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\hyphenation{group-oid group-oids}
\title{{\bf Lagrangian mechanics and groupoids}}
\author{Alan Weinstein\thanks{To appear in
proceedings of Mechanics Day, Fields Institute, June 1992. Dedicated
to Jerry Marsden on his 50th birthday. Research partially supported by NSF
Grant DMS-90-01089 and DOE contract AT 381-ER 12097.}
\\Department of Mathematics\\
University of California\\
Berkeley, CA 94720 USA\\
{\small(alanw@@math.berkeley.edu)}}
\date{}
\begin{document}
\maketitle
%\tableofcontents
\section{Introduction}
\label{sec-intro}
A recent paper of Moser and Veselov \cite{mo-ve:discrete} on the
complete integrability of certain discrete dynamical systems
uses the lagrangian and hamiltonian formalisms for discrete mechanics
in two different settings. In the first, one
begins with a suitably nondegenerate lagrangian function
$L$ on the cartesian square $M\times M$ of a configuration space $M$
and obtains a second order recursion relation on $M$; i.e. a map which
assigns to each pair $(x,y)$ a pair $(y,z)$. The corresponding
hamiltonian system is the canonical transformation of $T^{*}M$ for
which $L$ is the generating function in a very classical sense.
In the second setting, the lagrangian $L$ is defined on a Lie group
$G$, and the dynamical system is given by a diffeomorphism from $G$ to
itself. The corresponding hamiltonian system is the mapping from the
dual Lie algebra $\gstar $ to itself for which $L$ is the generating
function in the sense of Ge and Marsden \cite{ge-ma:lie}.
The purpose of this paper is to describe versions of the lagrangian
formalism for discrete and continuous time which are general enough to
include both constructions used by Moser and Veselov, as well as a
lagrangian formalism on Lie algebras due essentially to Poincar\'{e}
\cite{po:forme}. In the discrete version, $L$ is
defined on a Lie groupoid \cite{ma:lie} $\Gamma $, and the resulting
dynamical system maps each groupoid element $g$ to an element $h$ for
which the product $gh$ is defined. The corresponding hamiltonian
system is a Poisson automorphism of the dual of the Lie algebroid of
$\Gamma $. In the continuous version, $L$ and its lagrangian vector
field live on a Lie algebroid $\cala $, and the Legendre-transformed
system is a hamiltonian system on the Poisson manifold $\cala ^{*}$.
The Lie groupoids relevant to \cite{mo-ve:discrete} are the cartesian
square $G\times G$ of a Lie group $G$, with the multiplication
$(g,h)(h,k)=(g,k)$, and the group $G$ itself. The lagrangian on
$G\times G$ is invariant under the diagonal action of $G$ from the
right, and that on $G$ is just its image under the projection map
$(g,h)\mapsto gh\inverse $. Since $(gh\inverse
)(hk\inverse)=gk\inverse $, this projection is in fact a morphism of
groupoids, and the relation between the two systems is a consequence
of the naturality of the lagrangian formalism.
There is an analogous version of the discussion in the previous
paragraph in the algebroid setting. Given a Lie algebra $\frakg $ and
a function $L:\frakg \rightarrow \reals $ whose hessian is nowhere
degenerate, one may construct a lagrangian vector field on $\frakg $.
This vector field is the image under the right-translation projection
$r:TG\rightarrow \frakg $ of the ordinary lagrangian vector field on
$TG$ associated to the right-invariant function $L\smalcirc r$. Thus,
the theory of Lie-Poisson hamiltonian systems on $\gstar $ as
reductions of right-invariant hamiltonian systems on $T^{*}G$ has a
completely analogous lagrangian counterpart.\footnote{Jerry Marsden
has pointed out to me that a lagrangian formalism for Lie algebras acting on
manifolds was already found by Poincar\'{e} \cite{po:forme}.}
Since the continuous-time lagrangian formalism is more familiar, we
begin the body of this paper with a discussion of Lie algebroids and
their duals. When then go on to a general discussion of Lie groupoids
and their cotangent bundles, leading up to the general discrete-time
lagrangian formalism. In Section \ref{sec-reduction}, we describe
reduction procedures for lagrangian systems in both the continuous and
discrete cases.
The origin of lagrangian dynamics is, of course, the calculus of
variations. Integral curves of lagrangian systems of $TM$ are the
tangent lifts of extremals for functionals on spaces of paths on $M$.
In the same way, as Moser and Veselov note, ``integral sequences'' for
discrete lagrangian systems on $M\times M$ are connected to extremals
for functionals on spaces of sequences in $M$. Extensions of this
idea to general Lie algebroids and Lie groupoids are discussed in
Section \ref{sec-variational}.
I would like to think Clint Scovel for originally calling my attention
to the paper of Moser and Veselov, and Pierre Dazord, Kirill
Mackenzie, Jerry Marsden, J\"{u}rgen Moser, and Ping Xu for some
useful discussions.
\section{Lie algebroids}
\label{sec-algebroids}
A Lie algebroid over a manifold $M$ may be thought of as a
``generalized tangent bundle'' to $M$. Here is the definition (see
\cite{ma:lie} for more details on the theory).
\begin{dfn}
\label{dfn-algebroid}
A Lie algebroid over a manifold $M$ is a vector bundle $\cala $ over
$M$ equipped with a Lie
algebra structure $[~,~]$ on its space of sections and a bundle map
$\rho : \cala \rightarrow TM$ (called the {\em anchor})
which induces a Lie algebra homomorphism (also denoted $\rho $)
from sections of $\cala $ to vector fields on $M$.
The identity $$ [f\xi ,\eta ] = f[\xi ,\eta ]-\left(\rho \left(\eta \right)f\right)\xi \enspace $$
must be satisfied for every smooth function $f$ on $M$.
\end{dfn}
The standard local coordinates on $\cala $ have the form $(q,\lambda)$
where the $q_{i}$'s are coordinates on the base $M$ and the $\lambda
_{i}$'s are linear coordinates on the fibres, associated with a basis
$\xi _{i}$ of sections of the Lie algebroid. In terms of such
coordinates, the bracket and anchor have expressions $[\xi _{i},\xi
_{j}]=\sum c_{ijk}\xi _{k}$ and $\rho (\xi_{i})=\sum a_{ij}\frac{\del
}{\del q_{j}}$, where the $c_{ijk}$ and $a_{ij}$ are ``structure
functions'' lying in $\cinf(M)$.
\subsection{Examples}
\label{subsec-firstexamples}
The basic example of a Lie algebroid over $M$ is the tangent bundle
$TM$ itself, with the identity mapping as anchor. With respect to the
usual coordinates $(q,\dot{q})$, the structure functions are $c_{ijk}=0$ and
$a_{ij}=\delta _{ij}$, but if we were to change to another basis for the
vector fields, the structure functions would become nonzero.
