the ambient metric on $Q$. It follows that $\|F\|^2_q=
\|J_q\frac{D^{J^2}q}{\pa t^2}\|^2_q$, and so the problem 
(\ref{e:2.89}) is equivalent to the variational problem
\begin{equation}\label{e:2.90}
\begin{array}{l}
\displaystyle\min_{q(\cdot)}\int^T_0\frac12\langle\frac{D^{J^2}q}{\pa t^2},J_q
\frac{D^{J^2}q}{\pa t^2}\rangle dt\\
\mbox{subject to: }q\in(M,g^J),\quad q(0)=q_0,\dot q(0)=\dot q_0,\\
q(T)=q_T,\quad \dot q(T)=\dot q_T.
\end{array}
\end{equation}
The problem (\ref{e:2.90}) was considered in Bloch and Crouch 
[1993]%\cite{}
, following the work of Crouch and Leite [1995]%\cite{}
, and Camarinha [1996]%\cite{}
,
for $J=id$. For $J\not\equiv id$ the necessary conditions are given
by
\[
\frac{D^{J^2}}{\pa t^2}J_q\frac{D^{J^2}q}{\pa t^2}+R\left(J_q
\frac{D^{J^2}q}{\pa t^2},\frac{D^Jq}{\pa t}\right)\frac{D^Jq}{\pa t}+
\frac12J'_q\left(\frac{D^{J^2}q}{\pa t^2},\frac{D^{J^2}q}{\pa t^2}
\right)=0
\]
where $\langle J'(X,Y),Z\rangle=\langle(\nabla^J_ZJ)(X),Y\rangle$,
$X,Y,Z\in\G(TQ)$ and $R$ is the curvature tensor corresponding to the
Riemannian connection $\nabla^J$.

\subsection{%2.6
A special case}\label{s:2.6}

We now specialize some of the discussions above to systems where the
configuration space $Q$ is a semisimple, compact Lie group $G$. Such
cases arise as the Lie groups corresponding to Lie algebras $
\mathcal{G}$ obtained as the real form of complex semisimple Lie
algebras. In such cases the Killing form $\mathcal{K}$ on $\mathcal{G}$ is
negative definite, which in turn defines a left and right invariant
metric on $G$, coinciding with $-\mathcal{K}$ on $\mathcal{G}$. We denote this
metric by $(\cdot,\cdot)$ or $g$, and call it the ambient metric. The 
unique (symmetric) Riemannian connection $\nabla$ on $G$
corresponding to $g$ is defined by
\[
\nabla_XY=\frac12[X,Y]
\]
where $X,Y$ are left (or right) invariant vector fields on $G$. We
let $J:\mathcal{G}\to\mathcal{G}$ be a positive definite linear 
mapping, relative to $g$. We may extend $J$ to a bundle isomorphism
$J:TG\to TG$, by left (or right) translation of $J$, to define left
(or right) invariant metrics on $G$, denoted $g^J$. These in turn
define Riemannian connections on $G$, which are left (or right)
translates of the following bilinear form on $\mathcal{G}$
\begin{equation}\label{e:2.91}
\nabla^J_XY=\frac12([X,Y]+J^{-1}[X,JY]+J^{-1}[Y,JX])
\end{equation}
for $X,Y\in\mathcal{G}$.

The Killing form allows us to identify ${\mathcal{G}}^*$ with $
\mathcal{G}$, and hence identify adjoint and co-adjoint orbits. Since
$\mathcal{G}$ is semisimple, for cases where $\mathcal{G}$ and $G$
are represented by matrices we may identify the Killing form with the
trace form. There is also another metric on the adjoint orbits called
the normal metric. If $A\in\mathcal{G}$ we have an orthogonal
decomposition
\[
{\mathcal{G}}={\mathcal{G}}^A\oplus{\mathcal{G}}_A
\]
where ${\mathcal{G}}_A=\mbox{ker}adA$, the centralizer of $A$, and $
{\mathcal{G}}^A=\mbox{Im}adA$. If $X\in{\mathcal{G}}$, then we decompose
$X$ accordingly as $X=X^A+X_A$. If $[X,A]$ and $[YA]$ are tangents to
the adjoint orbit of $A$, then the normal metric is defined by
\begin{equation}\label{e:2.92}
\langle[X,A],[Y,A]\rangle_N=(X^A,Y^A).
\end{equation}
For cases where $\mathcal{G}$ is simply semisimple $\langle\cdot,
\cdot\rangle_N$ is merely nondegenerate. When $\mathcal{G}$ is
compact, then $\langle\cdot,\cdot\rangle_N$ is also positive
definite.

The interesting point about the special case treated above is that
appropriate variational problems and optimal control problems posed by systems 
with
configuration space $G$, can be reduced via both of the
techniques outlined in section~\ref{s:2.5}. In particular, the
sub-Riemannian geodesic problem (2.50), with constraints 
(2.51), may be treated in the situation where the manifold
$M$ is a Lie group $G$ as above, the metric is a right invariant
metric as described above, and the constraints are defined by right
invariant one forms. We will denote this as the right invariant
sub-Riemannian geodesic problem and clearly this is an example of
(2.47 ii), in the connection between mechanics and optimal control
theory. This problem is equivalent to the
optimal control problem on $G$, defined by (\ref{e:2.86}) and 
(\ref{e:2.87}), which we may rewrite in the form
\begin{equation}\label{e:2.93}
\begin{array}{l}
\displaystyle\min_u\int^T_0\frac12\langle\dot q,\dot q\rangle_qdt\\
\mbox{subject to: }\dot q=\sum^n_{i=m+1}R_{q*}X_iu_i,\quad q\in G^n,\\
q(0)=q_0,q(T)=q_T,X_1,\ldots,X_n\mbox{ is a basis for }\mathcal{G}.
\end{array}
\end{equation}
The nonholonomic constraints are defined by the right invariant one
forms
\begin{equation}\label{e:2.94}
\omega_i(X_q)=(R_{q*}X_i,X_q),X_q\in T_qQ,\quad 1\le i\le m,
\end{equation}
and the metric $g^J=\langle\cdot,\cdot\rangle$ is the right invariant
mechanical metric defined by a positive definite linear mapping $J:
\mathcal{G}\to\mathcal{G}$. Clearly both assumption (\ref{e:2.51}),
group invariance, and assumption (\ref{e:2.72}), invariance by
isometry, may be applied to the right invariant sub-Riemannian
geodesic problem.

To obtain the normal extremals we may simply appeal to the problem's 
variational formulation and note that this is simply an example of
constrained variational mechanics (\ref{e:2.26}) discussed in 
section~\ref{s:2.3}, without the
extremal force field $F$. Thus the normal extremals are governed by
system (\ref{e:2.28}). However, applying the reduction
technique, utilizing the invariance by isometry, we obtain the system
(\ref{e:2.81}). Utilizing the right invariance once more, we may
simplify even further. If $Z\in\G(TG)$ we may write
\[
Z(q(t))=\sum^n_{i=1}z_i(t)R_{q*}X_i.
\]
In particular, we write $\dot q(t)=\sum^n_{i=1}v_i(t)R_{q*}X_i$. We set
$Z_t=\sum^n_{i=1}z_i(t)X_i$, $V_t=\sum^n_{i=1}v_i(t)X_i$. It follows
that
\[
\frac{D^JZ}{\pa t}=R_{q*}(\dot Z_t+\nabla^J_{V_t}Z_t),
\]
where $\nabla^J$ is given by the expression (\ref{e:2.91}). It
follows once more that system (\ref{e:2.81}) may be expressed as
\begin{equation}\label{e:2.95}
\begin{array}{c}
\dot V_t+\nabla^J_{V_t}V_t+\displaystyle\sum^m_{i=1}\dot\la_iX_i\\
+\displaystyle\sum^m_{i=1}\la_i(\nabla^J_{V_t}X_i-\hat J^{-1}(\hat J(V_t)
(\nabla^JX_i)))=0\\
\langle V_t,X_i\rangle=0,\qquad 1\le i\le m.
\end{array}
\end{equation} 

For compact semisimple Lie algebras $\mathcal{G}$ we have
\[
(adX(Y),Z)+(Y,adX(Z))=0,\qquad X,Y,Z\in\mathcal{G}.
\]
It follows from (\ref{e:2.91}) that
\begin{eqnarray*}
\langle X,\nabla^J_{V_t}X_i\rangle-\langle X,\hat J^{-1}(\hat J(V_t)
\nabla^JX_i)\rangle&=&\langle X,(\nabla^J_{V_t}X_i)\rangle-\langle V_t,\nabla^J
_XX_i
\rangle\\
&=&\langle X,J^{-1}[V_t,JX_i]\rangle.
\end{eqnarray*}
Thus, (\ref{e:2.95}) may be written as
\[
\begin{array}{c}
\dot V_t+J^{-1}[V_t,JV_t]+\displaystyle\sum^m_{i=1}\dot\la_iX_i+
\la_iJ^{-1}[V_t,JX_i]=0,\\
\langle V_t,X_i\rangle=0,\qquad 1\le i\le m.
\end{array}
\]
Setting $L_t=\sum^n_{i=m+1}v_i(t)X_i+\sum^m_{i=1}\la_i(t)X_i=V_t+
\sum^m_{i=1}\la_i(t)X_i$, we obtain
\begin{equation}\label{e:2.96}
\dot L_t+J^{-1}[V_t,JL_t]=0.
\end{equation}
This equation defines the evolution of $v_i(t)=u_i(t)$, $m+1\le i
\le n$ and $\la_i(t)$, $1\le i\le m$. The case $J=I$ was obtained by
Brockett [1973] %\cite{ }
in the context of the rotation group, although he dealt with a more complicated
 case
where a drift term is included in the control dynamics.

The equations (\ref{e:2.96}) may be investigated further when we
assume $\mathcal{G}$ has additional structure, inherited from the
assumption that there is a Lie subgroup $K$ of $G$ such that the
$G/K$ is a Riemannian symmetric space. This case was investigated by
Bloch, Crouch and Ratiu [1994]%\cite{ }
. We may assume in this situation
that
\[
\mathcal{G}=P+K
\]
where
\begin{equation}\label{e:2.97}
[\mathcal{PP}]\subset \mathcal{K},\qquad [\mathcal{P},\mathcal{K}]\subset 
\mathcal{P},\qquad [\mathcal{K},\mathcal{K}]\subset \mathcal{K},
\end{equation}
and
\[
({\mathcal P},{\mathcal K})=0.
\]
If the basis $X_1,\ldots,X_n$ of $\mathcal{G}$ is such that $X_1,
\ldots,X_m$ is a basis of ${\mathcal K}$ and $X_{m+1},\ldots,X_n$ is a basis 
for
$\mathcal{P}$, with $\mathcal{P}$ and $\mathcal{K}$ being invariant subspaces 
for $J$, setting $
\La_t=\sum^m_{i=1}\la_i(t)X_i$ we obtain from (\ref{e:2.96})
\begin{eqnarray}\label{e:2.98}
\dot V_t&=&-J^{-1}[V_t,J\La_t]\\
\dot\La_t&=&-J^{-1}[V_t,JV_t].\nonumber
\end{eqnarray}

Setting
\[
P_t=JV_t,\qquad Q_t=J\La_t,
\]
we obtain
\begin{eqnarray}\label{e:2.99}
\dot P_t&=&[Q_t,J^{-1}P_t]\\
\dot Q_t&=&[P_t,J^{-1}P_t].\nonumber
\end{eqnarray}

If $H(P,Q)=\frac12(P,J^{-1}P)=H(P)$, the equations (\ref{e:2.99}) are
simply the Lie-Poisson equations (\ref{e:2.83}), obtained by
identifying $\mathcal{G}$ with $\mathcal{G}^*$.

Thus, the normal extremals of the right invariant sub-Riemannian
geodesic problem on a compact semisimple Lie group are governed by
Lie-Poisson equations but with a degenerate Hamiltonian. The general
right (or left) invariant geodesic problem on a compact semisimple
Lie group is known to yield the generalized rigid body flow. This
flow reduces to the Lie Poisson equations corresponding to a
non-degenerate Hamiltonian function on $\mathcal{G}^*$ $H(M)=
\frac12(M,J^{-1}M)$,
\begin{equation}\label{e:2.100}
\dot M=[M,J^{-1}M].
\end{equation}

If $M_t=JV_t+Q_t$, and we impose the additional conditions where
$G/K$ is a Riemannian symmetric case so that $Q_t\in \mathcal{K}$, $V_t\in 
\mathcal{P}$,
equations (\ref{e:2.100}) may be decomposed as:
\begin{eqnarray*}
\dot Q_t&=&[Q_t,J^{-1}Q_t]+[JV_t,V_t]\\
J\dot V_t&=&[Q_t,V_t]+[JV_t,J^{-1}Q_t].
\end{eqnarray*}
Thus, if $J|_K\to\infty$, we obtain the equations (\ref{e:2.99}) by
setting $P_t=JV_t$.

This result may also be viewed another way, as pointed out by
Brockett. Writing the right invariant Riemannian geodesic problem in
the form (\ref{e:2.86}) and (\ref{e:2.87}) and using the terminology
above we obtain the equivalent formulation:
\[
\begin{array}{l}
\displaystyle\min_{V_t}\int^T_0\frac12(V_t,JV_t)dt\\
\mbox{subject to $\dot q=R_{q*}V_t$, $q(0)=q_0$, $q(T)=q_T$}. 
\end{array}
\]
Writing $V_t=J^{-1}(P_t)+\La_t$, we obtain
\[
\begin{array}{l}
\displaystyle\min_{V_t}\int^T_0\frac12((\La_t,J\La_t)+(P_t,J^{-1}P_t))dt\\
\mbox{subject to $\dot q=R_{q_*}(J^{-1}P_t+\La_t)$, $q(0)=q_0$,
$q(T)=q_T$.}
\end{array}
\]
Hence $J|_K\to\infty$ can only be achieved in the limit if $\La_t
\equiv0$. Thus the dynamics is constrained to be in $\mathcal{P}$, and the
Hamiltonian of the extremal flow is $\frac12(P_t,J^{-1}P_t)$, as
already obtained above. These singular cases were originally studied
by Baillieul [1978]%\cite{ }
.
 
%\section{Analytical Mechanics}
\section{Optimization}
While optimal control, Lagrangian and Hamiltonian problems involve
minimization over a class of curves, many applied problems of interest
are optimization problems that involve minimization of a function
over a set. Such a set is often a smooth manifold but is sometimes
a set with piecewise smooth boundary (as in linear programming
problems) or a discrete set (in problems such as sorting
numbers or integer programming). While such problems appear at first
inspection to be static in nature, they can often be cast
in a dynamic context -- in the smooth setting by considering
the steepest descent (gradient) flow associated with the problem
and in the discrete setting by considering an iteration or
by embedding the iteration in a smooth flow.


An example which illustrates many of these ideas
was introduced by Brockett [1988] in the context of matching problems
in computer vision. He showed that 
while the least squares matching function could not be evaluated
explicitly in general, its equations of steepest descent were elegant
equations on an orthogonal group whose qualitative analysis was
particularly tractable. The analysis was in fact a dynamic analysis
of a type of total least squares problem (see Bloch [1990], [1990b]). 

The discrete optimization problem is easily described:
one is given 2 sequences of points in $\mathbf{R}^n$ and one
wishes to find an orthogonal transformation that maps one
to the other so as to minimize the least squares distance
between the points. This problem cannot be solved
explicitly in general, but one can right down a steepest
descent equation on the Lie group $SO(n)$ which
can be easily analyzed. We will write down this flow
explicitly below.

Brockett showed that his equations descended in natural fashion
to the so called ``double bracket'' equations $\dot L=[L,[L,N]]$
which we discuss further below. These equations perform 
in a smooth fashion the analysis for some interesting discrete
problems: in particular they can viewed as sorting sequences of
numbers, computing eigenvalues of symmetric matrices
and carrying out certain linear programming problems. These
ideas are in the spirit of Symes's observation that the
QR algorithm for finding the eigenvalues of symmetric 
tridiagonal matrices is the time-1 map of the Toda lattice flow,
Symes [1981], the subsequent work of
Deift, Li and Tomei [1983] on the Toda lattice, and
the work of Bayer and Lagarias [1989] on the relationship
between integrable Hamiltonian flows and Karmarkar's algorithm for
linear programming (see also later work by Faybusovich [1991]).
The double bracket equations in fact make contact with all these
ideas as well shall indicated below. Other interesting
extensions of these ideas are discussed in the collection
Bloch [1994], the book by Helmke and Moore [1994] and
the thesis of Steve Smith [1993].

We begin here by discussing a natural theoretical
generalization of these ideas -- see Bloch, Brockett and Ratiu [1990],
[1992] -- to gradient flows on compact Lie groups.  We derive an explicit 
equation for a particular gradient flow relative to a left-invariant
 Riemannian metric.  We then show 
that projecting the flow onto an adjoint orbit of the Lie algebra by
 taking the natural projection 
yields a flow which is itself gradient with respect to the normal metric
 on the orbit.  This flow 
is shown to be of the ``double bracket'' form
 $\dot L(t)=\lbrack L(t),\lbrack L(t),N\rbrack\rbrack$ where N is a
 given constant matrix (see Brockett [1988]
 and Bloch, Brockett and Ratiu [1992]). This flow 
is isospectral and we show that in fact this double bracket flow is the
 compatibility condition for 
the ``spectral equations'' for the flow at the group level.
In the tridiagonal matrices in $sl(n)$ it was shown (see Bloch [1990])
that this double
bracket flow reduces for appropriate $N$ to the Toda lattice flow --
an integrable Hamiltonian system. This was analyzed for general 
Lie algebras in Bloch, Brockett and Ratiu [1990], [1992] and Bloch, Flaschka
and Ratiu [1990]. More precisely what one is showing is that the Toda flow
is gradient on a level set of its integrals.