Lie algebroids can also be smaller or larger than $TM$. Any integrable
subbundle of $TM$ is a Lie algebroid with the inclusion as anchor and
the induced bracket. On the other hand, any Lie algebra $\frakg $ is
a Lie algebroid over a point. More generally, if $P$ is a principal
$G$-bundle over $M$, then $TP/G$ is a vector bundle over $M$ whose
sections are the $G$-equivariant vector fields (``infinitesimal gauge
transformations'') on $P$. These sections inherit the bracket from
$TP$, and the derivative $TP\rightarrow TM$ of the projection from $P$
to $M$ passes to a map from $TP/G$ to $TM$ which is the anchor of a
Lie algebroid structure on $TP/G$, which we call the {\bf gauge
algebroid} of $P$.
The kernel of the anchor of a gauge algebroid $TP/G$ is the
``adjoint'' bundle of the principal bundle $P$, with fibres isomorphic
to the Lie algebra $\frakg $ of $G$. This bundle of Lie algebras is a
Lie algebroid with the zero anchor. The Lie algebroid $TP/G$ plays a
basic role in the theory of connections on $P$ \cite{ma:lie}. We shall
see below that it is also the natural setting for the lagrangian
mechanics of classical particles in Yang-Mills fields associated with
such connections.
Another class of examples comes from actions of Lie algebras on
manifolds. If $\phi $ is a homomorphism from the Lie algebra $\frakg
$ to the vector fields on $M$, the {\bf action algebroid} is the
trivial bundle $M \times \frakg $ with the
the anchor $\rho (m,v)=\phi (v)(m)$ and the bracket $[\xi
,\eta ](m)=[\xi (m),\eta (m)]+\phi (\xi )\cdot \eta -\phi (\eta )\cdot
\xi.$ (In the bracket formula, we have identified sections of $M\times
\frakg $ with $\frakg $-valued functions on $M$.)
A final interesting class of Lie algebroids are the cotangent bundles
of Poisson manifolds. These carry a bracket
\cite{do:deformations}\cite{ma-mo:geometrical} characterized by the
rule $\{df,dg\}=d\{f,g\}$. The anchor is the map $\tilde{\pi
}:T^{*}M\rightarrow TM$ associated to the Poisson bivector field $\pi $.
\subsection{Poisson structure on the dual}
\label{subsec-algebroids}
The dual bundle $\cala ^{*}$ to a Lie algebroid carries a natural
Poisson structure. To describe this structure, it suffices to give
the Poisson brackets of a class of functions whose differentials span
the cotangent space at each point of $\cala ^{*}$. Such a class is
given by the functions which are affine on fibres. The functions
constant on fibres are just the functions on $M$, lifted to $\cala
^{*}$ via the bundle projection, while the functions linear on fibres
may be identified with the sections of $\cala $. If $f$ and $g$ are
functions on $M$, and $\xi $ and $\eta $ are sections of $\cala $,
their bracket relations as functions on $\cala ^{*}$ are: $$
\{f,g\}=0,~\{f,\xi \}=\rho\left(\xi
\right)\cdot f,\mbox{ and }\{\xi ,\eta \}=[\xi
,\eta ] \enspace .$$
Given standard coordinates $(q,\lambda )$ on $\cala $, we may
introduce dual coordinates $(q,\mu )$ on $\cala ^{*}$. In terms of
such coordinates and the structure functions introduced at the
beginning of Section \ref{sec-algebroids}, the Poisson bracket
relations on $\cala ^{*}$ are $\{q_{i},q_{j}\}=0$, $\{\mu _{i},\mu
_{j}\}=c_{ijk}\mu _{k}$, and $\{q_{i},\mu _{j}\}=a_{ji}$.
When $\cala =TM$, the Poisson structure on its dual is just the usual
cotangent bundle structure. The dual to an integrable subbundle of
$TM$ is the cotangent bundle along the leaves of the corresponding
foliation $\calf $; the symplectic leaves of the Poisson structure are
the cotangent bundles of the leaves of $\calf $.
When $\cala $ is a Lie algebra, the Poisson structure on its dual is
the standard Lie-Poisson structure. For a gauge Lie algebroid
$TP/G$, its dual is the phase space $T^{*}P/G$ for the
hamiltonian mechanics of a classical particle in a gauge field on $M$
\cite{we:universal}. The dual $M \times \gstar $ of an action
algebroid carries the ``semidirect product'' Poisson structure of
\cite{kr-ma:hamiltonian}.
When $\cala $ is the cotangent bundle of a Poisson manifold, its dual
is the tangent bundle with the ``tangent Poisson structure'', see
\cite{al:geometric}\cite{co-da-we:groupoides}\cite{co:dirac}. If the
Poisson bracket
relations in coordinates $q$ on $M$ are $\{q_{i},q_{j}\}=\pi
_{ij}(q)$, the brackets in coordinates $(q,\dot{q})$ on
$\cala ^{*}=TM$ are: $$ \{q_{i},q_{j}\}=0,~\{q_{i},\dot{q}_{j}\}=\pi
_{ij}(q),\mbox{and~} \{\dot{q}_{i}, \dot{q}_{j}\}=\sum \frac{\del \pi
_{ij}}{\del q_{k}} \dot{q}_{k}.$$
\subsection{Lagrangian formalism}
\label{subsec-algebroid lagrangian}
Our basic observation here is simply that all the ingredients used in
creating a
dynamical system from a lagrangian function on a tangent bundle (as
in \cite{ab-ma:foundations}, for instance) are
already present on an arbitrary Lie algebroid.
Let $L$ be a real-valued function on the Lie algebroid $\cala $ over
$M$. The {\bf Legendre mapping} $FL:\cala \rightarrow \cala ^{*}$ of
$L$ is defined as the fibre derivative of $L$. We define the {\bf
action} $A$ to be the function on $\cala $ defined by
$A\left(v\right)=$ (using the natural dual
pairing) and the {\bf energy} $E$ to be $A-L$.
If $L$ is a {\bf regular lagrangian} in the sense that $FL $ is a
local diffeomorphism, we can pull back the Poisson
structure on $\cala ^{*}$ to a Poisson structure on $\cala $ which we
call the {\bf Lagrange Poisson structure}. Using this Poisson
structure, we can form the hamiltonian vector field of the energy
function $E$; this vector field is called the {\bf lagrangian vector
field} associated with the regular lagrangian $L$.
In standard coordinates, we may write the lagrangian as
$L(q,\lambda )$. Its fibre derivative $FL $ is
defined by $\mu _{i}=\frac{\del L}{\del \lambda _{i}}$,
so the bracket relations for the Lagrange Poisson structure on $\cala
$ are $$\{q_{i},q_{j}\}=0, ~\{\frac{\del L}{\del \lambda
_{i}},\frac{\del L}{\del \lambda _{j}}\}=c_{ijk}\frac{\del L}{\del
\lambda _{k}}, ~\mbox{and}~ \{q_{i},\frac{\del L}{\del \lambda
_{j}}\}=a_{ji}. $$
The action function is $A=\sum \lambda _{i}\frac{\del L}{\del \lambda
_{i}}$, and the energy is $E=\sum \lambda _{i}\frac{\del L}{\del \lambda
_{i}} - L$. Lagrange's equations may, therefore, be put in Poisson
bracket form $\frac{dq_{i}}{dt}=\{q_{i},E\}$ and $\frac{d\lambda
_{i}}{dt}=\{\lambda _{i},E\}$.
Using the bracket relation $\{q_{i},L\}=\sum \{q_{i},\lambda
_{j}\}\frac{\del L}{\del \lambda _{j}}$, we can rewrite the first
Lagrange equation as follows:
\begin{eqnarray}
\label{eq-brack}
\frac{dq_{i}}{dt} &= & \{q_{i},E\}=\{q_{i},\sum \lambda _{j}\frac{\del
L}{\del \lambda _{j}} - L\} \nonumber \\
& = & \sum \{q_{i},\frac{\del
L}{\del \lambda _{j}}\} \lambda _{j} + \sum \{q_{i},\lambda _{j}\}
\frac{\del L}{\del \lambda _{j}} -\{q_{i},L\} \nonumber \\
& =& \sum \{q_{i},\frac{\del L}{\del \lambda _{j}}\}\lambda _{j}\nonumber \\
& =& \sum a_{ji}\lambda _{j}=\sum \left(\rho \left(\xi
_{j}\right)\cdot q_{i}\right) \lambda _{j} \nonumber \\
& = & \rho \left(\sum \lambda _{j}\xi _{j}\right)\cdot q_{i} \enspace .
\end{eqnarray}
The equation $\frac{dq_{i}}{dt}= \rho \left(\sum \lambda _{j}\xi _{j}\right)\cdot
q_{i} $ which we have just derived shows that the lagrangian vector
field associated with any regular lagrangian is a {\bf second order
differential equation} in the sense of the following definition.
\begin{dfn}
\label{dfn-second order}
A tangent vector $v$ to a Lie algebroid $\cala $ at a point $\lambda $
is called {\bf admissible} if $T\pr _{\cala }(v)= \rho (v)$ where
$T\pr _{\cala }:T\cala
\rightarrow TM$ is the derivative of the vector bundle projection $\pr
_{\cala }:\cala \rightarrow M$. A curve in $\cala $ is called
admissible if its tangent vectors are all admissible.
A vector field $X$ on $\cala $ is called a
second order differential equation if its values are all admissible
vectors. (Thus, a vector field on $\cala $ is a second order
differential equation iff all its integral curves are admissible.)
\end{dfn}
We can make the first Lagrange equation look even more like the usual
one if we introduce the coordinates $\left(q,\dot{q}\right)$ on $TM$, so that the
map $\rho $ is defined by $\dot{q}_{i}=\sum a_{ji} \lambda _{j}$, in
which case we get simply $$\frac{dq_{i}}{dt}=\dot{q}_{i}\enspace .$$
As a corollary of the fact that the lagrangian vector field is a
second order differential equation, we see immediately that each
solution of Lagrange's equations, projected into $M$, lies in a leaf
of the (possibly singular) foliation of $M$ determined
by the image of $\rho $ in $TM$. (These leaves are called {\bf
orbits} of the Lie algebroid $\cala $.) Another way to see that
trajectories are contained in orbits is to observe that the orbits are
the projections into $M$ of the symplectic leaves for the Poisson
structure on $\cala ^{*}$.
Passing now to the second Lagrange equation, we follow the usual custom
and compute the time derivative of $\frac{\del L}{\del \lambda _{i}}$
rather than that of $\lambda _{i}$:
\begin{eqnarray}
\label{eq-?}
\frac{d}{dt}\left(\frac{\del L}{\del \lambda _{i}}\right) & = & \{\frac{\del L}{\del \lambda _{i}},E\}
=\{\frac{\del L}{\del \lambda _{i}},\sum \lambda _{j}\frac{\del
L}{\del \lambda _{j}} - L\} \nonumber \\
& = & \sum \{\frac{\del L}{\del \lambda _{i}},\lambda _{j}\}\frac{\del
L}{\del \lambda _{j}} + \sum \{\frac{\del L}{\del \lambda
_{i}},\frac{\del L}{\del \lambda _{j}}\} \lambda _{j}
-\{\frac{\del L}{\del \lambda _{i}},L\} \nonumber \\
& = & \sum \{\frac{\del L}{\del \lambda _{i}},\lambda _{j}\}\frac{\del
L}{\del \lambda _{j}}+
\sum c_{ijk}\frac{\del L}{\del \lambda _{k}}\lambda _{j} \nonumber \\
& & -\sum \{\frac{\del L}{\del \lambda _{i}},q_{j}\}\frac{\del L}{\del q_{j}}
-\sum \{\frac{\del L}{\del \lambda _{i}},\lambda _{j}\}\frac{\del
L}{\del \lambda _{j}} \nonumber \\
&=& \sum a_{ij}\frac{\del L}{\del q_{j}}
+\sum c_{ijk}\frac{\del L}{\del \lambda _{k}}\lambda _{j}\enspace ,
\end{eqnarray}
i.e.
$$ \frac{d}{dt}\left(\frac{\del L}{\del \lambda _{i}}\right)
=\sum a_{ij}\frac{\del L}{\del q_{j}}
+\sum c_{ijk}\lambda _{j}\frac{\del L}{\del \lambda _{k}}\enspace . $$
We note that these equations are meaningful even if the lagrangian is
not regular. In that case, they define an inhomogeneous linear
equation on tangent vectors to $\cala $ whose solution at each point
is an affine subspace or the empty set. We can understand this
situation geometrically in the following way. When $L$ is not
regular, the pullback by $FL $ of the
Poisson structure on $\cala ^{*}$ is not a Poisson structure but only
a (possibly discontinuous) Dirac structure.\footnote{A Dirac structure
\cite{co:dirac} on a
manifold $A$ is a subbundle of $TA\oplus T^{*}A$ satisfying algebraic
and differential properties shared by the graphs of closed 2-forms and
Poisson structures. The pullback of a Poisson structure on $B$ by a
smooth mapping from $A$ to $B$ is a subset of $TA\oplus T^{*}A$ which
has the algebraic property of a Dirac structure in each fibre, but
which may vary discontinuously with the basepoint in $A$.} Such a
structure gives a relation between vector fields and 1-forms, but not
a mapping from
vector fields to 1-forms or vice-versa. In this situation, we cannot
define the
lagrangian vector field, but we can still define a
``solution to Lagrange's equations'' to be a curve in $\cala $ whose
derivative is everywhere related by the Dirac structure to the
differential of $L$.
In the case of tangent bundles, there is a formalism due to J. Klein
\cite{kl:espaces} (see \cite{go:geometrie} for an exposition) which
produces the lagrangian vector field without using the cotangent
bundle at all. It might be interesting to see whether this formalism
can be extended to general Lie algebroids.