Let $G_u$ be a compact Lie group with Lie algebra ${\mathfrak g}_u$ and
 let $\kappa (~~,~~)$ denote the 
Killing form on ${\mathfrak g}_u$.  Define the smooth function $F(\theta )$
 on $G_u$ by
\begin{equation}
F(\theta ) =\kappa (Q,Ad_\theta N) 
\end{equation}

\noindent where $Q$ and $N$ are fixed elements of ${\mathfrak g}_u$ and
 $Ad$ denotes the adjoint action 
of $G_u$ on ${\mathfrak g}_u$.  Then we have (see Brockett [1988]
 for the unitary case which applies to the matching problem 
described above):

\begin{prop}  The gradient flow of $F(\theta )$ relative to the left-invariant 
Riemannian metric on $G_u$ whose value at the identity is minus the Killing
 form is given by
\begin{equation} 
\dot \theta (t) = \theta (t) \cdot\lbrack Ad_{{\theta (t)}^{-1}}Q,N\rbrack
\label{upstairs}
\end{equation}
where $\theta\cdot P$ denotes the left translation of $P$ by $\theta$.
\end{prop}

For the proof see Bloch, Brockett and Ratiu [1992]. 

Let ${\mathfrak g}_u$ be the compact real form of a complex semisimple Lie
 algebra ${\mathfrak g}$ with Lie group $G$.  Then the 
equations have the normal real form of $G$ intersect 
as an invariant 
submanifold.  The gradient flow can then be formulated as a gradient flow
 with respect to the normal 
metric on this intersection.  The equation first appeared in this form
 on $SO(n)$ (see Brockett [1988]) and indeed is the flow
that arises in the computer vision matching problem described
above.

\begin{cor} The projection of the flow (\ref{upstairs}) onto the adjoint orbit
 containing $Q$, 
obtained by setting $L(t)=Ad_{\theta (t)^{-1}}Q$ is given by
\begin{equation}
\dot L(t) = \lbrack L(t),\lbrack L(t),N\rbrack\rbrack\,.
\label{downstairs}
\end{equation}
\end{cor}

Indeed, by the Chain Rule we get
$\dot L = \lbrack L,\theta^{-1}\dot\theta\rbrack =
 \lbrack L,\lbrack L,N\rbrack\rbrack\,.$

It is interesting to notice that Corollary~3.2 
can be interpreted as a compatibility condition between an ``eigenvalue
 equation'' and the time 
evolution of the ``eigenfunction''. 
 Indeed $L(t)=Ad_{\theta (t)^{-1}}Q$ says that $L(t)$ and $Q$ 
have the same spectrum, if they are matrices.  This relation can be thought
 of as the 
totality of eigenvalue equations for $L$, with the columns of $\theta$ as
 eigenfunctions.  The time 
evolution for $\theta$ is given by (\ref{upstairs}).

Note that the flow (\ref{downstairs}) is the image of (\ref{upstairs}) 
under the quotient map $G_u\to G_u/H$, 
 $H$ the stabilizer of $L(0)=L_0$, where $G_u/H$ is identified with the
 adjoint orbit ${\mathcal O}$ through $L_0$.  This ``projected'' flow 
turns out to be itself a gradient flow on the adjoint orbit ${\mathcal O}$
 endowed with the normal metric..


It was shown in Bloch, Brockett and Ratiu [1990, 1992] 
that the
following holds:


Then
\begin{prop}
The gradient vector field of the function $H(L)=\kappa(L,N)$
on the adjoint orbit $\cal O$ of $\mathfrak g_u$ containing the
initial condition $L(0)=L_0$ with respect to the normal
metric $<,>_N$ on $\cal O$ is given by

\begin{equation}
\dot L(t)=[L(t),[L(t),N]]\,.
\label{db}
\end{equation}

\end{prop}

Again one may restrict to the normal real form.

The connection with the Toda lattice flow is a follows:

Now let $\mathfrak a $ be a maximal abelian subalgebra of ${\mathfrak
g}_u$
and choose ${\mathfrak h}={\mathfrak a}\oplus i{\mathfrak a}$
 as the Cartan subalgebra
of the complexification ${\mathfrak g}$ of ${\mathfrak g}_u$.
 Let $\Phi$ denote the
 system of roots
of $\mathfrak g$ defined by $\mathfrak h$ and let $\Delta$ denote the 
simple roots. Choose $\{h_j,e_{\alpha}|j=1,\cdots ,l,\alpha\in\Phi\}$
to be a Chevalley basis of $\mathfrak g$ with $\alpha_1,\cdots ,\alpha_n$
denoting the simple roots. Here $h_j$ lie in the real part $\mathfrak a$ of 
$\mathfrak h$. 


Bloch, Brockett and Ratiu ([1990,1992]) proved the following result
for the symmetric tridiagonal generalized Toda flow (see Bloch [1990]
for the $sl(n)$ case):

\begin{thm}
If $N$ is $i$ times the simple coweights of $\mathfrak g$ then for
\begin{equation}
L=\sum_j^lib_jh_j+\sum_j^lia_j(e_{\alpha_j}+e_{-\alpha_j})\,,
\end{equation}  
equation (\ref{db}) gives the generalized (tridiagonal, symmetric)
Toda lattice equations on the level set of all integrals
of the Toda flow. Explicitly, $N$ is given by
\begin{equation}
N=\sum_j ix_jh_j\,,
\end{equation}
where $(x_1,\cdots ,x_l)$ is the unique solution of the system
\begin{equation}
\sum_jx_j\alpha_i(h_j)=-1,\,\,\,i=1\cdots l\,.
\end{equation}
\end{thm}

We remark that this flow is Hamiltonian on a coadjoint orbit
of the lower Borel subalgebra of the normal real form of $\mathfrak g$.


As indicated above the double bracket equations have a number of
very interesting properties. We will just summarize some here
briefly for the $sl(n)$ case. For generalizations to arbitrary
Lie algebras see Bloch, Brockett and Ratiu [1992] and Bloch, Flaschka and
Ratiu [1990].

The most basic case is obtained by setting $N$ to be a diagonal
matrix with integer entries being some permutation of
$(1,2,3,\cdots ,n)$ and by taking  $L$ to be in the space of symmetric
matrices. (The classical Toda flow just corresponds 
to the case where L is taken to be tridiagonal and is obtained
by Flaschka's transformation from a canonical Hamiltonian
system describing particles on a line interacting
under an exponential potential (see Moser [1975]).) 

Then we have (for L symmetric)

1. The flow has $n!$ equilibria corresponding to the
$L$ being diagonal with entries one of the $n$ permutations
of the $n$ eigenvalues of N. The flow has 
a unique stable equilibrium corresponding to
the case where the ordering is compatible with the permutation
 in the matrix $N$.
This can be checked by a linearization about the equilibria.

2. As indicated above, in the case of $L$ tridiagonal and 
$N={\rm diag}(1,2,3,\cdots ,n)$ we obtain the $sl(n)$ Toda 
lattice.

3. The equilibria lie at the vertices of a convex polytopes and the
diagonal entries of $L$ flow in the interior of the polytope. 

The last result follows from the classical Schur-Horn theorem, 
Schur [1923], Horn [1954] which
has a beautiful Lie algebraic generalization due to Kostant [1973]
and which generalizes to the convexity of the momentum map
(see Atiyah [1982] and Guillemin and Sternberg [1982]).

We recall the Schur-Horn theorem:

\begin{thm}
Let $A$ be a Hermitian matrix with diagonal entries 
$\mu=(\mu_1,\cdots , \mu_n)$
where $\mu_1\le \mu_2\cdots\le \mu_n$ and
 eigenvalues $\lambda=(\lambda_1,\cdots ,
\lambda_n)$ similarly ordered. Let the symmetric group act
on $\lambda$ by permuting the eigenvalues. Then $\mu$ lies in the convex
hull of these permutations. Further given any element in this
convex hull there exists a Hermitian matrix with eigenvalues $\lambda$
and the given element as diagonal entries.
\end{thm}

Point 3 above clearly follows from this result, since the
flow is isospectral and its equilibria occur precisely when
$L$ and $N$ commute. It must therefore have $n!$ equilibria
which coincide the vertices of the Schur-Horn polytope. In addition, since
the flow is the gradient flow of ${\rm Tr} LN$, we see that the flow
carries out a linear programming problem: minimizing this function
over the Schur-Horn polytope.

It is also straightforward to analyze the stability of the equilibria
by linearization. Denote the linearization of the flow at the
equilibrium $L_0$
by
\begin{equation}
\dot{\delta L}=[L_0,[\delta L,N]]\,.
\end{equation}

(The other term is zero since $L_0$ is diagonal.)