\subsection{Examples}
\label{subsec-algebroid examples}
When $\cala $ is the tangent bundle with its usual coordinates,
$a_{ij}$ is the identity matrix and all the $c_{ijk}$'s are zero, so
$\lambda _{i}=\dot{q}_{i}$, and we recover the usual form of
Lagrange's equations. (Notice that our formalism would also allow us
to write Lagrange's equation with respect to a ``moving frame''; the
brackets of the vector fields in the frame would then enter the
equations.) When the base manifold of the Lie algebroid is just a
point, the first term in the equation above disappears, and we recover
the Euler equations on a Lie algebra. For a general Lie algebroid, we
get a ``mixture'' of the two kinds of equations.
Whenever $\cala $ is {\em regular} in the sense that $\rho $ has constant
rank, the orbits of $\cala $ form a regular foliation, and a
lagrangian system simply breaks up into a family of systems lying over
the leaves. (Of course, this decomposition is only local, in
general.) In particular, when $\cala $ is the tangent bundle to a
foliation $\calf $, we obtain the lagrangian formalism for motion
constrained along the leaves of $\calf $. It would be
interesting to extend our theory to encompass
the equations of motion for systems with {\em nonintegrable}
constraints \cite{mo:isoholonomic}.
If $P$ is a principal $G$-bundle, a lagrangian on $TP/G$ is just a
$G$-invariant lagrangian on $TP$. In
this case, if we apply the usual lagrangian formalism on $TP$, we get
a $G$-invariant lagrangian vector field. This vector field projects
to a vector field on $TP/G$ which is precisely the one obtained by our
construction. Thus, one gets on the gauge algebroid $TP/G$ the
lagrangian vector field for the motion of a particle on $M$ in a
Yang-Mills field. In particular, if $G$ is 1-dimensional, the
resulting motion could be that of a particle in an electromagnetic
field.
When $P=G$, our lagrangian formalism on a Lie algebra $\frakg $
produces the Euler equations for right-invariant lagrangians on
$TG$. (Of course, one can handle left-invariant lagrangians as well
by changing some signs.)
\begin{rmk}
\label{rmk-right invariant}
In Section \ref{sec-reduction} (see Example \ref{ex-universal}), we
will see how lagrangian systems on a large class of Lie algebroids can
be obtained by reduction from subbundles of tangent bundles (of Lie
groupoids).
\end{rmk}
We will analyze lagrangian systems on gauge Lie
algebroids by using local coordinates. A local
trivialization of the bundle also trivializes the Lie algebroid as the
direct sum of the tangent bundle $TM$ and the bundle of Lie algebras
$M\times \frakg $; the anchor $\rho $ is the projection onto the
summand $TM$. The coordinates on $\cala $ are better written in this
case as $(q,\dot{q},\lambda )$, where $\lambda $ now refers only to
coordinates on the $\frakg $ summand; the dual coordinates on $\cala
^{*}$ are $(q,p,\mu)$; and the nonzero Poisson bracket relations on
$\cala ^{*}$ are simply $\{q_{i},p_{j}\}=\delta _{ij}$ and $\{\mu
_{i},\mu _{j}\}=\sum c_{ijk}\mu _{k},$ where the $c_{ijk}$'s are now
constants.
Lagrange's equations take the form $$
\frac{dq_{i}}{dt}=\dot{q}_{i}\enspace , $$ $$ \frac{d}{dt}\left(\frac{\del
L}{\del \dot{q}_{i}}\right) =\frac{\del L}{\del q_{i}}\enspace, $$ and $$
\frac{d}{dt}\left(\frac{\del L}{\del \lambda _{i}}\right) = \sum c_{ijk}\lambda
_{j}\frac{\del L}{\del \lambda _{k}}\enspace . $$
The equations for $q$ and $\dot{q}$ are coupled to those for $\lambda$
through the fact that $\frac{\del L}{\del \dot{q}_{i}}$ generally
depends on $\lambda $. It is already interesting to look at the case
where $\frakg $ is abelian, in which case the $\frac{\del L}{\del
\lambda _{i}}$'s are constants of the motion. For instance, suppose
that $\frakg $ is 1-dimensional and that $L\left(q,\dot{q},\lambda \right)=\half
\sum \dot{q}_{i}^{2}+\half \left(\lambda -\sum
A_{i}\left(q\right)\dot{q}_{i}\right)^{2}-V\left(q\right) $. The constant of motion is
then $\lambda - \sum A_{i}\left(q\right)\dot{q}_{i},$ which we denote by $e$.
$\frac{\del L}{\del \dot{q}_{i}}$ becomes $\dot{q}_{i}$, and
Lagrange's equation becomes $\ddot{q}_{i}=-\frac{\del V}{\del
\dot{q}_{i}}
+ e\left(\frac{\del A_{i}}{\del q_{j}}-\frac{\del A_{j}}{\del
q_{i}}\right)\dot{q}_{j}$, the equation of motion for a
particle of unit mass and charge $e$ in an electromagnetic field with
scalar potential $V$ and vector potential $\sum A_{i}dq_{i}$.
\begin{rmk} \label{rmk-electromagnetic} The lagrangian system above is
essentially the geodesic flow for a riemannian metric on $M\times
\reals $, with the cyclic variable in $\reals $ ignored. In order to
globalize this viewpoint, one usually replaces
$M\times \reals $ by a bundle over $M$ whose fibre is $\reals $ or
$S^{1}$, but even this is possible only when the electromagnetic field
(considered as a closed 2 form on $M$) has a deRham cohomology class
which is an integral multiple of an integer class. On the other hand,
our formulation in terms of Lie algebroids applies without this condition.
As noted in \cite{al-mo:suites}, any closed 2-form on $M$
gives rise to a Lie algebroid structure on $TM \times \reals $. (This
is the Lie algebroid version of the central extension corresponding to
a cocycle, well known in the case of Lie algebras.) If the
form is integral, the resulting Lie algebroid is the gauge algebroid of the
corresponding principal $S^{1}$ bundle. If the class is not a
multiple of an integral one, the Lie algebroid does not come from a
bundle, but it still exists and can be the domain of a lagrangian
system. Such a Lie algebroid is {\bf nonintegrable} in the sense that
it is not the Lie algebroid of any Lie groupoid.
\end{rmk}
\bigskip
To understand the lagrangian vector field when $\cala$ is the
cotangent Lie algebroid of a Poisson manifold $M$, we first give a
description in local coordinates.
On a general Poisson manifold with $\{q_{i},q_{j}\}=\pi _{ij}(q)$, the
structure functions of the Lie algebroid $T^{*}M$ are
$c_{ijk}=\frac{\del \pi _{ij}}{\del q_{k}}$ and $a_{ij}=\pi _{ij}$, so
Lagrange's equations are:
$$ \frac{dq_{i}}{dt} = - \sum \pi _{ij}p_{j} $$
and
$$ \frac{d}{dt}\left(\frac{\del L}{\del p_{i}}\right)
=\sum \pi _{ij}\frac{\del L}{\del q_{j}}
+ \sum \frac{\del \pi _{ij}}{\del q_{k}}p_{j}\frac{\del L}{\del
p_{k}}\enspace . $$
When the Poisson structure is zero, Lagrange's equations just say that
$q_{i}$ and $\frac{\del L}{\del p_{i}}$ are constants. Among the
solutions are all the constant curves; there may be others if $L$ is
not regular.
For a Lie-Poisson structure, where $\pi _{ij}=\sum
c_{ijk}q_{k}$, the equations become:
$$ \frac{dq_{i}}{dt} = - \sum c_{ijk}p_{j}q_{k} $$
and
$$ \frac{d}{dt}\left(\frac{\del L}{\del p_{i}}\right)
=\sum c_{ijk}\left(\frac{\del L}{\del q_{j}}q_{k}- \frac{\del L}{\del
p_{j}}p_{k}\right) \enspace .$$
In the symplectic case, the symplectic structure gives an isomorphism
between the Lie algebroid $T^{*}M$ and the tangent bundle. Modulo
this isomorphism, the lagrangian formalism in this case is just the
usual one.
For a general regular Poisson manifold, its Lie algebroid $T^{*}M$
fits into an exact sequence $ 0\rightarrow N^{*}\calf \rightarrow
T^{*}M\rightarrow T^{*}{\calf }\rightarrow 0$, where $T^{*}{\calf }$
is the cotangent bundle along the symplectic leaves, and the kernel
$N^{*}\calf $ is the conormal bundle of the leaves, carrying trivial
bracket and anchor. Locally, this exact sequence splits, and we are
essentially in the situation of gauge Lie algebroids as discussed
above. Globally, there is an obstruction \cite{da-he:integration} to
splitting the sequence which is, roughly speaking, the transverse variation
of the cohomology classes
of the symplectic structures along the leaves. This obstruction should
manifest itself as an unavoidable ``magnetic term'' in a lagrangian
system on $T^{*}M$.
\section{Lie groupoids}
\label{sec-groupoids} \subsection{Definition and basic properties}
\label{subsec-basic} If we think of a Lie algebroid over $M$ as a
generalized tangent bundle of $M$, and if we think of tangent vectors
to $M$ as ``pairs of infinitesimally near points'' of $M$, then the object
of which the general Lie algebroid is the infinitesimalization should
be a generalization of the product $M \times M$.
We will think of the elements of a this object as reversible
``arrows'', each with a
designated ``source'' and ``target'' in $M$. There is also an
associative composition law defined when the target of one arrow
equals the source of the next. Here is the formal definition.
\begin{dfn}
\label{dfn-groupoid}
A groupoid over a set $M$ is a set $\Gamma $ equipped
with source and target mappings $\alpha$ and $\beta $ from $\Gamma $ to $M$, a
multiplication map $m$ from $\Gamma _{2}\defequal \{(g,h)\in
\Gamma \times \Gamma |\beta (g)=\alpha (h)\}$ to $\Gamma $, a
units mapping $\epsilon :M\rightarrow \Gamma $, and an
inversion mapping $\iota :\Gamma \rightarrow \Gamma$
satisfying the following properties, where we write $gh$ for $m(g,h)$
and $g\inverse$ for $\iota (g)$:
(i) (associativity) $g(hk)=(gh)k$ in the sense that, if
one side of the equation is defined, so is the other, and then they
are equal;
(ii) (identities) $\epsilon (\alpha (g))g=g=g\epsilon (\beta (g))$;
(iii) (inverses) $gg\inverse =\epsilon(\alpha (g))$ and $g\inverse
g=\epsilon (\beta (g))$. \end{dfn}
Some basic properties of groupoids follow directly from this
definition. For instance, $\alpha(gh)=\alpha(g)$ and $\beta
(gh)=\beta (h)$, $\alpha (\epsilon(x))=x=\beta(\epsilon (x))$, and
$(g\inverse )\inverse =g$. We frequently identify $M$ with its image
$\Gamma _{0}$ in $\Gamma $, so that $\alpha $ and $\beta $ may be
considered as retractions of $\Gamma $ onto $\Gamma _{0}$.
The elements of $\Gamma _{2}$ are sometimes referred to as composable
pairs. By analogy with Definition \ref{dfn-second order}, we may also
refer to them as admissible pairs.
If $\Gamma $ and $M$ are differentiable manifolds, $\alpha $ and
$\beta $ are submersions (so that $\Gamma _{2}$ is a manifold), and
$m$ is differentiable, then $\Gamma $ is a {\bf Lie
groupoid}.\footnote{The term ``differentiable groupoid'' was used
\cite{ma:lie} until a
few years ago, with the term ``Lie groupoid'' reserved for the case
where $(\alpha ,\beta )$ is a locally trivial fibration from $\Gamma $
to $M\times M$.} Differentiability of $\epsilon $ and $\iota $ are
consequences of the groupoid properties and need not be assumed
separately.
The inverse images of points under the source and target maps are
called $\alpha $- and $\beta $-fibres. The fibres through a point $g$
will be denoted by $\calf ^{\alpha }(g)$ and $\calf ^{\beta }(g)$.
These fibres form two foliations tangent to the subbundles $\ker
T\alpha $ and $\ker T\beta $ of $T\Gamma $; both foliations are
transverse to $\Gamma _{0}$. Each element $g$ of $\Gamma
$ determines left and right translation maps $\ell_{g}:\calf ^{\alpha
}(\beta (g))\rightarrow \calf ^{\alpha }(\alpha (g))$ and
$r_{g}:\calf ^{\beta }(\alpha (g))\rightarrow \calf ^{\beta }(\beta
(g)).$ These enable us to define the spaces of left-invariant
sections of $\ker T\alpha $ and right-invariant sections of $\ker
T\beta $, both of which turn out to be closed under bracket.
\begin{rmk}
\label{rmk-diffeo}
Note that the map $(g,h)\mapsto (g,gh)$ is a diffeomorphism between
$\Gamma _{2}$ and the set of pairs $(g,k)$ such that $\alpha
(g)=\alpha (k)$. (Since $\beta (g\inverse )=\alpha (g)$, the product
$h=g\inverse k$ is always defined for such pairs.) It follows easily
that the map $m$ is a submersion.
\end{rmk}
The Lie algebroid $\cala _{\Gamma }$ of a Lie groupoid $\Gamma $ may
be thought of as consisting of the elements of $\Gamma $ which are
infinitesimally close to identity elements. More precisely, we define
$\cala _{\Gamma }$ as the normal bundle $T_{\Gamma _{0}}\Gamma
/T\Gamma _{0}$ to $\Gamma _{0}$ in ${\Gamma }$.
To define a bracket structure on the sections of $\cala _{\Gamma }$,
we identify them with the right-invariant sections of $\ker T\beta $
on $\Gamma $. The anchor is defined by $T\alpha :\ker T\beta
\rightarrow TM$ along $\Gamma _{0}$.
For our ultimate applications to lagrangian mechanics, it will be more
important to have the Poisson structure on the dual bundle, which is
the {\em conormal} bundle to $\Gamma _{0}$ in ${\Gamma }$, so we will
also define this Poisson structure directly.