Denote the diagonal entries of $L_0$ by ${\rm diag}(a_1,\cdot,a_n)$
(a rearrangement of eigenvalues of $L$) and the diagonal entries
of $N$ by ${\rm diag}(n_1,\cdot,n_n)$.
Then the linearized flow is given by
\begin{equation}
\dot{\delta l}_{ij}={\delta l}_{ij}(n_i-n_j)(a_i-a_j)\,.
\end{equation}

Hence the only stable equilibrium occurs when the eigenvalues of 
are similarly ordered -- this is the sorting
of numbers property referred to earlier. This is consistent
with the behavior of
the Toda lattice flow observed by Symes [1981]. In the nontridiagonal
case we have more freedom than Toda -- we may move to an equilibrium
that consists of any ordering of the eigenvalues of $L$. These equilibria
actually lie at the closure of the classical Toda phase space.

The generalization of these ideas to arbitrary
compact Lie algebras can be found in Bloch, Brockett and
Ratiu [1992]. One can also prove a nontrivial convexity result
for tridiagonal matrices which
considerably strengthens the picture above by using the theory
of the momentum map for Kahler manifolds. In Bloch, Flaschka and
Ratiu [1990] we showed in fact that an isospectral set of tridiagonal
symmetric matrices can be mapped in a 1-1 fashion onto a convex
polytope via the momentum map. This extended a topological
observation of van Moerbeke [1976] and Deift, Nanda and Tomei [1983]
to the symplectic context. 


There is actually another rather instructive method of writing the
tridiagonal Toda lattice flow. This makes use of Moser's ([1975])
variables $r_j$. Since the eigenvalues of the matrix $L$ are conserved
in the Toda flow, a natural complementary set of variables is the 
set of top components of the normalized eigenvalues of $L$ -- these
are the $r_j$, which evolve according to a gradient flow.

In Bloch [1990b] and Bloch, Flaschka and Ratiu [1990] we showed how
this flow also may be written in double bracket form: 
Consider projective space $\mathbf{CP}^n$ with elements represented
as rank 1 projection matrices $r\otimes r$ where 
$r=(r_1,\cdots,r_{n+1})$ and $\sum_1^{n+1}|r_j^2|=1$.
Then the flow of $r_j$ for Toda and its commuting higher
flows is given by
\begin{equation}
i\dot{P}=[iP,[iP,c\Lambda^k]]
\end{equation}
where $c$ is $1$ or $i$ chosen to make the gradient
function $-c{\rm Tr}(iP\Lambda^k)$ real and $\Lambda$
is the diagonal matrix of eigenvalues of $L$. This is a gradient flow
on a low rank orbit of $su(n)$ with respect to the normal or K\"ahler
metrics which in this case coincide.  This flow is also
connected to Total Least Squares problems as described in 
Bloch [1990b].

There is also a beautiful generalization of many of the ideas above
to partial differential equations and flows on diffeomorphism groups.
We shall not discuss this here but refer the reader to 
Brockett and Bloch [1990] and Bloch, Brockett and Ratiu [1993], [1996].

We shall discuss the Toda flow again in the context of optimal control.

\section{%4.
Optimal Control Problems and Integrable Systems}\label{S:4}
\subsection{%4.1
Introduction}\label{s:4.1}

In this section we discuss Hamiltonian flows and related geometry
arising in three classes of optimal control problems, defined on
adjoint orbits, coadjoint orbits, and certain symmetric spaces. 
These are all examples of (2.47 i) in the connection between
mechanics and optimal control theory.
The first problem we consider was originally considered by Brockett 
[1994] %\cite{}
from the Lagrangian point of view, and concerns flows on adjoint orbits
of semisimple Lie algebras, of the form $\dot x=[x,u]$. We consider
Brockett's problem via the corresponding Hamiltonian equations and
the maximum principle. In the case where the flow corresponds to a
bi-invariant geodesic flow, we obtain a pair of coupled double
bracket equations which are explicitly soluble. These equations are
quite different from the double bracket equations discussed in
Brockett [1994], %\cite{}
Bloch, Brockett and Ratiu [1992] %\cite{}
and Bloch, Flaschka and Ratiu [1990]%\cite{}
. While the double bracket flow in these papers was a gradient flow
in general and a Hamiltonian flow for the Toda lattice on an
isospectral set (see also Bloch [1990]%\cite{}
), the flow we discuss here is always Hamiltonian. In fact, in many
cases it turns out that the flow is completely integrable and is
related to the work by Thimm [1981] %\cite{} 
on the integrability of flows on symmetric spaces. In a related
paper, Bloch, Brockett and Crouch [1997] %\cite{}
discuss the use of the general framework presented here to prove
integrability via the method of Thimm and to provide an explicit
realization of Thimm's commuting flows.

The second problem we consider was discussed in Bloch and Crouch 
[1995]%\cite{}
, [1996]%\cite{}
, and generalizes the work above to certain symmetric spaces, such as
real and complex Grassmannians, spheres and indeed Lie groups. The 
paper by Bloch and Crouch [1996] %\cite{ }
treats a general optimal control problem, whose extremal flow
specializes to each of these cases yielding in particular Moser's 
[1974] %\cite{}
flow of geodesic  motion on $S^n$, and a particularly interesting new
form for (generalized) rigid body dynamics.

The third problem we consider is posed by an optimal control problem 
whose extremals yield the full Toda lattice equations. 
The full Toda lattice equations, a dynamical system on the space of
symmetric matrices, was shown to be completely integrable (in the
case of generic initial data) in the work of Deift, Li, Nanda and
Tomei [1983]%\cite{}
. (For more work in this regard see Ercolani, Flaschka and Singer 
[1996] %\cite{}
and Gekhtman and Shapiro [1997]%\cite{}
Bloch and Gekthman [1997].) We show
 here that the optimal control approach leads to a
formulation of the full Toda flow due to Symes [1980]%\cite{}
, and is described in Bloch and Crouch [~1997~]%\cite{}
. The full Toda flow is a generalization of standard tridiagonal Toda
lattice equations as analyzed in the work of Flaschka [1976]%\cite{}
, Moser [1974] %\cite{}
and Kostant [1979] %\cite{}
and Toda [1975]%\cite{}
, among others, and which has spawned a great deal of
interesting theory and related work.

The original tridiagonal Toda lattice flow models the interaction of
a one-dimensional array of particles under a nearest neighbor
exponential potential. One can impose various boundary
conditions---viewing the system on the line, or on the circle
(periodic boundary conditions) or on a fixed interval (the so-called
finite Toda lattice as introduced in the work of Moser [1974]%\cite{}
). Flaschka showed how a transformation of variables enables the
nonperiodic lattice to be written in Lax pair form, i.e., in the form
$\dot L=[B,L]$, with the Lax pair $L,B$ being tridiagonal symmetric
matrices (reflecting the nearest neighbor interaction) as in the
previous section. A natural
generalization of this is to full symmetric matrices, reflecting a
kind of long range interaction (although this system does not
correspond, as far as we know, to an actual physical lattice).

We remark, however, that the full (and tridiagonal) finite Toda flows
have been of great interest for numerical reasons due to an
observation of Symes [1982], %\cite{}
that the time-1 map of the Toda flow gives the QR algorithm for
diagonalizing matrices. (See also Deift, Nanda and Tomei [1983]%\cite{}
and also earlier observations by Kostant [1979]%\cite{}
.) The Toda flows are also related to the double bracket flows, which
also carry out diagonalization procedures, as well as other
algorithms such as sorting---see Brockett [1988]%\cite{}
, Bloch [1990]%\cite{}
, Bloch, Brockett and Ratiu [1992]%\cite{}
.

The full Toda flow is straightforward to write down. It is given in
the $gl(n)$ case by the equations:
\[
\dot A=[A,\pi_kA]
\]
where $A$ is a symmetric matrix and $\pi_k$ is the projection of $A$
onto its skew symmetric part in the lower triangular-skew symmetric
decomposition of $A$. (It generalizes very naturally to other Lie
algebras, but we shall not consider these cases here.) There is also,
however, an accompanying ``kinematic'' equation which arises very
naturally in our formulation and which we shall describe below.

The extremal flows resulting from our optimal control problem have a
lot of structural similarity with the extremal flows which describe
the generalized rigid body dynamics, and naturally evolve as a flow on $T^*L$, 
the form given in
Symes [1980]%\cite{}
, where $L$ is the space of lower triangular matrices. This flow
naturally reduces to a flow on a coadjoint orbit of $L$. We thus
obtain a highly nontrivial optimal control problem of geometric
interest which is completely solvable via the accompanying
integrability theory.