First of all, we recall (see \cite{co-da-we:groupoides}) that the
cotangent bundle $T^{*}\Gamma $ is, in
addition to being a symplectic manifold, a groupoid itself, the base
being $\cala _{\Gamma }^{*}$. The source mapping $\tilde{\alpha
}:T^{*}\Gamma \rightarrow \cala _{\Gamma }^{*}$ is defined as follows.
Let $\mu $ be a cotangent vector to $\Gamma $ at the element $g$. We
restrict $\mu $ to $T_{g}\calf^{\beta}(g)$ and then pull back by
$r_{g}$ to move it to $T_{\alpha (g)}\calf ^{\beta}(\alpha (g))$.
Finally, we identify the tangent space $T_{\alpha (g)}\calf
^{\beta}(\alpha (g))$ with the conormal space to $\Gamma _{0}$, since
the $\beta $-fibre is transverse to $\Gamma _{0}$.
By interchanging $\alpha $ and $\beta $ and ``right'' and ``left'' in
the previous paragraph, we construct in a similar way the target map
$\tilde{\beta }$. We refer to \cite{co-da-we:groupoides} for the
definition of multiplication in $T^{*}\Gamma $, which will not concern
us in this paper. It turns out (again see
\cite{co-da-we:groupoides}) that the fibres of the mappings
$\tilde{\alpha}$ and $\tilde{\beta }$ are symplectically orthogonal to
one another, so they form a ``dual pair'', and there is a unique
Poisson structure on $\cala ^{*}$ for which $\tilde{\alpha }$ is a
Poisson map and $\tilde{\beta }$ is anti-Poisson. It can also be
shown that the Poisson bracket of functions linear on fibres is again
linear on fibres, so that the
bundle $\cala $ has a Lie algebroid structure for which this is the
dual.
\subsection{Examples}
\label{subsec-groupoidexamples}
There are Lie groupoids corresponding to many of the
Lie algebroids in Section \ref{sec-algebroids}.
The first example is the {\em pair groupoid} $M \times M$, with the
multiplication law $(x,y)(y,z)=(x,z)$. Its Lie algebroid
may be identified with the tangent bundle $TM$.
To get a groupoid corresponding to an integrable subbundle of $TM$, it
is necessary to form the holonomy groupoid of the corresponding
foliation, or one of its close relatives \cite{wi:graph}. This example shows
that it may be necessary to admit non-Hausdorff manifolds in order to
integrate a Lie algebroid to a Lie groupoid. (Worse yet, there are
Lie algebroids which cannot be integrated at all \cite{al-mo:suites}.)
Any Lie group $G$ is a Lie groupoid over a point; its Lie algebroid is
its Lie algebra. For a gauge Lie algebroid $TP/G$, the corresponding
groupoid is the {\bf gauge groupoid} $(P\times P)/G$, whose
elements may be thought of as $G$-equivariant maps from one fibre of
$P$ to another.
If we have a right action of the group $G$ on $M$, there is an {\bf action
groupoid} structure on $M\times G$ with $\alpha (x,g)=x,$ $\beta(x,g)=xg,$
and $(x,g)\cdot (xg,h)=(x,gh)$. Its Lie algebroid is the
action algebroid $M \times \frakg $ associated to the derived Lie
algebra action of $\frakg $. Given an action Lie
algebroid corresponding to a nonintegrable Lie algebra action on a
(noncompact) manifold, there may still be a corresponding Lie
groupoid; the exact conditions under which an action Lie algebroid is
integrable are not completely understood.
Finally, for the Lie algebroid structure on $T^{*}M$ corresponding to
a Poisson structure on $M$, the corresponding Lie groupoid is a {\bf
symplectic groupoid} \cite{co-da-we:groupoides}. It exists
whenever $M$ is symplectic or Lie-Poisson, and in many other cases as
well, but there are also simple examples of Poisson manifolds
admitting no symplectic groupoid \cite{da-he:integration}.
\subsection{Lagrangian formalism}
\label{subsec-groupoid lagrangian}
We are finally ready to define lagrangian mechanics on a Lie groupoid
$\Gamma $. Let $L$ be a smooth, real-valued function on $\Gamma $,
$L_{2} $ the restriction to
$\Gamma _{2}\subset \Gamma \times \Gamma $ of the function
$(g,h)\mapsto L(g)+L(h)$ . Define $\Sigma
_{L}\subset \Gamma _{2}$ to be the set of critical points of $L_{2}$
along the fibres of the multiplication map $m$; i.e. the points of
$\Sigma _{L}$ are stationary points of the function $L(g)+L(h)$ when
$g$ and $h$ are restricted to admissible pairs with the constraint
that the product $gh$ is fixed. (We saw in Remark \ref{rmk-diffeo}
that $m$ is a submersion.)
\begin{dfn}
\label{dfn-solution}
A {\bf solution of Lagrange's equations} for the lagrangian function
$L$ is a sequence $\ldots,g_{-2},g_{-1},g_{0},g_{1},g_{2},\ldots $ of
elements of $\Gamma $, defined on some ``interval'' (finite or
infinite) in $\integers $,
such that $(g_{j},g_{j+1})\in \Sigma _{L}$ for each $j$.
\end{dfn}
Lagrange's equations on a groupoid define a ``second order difference
equation'' in the sense that every solution is an admissible sequence
according to the following definition.
\begin{dfn}
\label{dfn-admissible groupoid}
A sequence
$\ldots,g_{-2},g_{-1},g_{0},g_{1},g_{2},\ldots $ in the groupoid
$\Gamma $, defined on some ``interval'' (finite or
infinite) in $\integers $, is called {\bf
admissible} if $(g_{j},g_{j+1})\in \Gamma _{2}$ for each $j$ .
\end{dfn}
\begin{ex}
\label{ex-pairlagrangian}
Let $\Gamma =M\times M$ be a pair groupoid. Then $$\Gamma_{2}=
\{((x,y),(y,z))|(x,y,z)\in M \times M \times M\}$$ can be identified
with $M\times M \times M,$ and $L_{2}(x,y,z)=L(x,y)+L(y,z)$. $\Sigma
_{L}$ is defined by the equation $\del _{2}L(x,y)+\del _{1}L(y,z)=0$,
where $\del _{i}$ denotes the differential with respect to the $i$'th
argument. Solutions of Lagrange's equations can be identified with
finite or infinite sequences $\ldots, x_{-1},x_{0},x_{1},\ldots $ of
elements of $M$ such that $\del _{2}L(x_{j-1},x_{j})+\del
_{1}L(x_{j},x_{j+1})$ for each $j$.
\end{ex}
In the remainder of this section, we discuss the following two
general questions concerning the lagrangian formalism on groupoids.
\begin{enumerate}
\item What is the discrete hamiltonian formalism corresponding to this
lagrangian formalism?
\item When is a lagrangian $L$ ``regular'' in the sense that $\Sigma
_{L}\subset \Gamma \times \Gamma $ is (perhaps just locally) the graph
of a mapping from $\Gamma $ to $\Gamma $, so that each initial
condition $g_{0}$ can be extended to a solution of Lagrange's equations?
\end{enumerate}
\subsection{Hamiltonian formalism}
\label{subsec-hamiltoniangroupoid}
The hamiltonian formalism for discrete lagrangian systems is based on
the fact that each Lagrangian\footnote{To
distinguish the two completely different
meanings of ``lagrangian'' in this paper, we will use an upper case L
when referring to submanifolds rather than functions.} submanifold of
a symplectic groupoid determines a Poisson automorphism (or at least a
coisotropic relation) on the base Poisson manifold. Recall that the
cotangent bundle of a Lie groupoid $\Gamma $ is a symplectic groupoid
over the dual $\cala _{\Gamma }^{*}$ If $L$ is any smooth function on
$\Gamma $, the Lagrangian submanifold $dL(\Gamma )$ therefore
determines (under a suitable hypothesis of nondegeneracy) a Poisson
map from $\cala _{\Gamma }^{*}$ to itself, which may be said to be
{\bf generated} by $L$. (In the case where $\Gamma $ is a group, this
notion of generating function was used in \cite{ge-ma:lie}.)
How is this Poisson mapping related to the relation from $\Gamma $ to
itself given by $\Sigma _{L}$? The appropriate ``Legendre mapping''
$FL $ in the groupoid context is $\tilde{\alpha }\smalcirc
dL:\Gamma \rightarrow \cala ^{*}$, which turns out to intertwine these
two relations. We can also think of $FL $ as pulling back the
Poisson structure from $\cala ^{*}$ to $\Gamma $, so that the
resulting structure on $\Gamma $ is preserved by the mapping whose
graph is $\Sigma _{L}$.
\begin{ex}
\label{ex-pairhamiltonian}
We return to the pair groupoid $M\times M$ discussed in Example
\ref{ex-pairlagrangian}. The Legendre mapping $FL=\tilde{\alpha }\smalcirc
dL :M\times M\rightarrow T^{*}M$ is given by $(x,y)\mapsto
(x,L_{1}(x,y))$, from which it follows that the canonical relation
$\Lambda _{L}$ from $T^{*}M$ to itself is given by
$(x,L_{1}(x,y))\mapsto (y,L_{1}(y,z))$, where $(x,y,z)$ satisfies the
condition $L_{2}(x,y)+L_{1}(y,z)=0$. We may therefore describe
$\Lambda _{L}$ as the relation $(x,\xi )\mapsto (y,\eta )$, where $\xi
=\del L/ \del x$ and $\eta = - \del L/ \del y$. The relation may be
recognized as the graph of the canonical transformation for which the
generating function (of Type $F_{1}$ in the terminology of
\cite{go:classical}) is $L(x,y)$.
Suppose, for example, that $M$ is a
riemannian manifold with bijective exponential mappings, and
$L(x,y)=\half \mbox{dist}(x,y)^{2}$. The relation $\Lambda _{L}$ then
maps $(x,y)$ to that $(y,z)$ for which the geodesic segment from $y$ to $z$ is
the continuation, with the same length, of that from $x$ to $y$. The
corresponding hamiltonian picture is the time 1 map of the geodesic flow.
\end{ex}
\begin{ex}
\label{ex-groups}
As we have already noted, the generation of a Poisson map
$\gstar \rightarrow \gstar $ from a function $L$ on a Lie group $G$ is
a basic construction in \cite{ge-ma:lie}. The corresponding relation
on $G$ is $\{(g,h)|\ell_{g}^{*}dL(g)=r_{h}^{*}dL(h)\}$, which may or
may not (even locally) be the graph of a mapping. This is the same
relation to be found in \cite{mo-ve:discrete}, along with its
hamiltonian version.
\end{ex}
\begin{rmk}
\label{rmk-square}
There is another way to produce a Lagrangian submanifold of
$T^{*}\Gamma $ from a function on $\Gamma $, based on the
concept of a generating family for a Lagrangian
submanifold \cite{we:lectures}.
When $s:X\rightarrow Y$ is a submersion of manifolds and $E$ is a real
valued function on $X$, there is a mapping $\lambda_{E} $ from the critical set
$\Sigma _{E}$ of $E$ along the fibres of $s$ to the cotangent bundle
$T^{*}Y$ given by the ``horizontal'' differential of $E$, which is
well defined wherever the derivative along the fibres is zero. If $E$
is a {\bf Morse family} in the sense that the fibre derivative of $E$
is transverse to the zero section in the bundle $\ker ds$, then
$\lambda_{E} :\Sigma _{E}\rightarrow T^{*}X$ is a Lagrangian immersion,
and its image Lagrangian submanifold is said to be generated by $E$.
Applying the definition above to the function $L_{2}$ considered as a
family over $\Gamma $ via the submersion $m$, we get a subset
$\lambda _{L_{2}}(\Sigma _{L_{2}}) $ of $T^{*}\Gamma $ which is a
Lagrangian submanifold when $L_{2}$ is a Morse family. It turns out
that this subset is the {\em square} of $dL(\Gamma )$ in the algebra
of subsets of the groupoid $T^{*}\Gamma $.
\end{rmk}
\subsection{Regularity conditions}
\label{subsec-regularity}
In the case of lagrangian mechanics on tangent bundles, there are two
notions of regularity which are well known to be equivalent. The first is
that the Legendre map $FL :TM\rightarrow T^{*}M$ should be a local
diffeomorphism, so that the pullback of the symplectic structure on
$T^{*}M$ is nondegenerate; the second is the the solutions of
Lagrange's equations should be the trajectories of a well-defined
vector field on $TM$. The equivalent condition on the lagrangian
function $L$ is that the second differential of $L$ (with respect to
the standard affine structure on the fibres of $L$) should be
nondegenerate as a symmetric bilinear form.
There is also a more restrictive condition of
hyperregularity, where $FL $ is required to be a global
diffeomorphism, so that the motion described by Lagrange's equations
is also described by a global hamiltonian system on $T^{*}M $ with its
standard symplectic structure. Useful sufficient conditions for
hyperregularity are that $L$ be strictly convex on fibres or that it
be given by a nondegenerate quadratic form.
For the discrete analogue, lagrangian mechanics on a pair groupoid
$M\times M$, there are also two possible
regularity conditions. The first is that the Legendre map $FL
:M\times M\rightarrow T^{*}M$ should be a local diffeomorphism, so
that the pullback to $M\times M$ of the symplectic structure on $M$ is
nondegenerate. The second is that the equation $\del _{2}L(x,y)+\del
_{1}L(y,z)=0$ should define $\Sigma _{L}$ as a regular submanifold of
$M\times M\times M$ which projects by local diffeomorphism onto the
first two factors, so that $z$ is locally a smooth function of $x$ and
$y$. Both of these conditions again turn out to be equivalent to each
other and to a condition on the lagrangian $L$; the condition on $L$
in this case is that, for each $x$ and $y$ in $M$, the bilinear form
$T_{x}M\times T_{y}M$ defined by the second derivatives of $L$ be
nondegenerate. (It is a consequence
of this nondegeneracy condition that $\Sigma_{L}$ also defines $x$
locally as a smooth function of $y$ and $z$, so that the local
lagrangian dynamics is time-reversible.)