\subsection{%4.2
Optimal Control on Adjoint Orbits}\label{s:4.2}

We consider here the situation in section~\ref{s:2.6} where $
\mathcal{G}$ is a compact, semi-simple Lie algebra, with Lie group
$G$. A natural drift free control system on $\mathcal{G}$ is given by
\begin{equation}\label{e:4.1}
\dot x=[x,u]
\end{equation}
where $u$ is a Lie algebra valued control function. It is clear that
this system evolves naturally on adjoint orbits of $G$. We may
consider the following problem associated with the dynamics 
(\ref{e:4.1}):
\begin{equation}\label{e:4.2}
\begin{array}{l}
\displaystyle\min_{u(\cdot)}\int^T_0[\textstyle\frac12(u,u)-V(x)]dt\\
\mbox{subject to: the dynamics (\ref{e:4.1}) and $x(0)=x_0$,
$x(T)=x_T$.}
\end{array}
\end{equation}
Here $(\cdot,\cdot)$ is the metric on $\mathcal{G}$ defined by the
negative of the Killing form, and $V$ is a smooth function on $
\mathcal{G}$. We may use the metric to identify $\mathcal{G}^*$ with
$\mathcal{G}$, so that a covector $p$ on $\mathcal{G}$, an element
of $\mathcal{G}^*$, may also be denoted by an element of $
\mathcal{G}$ via
\[
p(x)=(p,x),\quad x\in\mathcal{G}.
\]
If $H$ is a function on $\mathcal{G}$, then $dH(x)/dx$ is an element
of $\mathcal{G}^*$. We may identify this as an element of $
\mathcal{G}$ and denote the resulting vector by $\nabla H(x)$
\[
\frac{\pa H(x)}{\pa x}(z)=(\nabla H(x),z),\quad z\in\mathcal{G}.
\]
Since $\mathcal{G}$ is a vector space, $T^*\mathcal{G}$ has a natural
symplectic structure. Denote by $(\rho^k_x,\rho^k_p)$
two vector fields on $T^*\mathcal{G}$, $k=1,2$, represented by
vectors in $\mathcal{G}\times\mathcal{G}^*$, and hence as elements of
$\mathcal{G}\times\mathcal{G}$ through the identification of $
\mathcal{G}$ with $\mathcal{G}^*$, where $(x,p)\in\mathcal{G}\times
\mathcal{G}$ represents an arbitrary element of $T^*\mathcal{G}$. The
natural symplectic form on $T^*\mathcal{G}$ takes the form
\[
\omega((\rho^1_x,\rho^1_p),(\rho^2_x,
\rho^2_p))=(\rho^1_x,\rho^2_p)-(
\rho^1_p,\rho^2_x).
\]
It follows that if $H$ is a function on $T^*\mathcal{G}\simeq\mathcal{G}
\times\mathcal{G}$ Hamilton's equation of motion corresponding to $H$
may be written as 
\begin{eqnarray}\label{e:4.3}
\dot x&=&(\nabla_pH)(x,p)\\
\dot p&=&-(\nabla_xH)(x,p)\nonumber
\end{eqnarray}
where $\nabla_xH$ and $\nabla_pH$ are the elements of $\mathcal{G}$
corresponding to the partial derivatives of $H$, with respect to $x$
and $p$ respectively.

\begin{thm}\label{t:4.1}
The extremal flow of the problem (\ref{e:4.2}) as given by the
maximum principle is
\begin{eqnarray}\label{e:4.4}
\dot x&=&[x,[p,x]]\\
\dot p&=&[p,[p,x]]-\nabla V(x).\nonumber
\end{eqnarray}
The optimal control is given by
\begin{equation}\label{e:4.5}
u^*=[p,x]
\end{equation}
and the extremal Hamiltonian function
\begin{equation}\label{e:4.6}
H^*(x,p)=\textstyle\frac12([p,x],[p,x])+V(x).
\end{equation}
\qed
\end{thm}

\noindent
{\bf Proof.  }Following the prescription given in Section~\ref{S:2},
we have the Hamiltonian:
\[
H(x,p,u)=(p,[x,u])-\textstyle\frac12(u,u)+V(x).
\]
Using the following identity for the compact and semi-simple Lie
algebra $\mathcal{G}$
\begin{equation}\label{e:4.7}
(adX(Y),Z)+(Y,adX(Z))=0,\quad X,Y,Z\in\mathcal{G},
\end{equation}
\begin{equation}\label{e:4.8}
H(x,p,u)=([p,x],u)-\textstyle\frac12(u,u)+V(x).
\end{equation}
Statements (\ref{e:4.5}) and (\ref{e:4.6}) are now clear. Computing
the Hamiltonian flow (\ref{e:4.3}), along either (\ref{e:4.6}) or 
(\ref{e:4.8}), we obtain (\ref{e:4.4}).\qed

\vspace{2ex}
We now consider the interesting special case (see Brockett [1994]%\cite{}
), where for a fixed element $n\in\mathcal{G}$,
\begin{equation}\label{e:4.9}
V(x)=-\textstyle\frac12([x,n],[x,n]).
\end{equation}

In this case the extremal flow (\ref{e:4.4}) becomes
\begin{eqnarray}\label{e:4.10}
\dot x&=&[x,[p,x]]\\
\dot p&=&[p,[p,x]]-[n,[n,x]].\nonumber
\end{eqnarray}

\begin{cor}\label{c:4.2}
In the case of the potential function (\ref{e:4.9}), the extremal
flow (\ref{e:4.4}) admits the special form
\begin{equation}\label{e:4.11}
\dot x=-[x,[x,n]].
\end{equation}
\end{cor}

\noindent
{\bf Proof.  }Computing the derivative of $[p,x]$ along the flow 
(\ref{e:4.10}) we find
\begin{eqnarray*}
\frac d{dt}[p,x]&=&+[x,[n,[n,x]],\\
&=&[n,[x,[n,x]].
\end{eqnarray*}
But we also have
\[
\frac d{dt}[n,x]=[n,[x,[p,x]].
\]
Hence, $[p,x]=[n,x]+c$, where $c\in\ker adx$ yields a particular
solution of the equations (\ref{e:4.10}), in the form of equations 
(\ref{e:4.11}).\qed

Thus the solution to the optimal control problem (\ref{e:4.2}), with
potential function (\ref{e:4.9}), has a very nice connection with the
double bracket equation as Brockett [1994] %\cite{}
noticed. In particular the classical Toda lattice flow is a special
case of the equations (\ref{e:4.11}).

The analysis above was obtained by Brockett using a Lagrangian
approach, which we briefly describe. Recalling the decomposition from
Section~\ref{s:2.6} of ${\mathcal G}={\mathcal G}^A\oplus
{\mathcal G}_A$, ${\mathcal G}_A=\ker adA$, ${\mathcal G}^A=\mbox{Im
}adA$ it follows that $ad_A^{-1}$ is well defined on $\mathcal{G}^A$,
for each $A\in{\mathcal G}$. First note that the extremal flow 
(\ref{e:4.10}) or (\ref{e:4.4}) is defined by a vector field on
adjoint orbits, $\dot x=[x,u]$, where $u\in{\mathcal G}^x$, and so $
ad^{-1}_x\dot x=u=u^x$. Thus, for these flows,
\[
(ad^{-1}_x\dot x,ad^{-1}_x\dot x)=(u^x,u^x)=\langle[x,u],[x,u]
\rangle_N
\]
where $\langle\cdot,\cdot\rangle_N$ is the normal metric. It follows
that Lagrangian function
\begin{equation}\label{e:4.12}
L(x,\dot x)=\textstyle\frac12(ad^{-1}_x\dot x,ad^{-1}_x\dot x)-V(x)
\end{equation}
is well defined on extremals of the problem (\ref{e:4.2}), and
motivates the variational problem:
\begin{equation}\label{e:4.13}
\begin{array}{l}
\displaystyle\min_{x(\cdot)}\int^T_0L(x,\dot x)dt;\\
\mbox{subject to $\dot x$, $ad^{-1}_x\dot x\in{\mathcal G}^x$,
$x(0)=x_0$, $x(T)=x_T$}.
\end{array}
\end{equation}

It is clear that the Legendre transform corresponding to the
Lagrangian $L$ is simply given by
\begin{equation}\label{e:4.14}
p=\nabla_{\dot x}L(x,\dot x)=-ad^{-2}_x\dot x
\end{equation}
which clearly exists, under the conditions of problem (\ref{e:4.13}).
It is fairly routine to check that under this mapping the extremal
equations (\ref{e:4.4}) become
\begin{equation}\label{e:4.15}
\ddot x=[\dot x,ad^{-1}_x(\dot x)]+\nabla V(x).
\end{equation}
It is much less routine to check the extremal equations 
(\ref{e:4.15}), directly from the variational problem (\ref{e:4.13}),
Brockett [1994]%\cite{}
.

If we now consider the equations (\ref{e:4.4}) or (\ref{e:4.15}) in
the case $V\equiv0$, it is clear we have recovered the geodesic
equations on an adjoint orbit of $\mathcal{G}$ with the metric given
by the normal metric. Now if we denote the adjoint action of $G$ on 
$\mathcal{G}$ by $(g,x)\mapsto g\cdot x$, and by $t\mapsto h(t)$ the
one parameter group with generator $a\in\mathcal{G}$, then
\[
\frac d{dt}h(t).x\bigg|_{t=0}=-[x,a].
\]
Thus
\[
\frac d{dt}(R_gh(t)).x\bigg|_{t=0}=-Adg([x,a]),\quad\frac d{dt}(L_gh(t)).x
\bigg|_{t=0}=-[Adg(x),a].
\]
Since $Adg(a^x)=(Adga)^{Adgx}$, and hence
$Adg((Ad^{-1}ga)^x)=a^{Adgx}$ it is clear that for the normal metric
$\langle Adg[x,u],Adg[x,u]\rangle_{g.x}=\langle[x,u],[x,u]\rangle_x$
and $\langle[Adgx,u],\allowbreak[Adgx,u]\rangle_{g.x}=\langle[x,u],[x,u]
\rangle_x$ as long as $u\in{\mathcal G}^x$, so the normal metric is
both left- and right-invariant on adjoint orbits.

We may define a right invariant metric by introducing a positive
definite linear mapping $J:\mathcal{G}\to\mathcal{G}$ by setting
\begin{equation}\label{e:4.16}
\langle Adg[x,u],Adg[x,u]\rangle_{g.x}=(u^x,Ju^x).
\end{equation}

By modifying the variational problem (\ref{e:4.2}) to include 
the mapping $J$, we obtain
\begin{equation}\label{e:4.17}
\begin{array}{l}
\displaystyle\min_{u(\cdot)}\int^T_0[\textstyle\frac12(u,Ju)-V(x)]dt\\
\mbox{subject to: the dynamics (\ref{e:4.1}) and $x(0)=x_0$,
$x(T)=x_T$},
\end{array}
\end{equation}
The extremal flow (\ref{e:4.4}) becomes
\begin{eqnarray}\label{e:4.18}
\dot x&=&[x,J^{-1}[p,x]]\\
\dot p&=&[p,J^{-1}[p,x]]-\nabla V(x)\nonumber
\end{eqnarray}
with extremal Hamiltonian
\[
H(x,p)=\textstyle\frac12([p,x],J^{-1}[p,x])+V(x).
\]
Thus the case $V\equiv0$ corresponds to the geodesic equations on an 
adjoint orbit, subject to the right invariant metric (\ref{e:4.16}).

The  bi-invariant geodesic flow on adjoint orbits
\begin{eqnarray}\label{e:4.19}
\dot x&=&[x,[p,x]]\\
\dot p&=&[p,[p,x]]\nonumber
\end{eqnarray}
are remarkable equations. As we have seen they are Hamiltonian with
respect to the natural symplectic structure on $T^*\mathcal{G}$.
However they are also Hamiltonian with respect to the natural
symplectic structure on the cotangent bundle of the adjoint orbit,
which may be identified with the tangent bundle. A tangent vector to
the tangent bundle to the orbit at the point $(x,[x,\xi])$ is of the
form
\[
(x,[x,\xi],[x,\eta],[[x,\eta],\xi]+[x,\zeta])
\]
for $\xi,\eta,\zeta\in\mathcal{G}$. Then, using the natural symplectic
structure as in Thimm [1981] %\cite{}
gives the result. The details of this standard but lengthy
computation are given in Bloch, Brockett and Crouch [1997]%\cite{}
.\qed

\subsection{%4.3
Optimal Control on Symmetric Spaces}\label{s:4.3}

The equations discussed above are not only well defined on adjoint
orbits but also on general symmetric spaces where the tangent vectors
to the space are given in the form of a suitable bracket. This includes
the complex and real Grassmannians of $q$-planes in $n+1$-space
$G_{q,n+1}(\C)$ or $G_{q,n+1}(\R)$ and in particular
the spheres. This may be seen as follows:

The complex Grassmannian is given by
\[
U(n+1)/U(q)\times U(p),\quad q+p=n+1,q\le p
\]
and the real Grassmannian by
\[
SO(n+1)/SO(q)\times SO(p),\quad q+p=n+1,q\le p
\]
where $U(n)$ is the unitary group and $SO(n)$ the special orthogonal
group. In either case let $\mathcal{G}=\mathcal{P}\oplus\mathcal{K}$
be the Lie algebra decomposition corresponding to $G/K$. We may thus
represent a point in the complex (real) Grassmannian by a matrix
\[
\hat Q=\left[\begin{array}{cc}
0&Q\\
-Q^*&0\end{array}\right]
\]
in $\mathcal{P}$ where $Q$ is a complex (real) $p\times q$ matrix of
full rank and $Q^*$ is its Hermitian conjugate (transpose). A point
in $\mathcal{K}$ may be represented by the matrix
\[
\hat K=\left[\begin{array}{cc}
K_1&O\\
O&K_2\end{array}\right]
\]
where $K_1\in u(p)(so(p))$ and $K_2\in u(q)(so(q))$. Define $\hat P$
to be a similarly partitioned matrix. It turns out that in this
representation of the Grassmannian, tangent vectors to the
Grassmannian at the point $\hat Q$ have the form:
\[
[\hat Q,\hat K].
\]

Since tangent vectors are given by brackets, just as in the case of
orbits, a normal metric may be defined and we obtain the following
result.

\begin{thm}\label{t:4.3}
Relative to a natural bi-invariant normal metric, the geodesic
equations on the real or complex Grassmannian may be represented by
the equations
\begin{eqnarray}\label{e:4.20}
\dot{\hat Q}&=&[\hat Q,[\hat P,\hat Q]]\\
\dot{\hat P}&=&[\hat P,[\hat P,\hat Q]].\nonumber
\end{eqnarray}
\end{thm}
\qed

Note that the relations for symmetric spaces 
\[
[\mathcal{P}, \mathcal{P}]\subset\mathcal{K},\quad [\mathcal{P},
\mathcal{K}]\subset\mathcal{P},\quad[\mathcal{K}, \mathcal{K}]\subset
\mathcal{K},
\]
ensure that these equations are well defined since $\hat Q,\hat P\in
\mathcal{P}$.

In fact the formalism developed here can be combined with the work of
Thimm [1981] %\cite{}
to give a very explicit proof of complete integrability of the
geodesic flow on symmetric spaces such as the real and complex
Grassmannians. In particular it is possible to derive explicitly a
complete set of commuting flows and to prove their involutivity. This
is the subject of the paper by Bloch, Brockett and Crouch [1997]%\cite{}
.

This should be contrasted with the observation of Brockett regarding
the optimal control problem (\ref{e:4.2}) and (\ref{e:4.9}). As we
have observed, the optimal flow under suitable conditions is given by
(\ref{e:4.11}) and in the particular case of $x$ being tridiagonal
one obtains the integrable Toda lattice equations in Flaschka's form
(see Bloch [1990]%\cite{}
). Integrability of optimal control problems is a rich subject (see,
for example, Faybusovich [1988]%\cite{}
).

As in the variational problem on adjoint orbits discussed in 
section~\ref{s:4.2}, the double, double bracket equations on Grassmannians
are solutions to a natural optimal problem. We consider a very general optimal
control problem on symmetric spaces which we will then reduce to
various special cases.

Let $Q$ be a $p\times q$ complex matrix and let $U\in u(p)$ and $V\in
u(q)$. Let $J_U$ and $J_V$ be constant symmetric positive definite
operators on the space of complex $p\times p$ and $q\times q$
matrices, respectively, and let $\langle\cdot,\cdot\rangle_T$ denote the trace
inner product $\langle A,B\rangle_T=\mbox{Tr}A^*B$, where $A^*$ is
the Hermitian conjugate.

Consider the optimal control problem
\begin{equation}\label{e:4.21}
\begin{array}{l}
\displaystyle\min_{U,V}\int^T_0\textstyle\frac14\{\langle U,J_UU
\rangle_T+\langle V,J_VV\rangle_T\}dt\\
\mbox{subject to $\dot Q=UQ-QV$, $Q(0)=Q_0$, $Q(T)=Q_T$}.
\end{array}
\end{equation}

As in Bloch and Crouch [1996] %\cite{ }
we have the following result.
\begin{thm}\label{t:4.4}
The optimal controls for the problem (\ref{e:4.21}) are given by
\begin{eqnarray}\label{e:4.22}
U&=&J^{-1}_U(PQ^*-QP^*)\\
V&=&J^{-1}_V(P^*Q-Q^*P)\nonumber
\end{eqnarray}
and the extremal flow of the states $Q$ and costates $P$ is given by
\begin{eqnarray}\label{e:4.23}
\dot Q&=&J^{-1}_U(PQ^*-QP^*)Q-QJ^{-1}_V(P^*Q-Q^*P)\\
\dot P&=&J^{-1}_U(PQ^*-QP^*)P-PJ^{-1}_V(P^*Q-Q^*P).\nonumber
\end{eqnarray}
\qed
\end{thm}

\begin{cor}\label{c:4.5}
Equations (\ref{e:4.23}) are given by the double, double bracket
equations
\begin{eqnarray}\label{e:4.24}
\dot{\hat Q}&=&[\hat Q,J^{-1}[\hat P,\hat Q]]\\
\dot{\hat P}&=&[\hat P,J^{-1}[\hat P,\hat Q]],\nonumber
\end{eqnarray}
where $J$ is the operator $\mbox{diag}(J_U,J_V)$.
\qed
\end{cor}

It is clear that we recover the result of Theorem~\ref{t:4.3}, the
bi-invariant geodesic flow on Grassmannians by simply setting $J=
\mbox{Id}$, in the equations (\ref{e:4.24}). We now consider some
other special cases of Theorem~\ref{t:4.3}. Note that the Hamiltonian
equations (\ref{e:4.23}) correspond to the natural symplectic
structure on $T^*gl(n)$, with the Hamiltonian function
\begin{eqnarray}\label{e:4.25}
H(Q,P)&=&\textstyle\frac14\langle(PQ^*-QP^*),J^{-1}_U(PQ^*-QP^*)
\rangle_T\\
&&+\textstyle\frac14\langle(P^*Q-Q^*P),J^{-1}_V(P^*Q-Q^*P)\rangle_T.
\end{eqnarray}
We restrict to the real case for the situation in which $p=n$, $q=1$,
$V=0$, $J_V=0$, $J_U=Id$, and represent $P$ and $Q$ by $n$ vectors
$p$ and $q$, respectively, assuming no confusion by the context. The
Hamiltonian function (\ref{e:4.25}) reduces to the form;
\begin{equation}\label{e:4.26}
H(q,p)=\textstyle\frac12(\|q\|^2\|p\|^2-(q^Tp)^2),
\end{equation}
and the equations (\ref{e:4.23}) reduce to the form:
\begin{eqnarray}\label{e:4.27}
\dot q&=&pq^Tq-qp^Tq\\
\dot p&=&pq^Tp-qp^Tp.\nonumber
\end{eqnarray}
It is easily verified that equations (\ref{e:4.27}) are indeed
Hamiltonian with respect to the natural symplectic form on $\R^{2n}$,
and the Hamiltonian (\ref{e:4.26}). Clearly these equations represent
the geodesic flow on projective $n$ space $\mathbf{P}^n$. Since $\mathbf{P}^n$ 
has a
double cover given by $S^n$ it is remarkable that we may also obtain
a global representation of the geodesic flow on $S^n$ by looking at
the same situation, by increasing the dimension of $p$ by 1. We then
note that the flow (\ref{e:4.27}) which admits $q^Tp$, $\|p\|^2$, $
\|q\|^2$ as invariants, restricts naturally to the case $q^Tp
\equiv0$, $\|q\|^2=1$, in which case we obtain the flow:
\begin{equation}\label{e:4.28}
\ddot q=-\|\dot q\|^2q,\quad q\in\R^{n+1},\quad \|q\|^2=1.
\end{equation}
This is precisely the geodesic flow on $S^n$, as in Moser [1974]%\cite{}
, with Hamiltonian obtained from (\ref{e:4.26}) $H(q,p)=\frac12\|p
\|^2\|q\|^2$.

Finally we consider another special case, the generalized rigid body. 
We recall the rigid body equations on $SO(3)$ (or generally on
$SO(n)$ or any compact Lie group---see, e.g., Ratiu [1980]%\cite{}
) may be written (in right invariant form) as
\begin{eqnarray}\label{e:4.29}
\dot Q&=&\Omega Q\\
\dot M&=&[\Omega,M]\nonumber
\end{eqnarray}
where $Q\in SO(3)$ denotes the configuration space variables, $\Omega
\in so(3)$ is the angular velocity, and $M=J\Omega=\La\Omega+\Omega
\La$ is the angular momentum. Here $J$ is a symmetric positive
definite operator defined by the diagonal positive definite matrix $
\La$.

We now consider the following equations:
\begin{eqnarray}\label{e:4.30}
\dot Q&=&\Omega Q\\
\dot P&=&\Omega P\nonumber
\end{eqnarray}
where $\Omega=J^{-1}M$ and $M=PQ^T-QP^T$.

It is easy to check that equation (\ref{e:4.30}) reduce to the
classical equations (\ref{e:4.29}) simply by differentiating
$M=PQ^T-QP^T$.

Conversely, given the rigid body equations (\ref{e:4.29}) we may
solve for the variable $P$ in the expression
\[
M=PQ^T-QP^T
\]
in a neighborhood of $M=0$. Locally
\[
P=(e^{\sinh^{-1}M/2})Q.
\]
This follows from the observation that
\[
M=e^{\sinh^{-1}M/2}-e^{-\sinh^{-1}M/2}.
\]

For $so(n)$, however, $\sinh$ is many to one, so the two
representations (\ref{e:4.29}) and (\ref{e:4.30}) are not entirely
equivalent. However, the form (\ref{e:4.30}) of the rigid body
equations does arise from the optimal control problem as follows.

Consider the optimal control problem, which is a special case of the
problem (\ref{e:4.21})
\begin{equation}\label{e:4.31}
\begin{array}{l}
\displaystyle\min_U\int^T_0\textstyle\frac14\langle U,JU\rangle_Tdt\\
\mbox{subject to: $\dot Q=UQ$, $Q(0)=Q_0$, $Q(T)=Q_T$}\\
\mbox{and $U\in so(n)$, $J\Omega=\La\Omega+\Omega\La$}.
\end{array}
\end{equation}

\begin{thm}\label{t:4.6}
The optimal control problem (\ref{e:4.30}) yields the rigid body
equations (\ref{e:4.30}) as an extremal flow. These equations are
Hamiltonian with respect to the natural symplectic structure on
$T^*gl(n)$, and the Hamiltonian function
\[
H(P,Q)=\textstyle\frac14\langle (PQ^T-QP^T),J(PQ^T-QP^T)\rangle_T.
\]
\end{thm}
\qed

It is interesting to note that the equations (\ref{e:4.30}) may also
be viewed as a singular limit of the double-double bracket flow in 
(\ref{e:4.24}). Here we restrict to the real case and set $J_U=J$ in
(\ref{e:4.31}) and $J^{-1}_V=0$ $(J_V\to\infty$). We know that
the reduced rigid body dynamics in (\ref{e:4.29}), the real dynamics
given by $\dot M=[\Omega,M]$, are completely integrable (see, e.g.,
Ratiu [1980]%\cite{}
). We also know that the double-double bracket equations
corresponding to a bi-invariant normal metric, equations 
(\ref{e:4.20}), are also completely integrable as pointed out
earlier. It is interesting to note that explicit corresponding proofs
of integrability of the right invariant double-double bracket
equations, and the corresponding full rigid body equations 
(\ref{e:4.29}) or (\ref{e:4.30}) do not seem to exist in the
literature.

\subsection{%4.4
Optimal Control and the Toda Flow}\label{s:4.4}

In this section we introduce an optimal control problem which yields
the full Toda flow as discussed in Bloch and Crouch [1997]%\cite{}
. We begin by introducing some specialized notation. Let $
{\mathcal G}=gl(n)$ denote the Lie algebra of $n\times n$ matrices
(with corresponding Lie group $GL(n)$ of invertible $n\times n$
matrices).

\vspace{2ex}
\noindent
{\bf Definition  }For $A\in gl(n)$, let $A=A_++A_0+A_-$ where
$A_+$ is the strictly upper part of $A$, $A_0$ the diagonal part, and
$A_-$ is the strictly lower part. Let
\begin{eqnarray*}
\pi_l(A)&=&A_-+A^T_++A_0\mbox{ (lower)}\\
\pi_k(A)&=&A_+-A^T_+\mbox{ (skew)}\\
\pi_{k^\perp}(A)&=&A_-+A_0+A^T_-\mbox{ (symmetric)}\\
\pi_{l^\perp}(A)&=&A_+-A^T_-\mbox{ (strictly upper)}.
\end{eqnarray*}
The reason for the above notation is as follows (see Symes [1980]
%\cite{}
):

Endow ${\mathcal G}$ with the inner product $\langle A,B\rangle_T=
\mbox{Tr}A^TB$. We observe
\[
{\mathcal G}={\mathcal L}\oplus{\mathcal K}
\]
where ${\mathcal L}$ is the subalgebra of lower triangular matrices
and ${\mathcal K}$ the subalgebra of skew-symmetric matrices. $
{\mathcal L}$ is the Lie algebra of $L$, the lower triangular group
and ${\mathcal K}$ is the Lie algebra of $K$, the group of orthogonal
matrices.

Now denote by $\perp$ the perpendicular subspace under the scalar
product. Then
\[
{\mathcal L}^\perp=\{x:\mbox{Tr}(x^Ty)=0,\forall y\in{\mathcal L}\}
\]
is the algebra of strictly upper triangular matrices and
\[
{\mathcal K}^\perp=\{x:\mbox{Tr}(x^Tz)=0,\forall z\in{\mathcal K}\}
\]
is the space of all symmetric matrices.

Hence we can write
\[
{\mathcal G}\simeq{\mathcal G}^*={\mathcal L}^\perp\oplus{\mathcal K}^
\perp,
\]
and we can make the identification
\begin{eqnarray*}
{\mathcal L}^*&\simeq&{\mathcal K}^\perp\\
{\mathcal K}^*&\simeq&{\mathcal L}^\perp.
\end{eqnarray*}

We now give some preliminary results that will be useful in the main
theorem. 

\begin{lem}\label{l:4.7}
For any matrix $S\in\mathcal{G}$ and $L\in{\mathcal L}$
\begin{eqnarray*}
\pi_{k^\perp}(L^TS)&=&\pi_{k^\perp}(L^T\pi_{k^\perp}S)\\
\pi_{k^\perp}(SL^T)&=&\pi_{k^\perp}(\pi_{k^\perp}(S)L^T).
\end{eqnarray*}
\qed
\end{lem}

Let 
$F:{\mathcal G}\to{\mathcal G}$ be analytic and consider
\[
f(PQ^T)=\mbox{Tr}F(PQ^T).
\]
Let
\[
{\mathcal F}_Af(A)(R)=\lim_{h\to0}\frac{f(A+hR)-f(A)}h=\langle R,
\nabla f(A))
\]
for $A,R\in gl(n)$. Hence
\begin{eqnarray*}
{\mathcal F}_Qf(PQ^T)(R)&=&\langle PR^T,\nabla f(PQ^T)\rangle\\
&=&\langle RP^T,\nabla f(PQ^T)^T\rangle\\
&=&\langle R,\nabla f(PQ^T)^TP\rangle\\
{\mathcal F}_Pf(PQ^T)(R)&=&\langle RQ^T,\nabla f(PQ^T)\rangle\\
&=&\langle R,\nabla f(PQ^T)Q\rangle.
\end{eqnarray*}
Hence, we have

\begin{lem}\label{l:4.8}
Let $f(PQ^T)=\mbox{Tr}F(PQ^T)$. Then the Hamiltonian flows with
respect to the canonical structure on $T^*{\mathcal G}$ are given by
\begin{eqnarray*}
\dot Q&=&\nabla f(PQ^T)Q\\
\dot P&=&-\nabla f(PQ^T)^TP,
\end{eqnarray*}
respectively.
\end{lem}
\qed

The main result in Bloch and Crouch [1997] %\cite{}
can now be stated in the following way. Consider the optimal control
problem
\begin{equation}\label{e:4.32}
\begin{array}{l}
\displaystyle\min_U\int^T_0\textstyle\frac12\langle U,U\rangle_Tdt\\
\mbox{subject to: $\dot X=\pi_l(U)X$, $X(0)=X_0$, $X(T)=X_T$},\\
\mbox{where $X\in L$, $U\in\mathcal{G}$}.
\end{array}
\end{equation}

\begin{thm}\label{t:4.9}
For the optimal control problem (\ref{e:4.32}), the optimal controls
are given by
\[
U=\pi_{k^\perp}(RX^T)
\]
where $R$ is the costate vector and the corresponding extremal flow
is given by
\begin{eqnarray}\label{e:4.33}
\dot X&=&\pi_l(\pi_{k^\perp}(RX^T))X\\
\dot R&=&-\pi_{k^\perp}(\pi^T_l(\pi_{k^\perp}(RX^T))R).\nonumber
\end{eqnarray}
Equations (\ref{e:4.33}) are Hamiltonian with respect to the
canonical structure on $T^*\mathcal{G}$, with corresponding
Hamiltonian function
\begin{equation}\label{e:4.34}
H(R,X)=\textstyle\frac12\langle\pi_{k^\perp}(RX^T),\pi_{k^
\perp}(RX^T)\rangle_T.
\end{equation}
\end{thm}
\qed

>From this result, and Lemma~\ref{l:4.7}, we observe that if $S=
\pi_{k^\perp}(R)$ we may rewrite $H$ in (\ref{e:4.34}) in the form:
\begin{equation}\label{e:4.35}
H=\textstyle\frac12\langle\pi_{k^\perp}(SX^T),\pi_{k^\perp}(SX^T)
\rangle_T.
\end{equation}
Using Lemma~\ref{l:4.7} once more, the Hamiltonian equations may be
rewritten in the form
\begin{eqnarray}\label{e:4.36}
\dot X&=&\pi_l(\pi_{k^\perp}(SX^T))X\\
\dot S&=&-\pi_{k^\perp}(\pi^T_l(\pi_{k^\perp}(SX^T))S).
\nonumber
\end{eqnarray}
Now set
\begin{equation}\label{e:4.37}
A=\pi_{k^\perp}(SX^T).
\end{equation}
Then the equations (\ref{e:4.36}) are of the form
\begin{eqnarray*}
\dot X&=&\pi_l(\nabla f(A))X\\
\dot S&=&-\pi_{k^\perp}(\pi^T_l(\nabla f(A))S)
\end{eqnarray*}
where $f(A)=\frac12\mbox{Tr}(A^2)$.

Now we compute
\begin{eqnarray*}%\label{e:4.38}
\dot A&=&\pi_{k^\perp}(\dot SX^T+S\dot X^T)\nonumber\\
&=&\pi_{k^\perp}(-\pi_{k^\perp}(\pi^T_l(\nabla f)S)X^T)\nonumber\\
&&+\pi_{k^\perp}(SX^T\pi^T_l(\nabla f))\nonumber\\
&=&-\pi_{k^\perp}(\pi^T_l(\nabla f)SX^T)\\
&&+\pi_{k^\perp}(SX^T\pi^T_l(\nabla f))\nonumber\\
&&\mbox{(by Lemma (\ref{l:4.7})}\nonumber\\
&=&\pi_{k^\perp}([A,\pi^T_l(\nabla f)]).\nonumber
\end{eqnarray*}
Hence
\begin{equation}\label{e:4.38}
\dot A=\pi_{k^\perp}([A,\pi^T_l(\nabla f)]).
\end{equation}
But since
\[
\nabla f(A)=\pi_l(\nabla f(A))+\pi_k(\nabla f(A))
\]
and $\nabla f(A)\in {\mathcal K}^\perp$ (for any invariant polynomial in $A$),
we have
\[
\nabla f(A)=\pi^T_l(\nabla f(A))-\pi_k(\nabla f(A)).
\]
Observing also that $[A,\nabla f(A)]=0$, we obtain
\begin{eqnarray*}
\dot A&=&\pi_{k^\perp}[A,\pi_k(\nabla f(A))]\\
&=&[A,\pi_k(\nabla f(A))]\\
&&\mbox{(since $[{\mathcal K}^\perp,{\mathcal K}]\subset{\mathcal K}^\perp$)}\\
&=&[A,\nabla f(A)-\pi_l\nabla f(A)]\\
&=&[\pi_l(\nabla f(A)),A].
\end{eqnarray*}
Thus for $f(A)=\frac12\mbox{Tr}(A^2)$, we obtain from (\ref{e:4.36})
and (\ref{e:4.37}) the following set of equations which are
equivalent to the extremal flow (\ref{e:4.33}), and represent the
(augmented) full Toda flow:
\begin{eqnarray}\label{e:4.39}
\dot X&=&\pi_l(A)X\\
\dot A&=&[A,\pi_k(A)]=[\pi_l(A),A].\nonumber
\end{eqnarray}
The equations (\ref{e:4.33}) and (\ref{e:4.36}) were originally
derived in a completely different fashion in the work of Symes 
[1980]%\cite{}
. We note the special form of the full reduced Toda flow equation 
(\ref{e:4.38})
\begin{equation}\label{e:4.40}
\dot A=\pi_{k^\perp}([A,\pi^T_l(\nabla f)]).
\end{equation}

We observe that this evolves naturally in a coadjoint orbit of $L$.
We can see this by considering tangent vectors to an orbit. These are
of the form $ad^*_WA$ for $W\in{\mathcal L}$ and $A\in{\mathcal L}^*
\simeq{\mathcal K}^\perp$. Let $A=\langle S,\cdot\rangle_T$ for $S\in
{\mathcal K}^\perp$, and let $V\in{\mathcal L}$. Then
\begin{eqnarray*}
ad^*_W(A)(V)&=&-A(ad_WV)=-A([W,V])\\
&=&-\langle S,[W,V]\rangle_T=-\langle[W^T,S],V\rangle_T\\
&=&-\langle[W^T,S],\pi_lV\rangle_T\\
&&\mbox{since $V\in{\mathcal L}$}\\
&=&\langle\pi_{k^\perp}[S,W^T],V\rangle_T.
\end{eqnarray*}
Hence indeed the right hand side of (\ref{e:4.40}) is a tangent
vector to a coadjoint orbit. The classical tridiagonal Toda flow lies
on a low rank nongeneric orbit of this type.



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