It would be interesting to formulate in terms of $L$ a
sufficient condition for some kind of hyperregularity, so that the
relation $(x,y)\mapsto z$ defined by $\Sigma _{L}$ is a function with
one or more of the following properties:
single-valued, globally defined, invertible.
We will leave to some future work the question of formulating
regularity and hyperregularity conditions for more general Lie
groupoids, and for that matter, for Lie algebroids more general than
tangent bundles. In those situations, it is not quite clear that the
two infinitesimal conditions which were equivalent in the cases above
are still equivalent.
\section{Reduction}
\label{sec-reduction}
Reduction theory for hamiltonian systems reflects a kind of
``functoriality'' with respect to Poisson maps. If $\psi :P\rightarrow Q$
is a mapping of Poisson manifolds and $H$ is a hamiltonian function on
$Q$, then $K={\psi ^{*}(H)}$ is a function on $P$, and the
trajectories of the hamiltonian vector field $\xi_{K}$ project under
$\psi $ to the trajectories of $\xi _{H}$.
Reduction is the application of this fact to the case where
$Q$ is the quotient of $P$ by a group action, and $K$ is an
invariant hamiltonian. In order to formulate reduction theory for
lagrangian systems, we will need to begin with appropriate notions
of mappings for Lie algebroids and Lie groupoids.
\subsection{Poisson relations}
\label{subsec-relations}
We remain for a moment in the Poisson category and
recall from \cite{we:coisotropic} a more general notion of morphism
between Poisson manifolds. For any Poisson manifold $P$, we denote by
$P^{-}$ the same manifold but with the negative of the original
Poisson structure. A map from $P$ to $Q$ is a Poisson map if an only
if its graph $\gamma $ is {\bf coisotropic} in $Q \times P^{-}$ in the sense
that the hamiltonian vector field of any function which vanishes on
$ \gamma $ is tangent to $\gamma $. We may therefore call a
coisotropic submanifold of $Q \times P^{-}$, whether or not it is a
graph, a {\bf Poisson relation} from $P$ to $Q$.
\begin{prop}
\label{prop-poissonreduction}
Let $\gamma $ be a closed Poisson relation from $P$ to $Q$, and
suppose that functions $H$ on $Q$ and $K$ on $P$ are $\gamma $-related
in the sense that the function $H(q)-K(p)$ on $Q\times P^{-}$ vanishes
when restricted to $\gamma $. If $\sigma _{H}$ and $\sigma _{K}$ are
trajectories of $\xi _{H}$ and $\xi _{K}$ respectively for which $(\xi
_{H}(t_{0}),\xi _{K}(t_{0})) $ lies on $\gamma $ for some $t$, then $(\xi
_{H}(t),\xi _{K}(t)) $ lies on $\gamma $ for all $t$.
\end{prop}
\pf
The hamiltonian vector field of $H(q)-K(p)$ on $Q\times P^{-}$ is
$(\sigma _{H},\sigma _{K})$. Since $\gamma $ is coisotropic and closed, all
trajectories of $(\sigma _{H},\sigma _{K})$ starting on $\gamma $
must remain on $\gamma $.
\qed
\subsection{Lie algebroid morphisms and lagrangian reduction}
\label{subsec-algebroid morphisms}
The following definitions are essentially to be found in
\cite{hi-ma:duality}, but the exact formulation is due to P. Xu
(unpublished).
If $\cala \rightarrow M$ and $\calb \rightarrow N$ are vector bundles,
and $\psi :\cala \rightarrow \calb $ is a bundle mapping covering a
mapping $\underline{\psi }:M\rightarrow N$, the {\bf dual comorphism} is the
relation\footnote{Higgins and Mackenzie \cite{hi-ma:duality} consider
this relation as a map
$\underline{\psi }^{*}\calb \rightarrow \cala $ of vector bundles over
$M$.} $\psi^{*}:\calb
^{*}\rightarrow \cala ^{*}$ whose graph is
\begin{equation}
\label{eq-psistar}
\{((m,\psi _{m}^{*}(\omega )),(\underline{\psi }(m),\omega) )|m\in M,
\omega \in \calb _{\underline{\psi }(m)}^{*} \} \subset \cala ^{*}\times
(\calb ^{*})^{-}.
\end{equation}
If $\cala \rightarrow M$ and $\calb \rightarrow
N$ are Lie algebroids, then the bundle mapping $\psi $ is called a
{\bf Lie algebroid morphism} when the dual comorphism $\psi ^{*}$ is a
Poisson relation.
Suppose now that $L$ is a lagrangian function on $\calb $; let
$K=L\smalcirc \psi $.
\begin{lemma}
\label{lemma-functorial legendre}
The fibre derivatives of $K$ and $L$ satisfy the relation
$$ (FK(m,v),FL(\psi (m,v)))\in \psi ^{*} $$
for all $(m,v)\in
\cala.$ The energy functions $E_{K}$ and $E_{L}$ satisfy $E_{K}=E_{L}\smalcirc
\psi $.
\end{lemma}
\pf
Since $K$ is the pullback of $L$ by $\psi $, $dK(m,v)$ is the pullback
by $T_{(m,v)}\psi $ of $dL(\psi (m,v))$. In the definition of the
fibre derivative, the tangent space to the fibre is identified with
the fibre itself. Under this identification, the restriction to
fibres of to $T_{(m,v)}\psi $ is just $\psi _{m}$, since $\psi $ is a
vector bundle map. It follows that $FK(m,v)=\psi _{m}^{*}(FL(\psi (m,v)))$,
so that $(FK(m,v),FL(\psi (m,v)))$ belongs to the relation $\psi ^{*}$.
To prove the second part of the lemma, we compute:
$$ E_{K}(m,v)=-K(m,v)
$$ $$=<\psi _{m}^{*}(FL(\psi
(m,v))),(m,v)>-L(\psi (m,v))=
$$ $$-L(\psi (m,v))=E_{L}(\psi (m,v)). $$
\qed
It is tempting to draw the following conclusion from Lemma
\ref{lemma-functorial legendre} and the fact that $\psi ^{*}$ is a
Poisson relation.
\begin{fakethm}
\label{fakethm-functorial}
Let $\psi :\cala \rightarrow \calb $ be a morphism of Lie algebroids,
and let $L$ be a function on $\calb $. Then the image under $\psi $
of any solution of Lagrange's equations for $L\smalcirc \psi $ is a
solution of Lagrange's equations for $L$.
\end{fakethm}
The reason why this ``Theorem'' fails is that Lemma
\ref{lemma-functorial legendre} involves a combination of $\psi $ and
$\psi ^{*} $ rather than either mapping separately.
\begin{ex}
\label{ex-fake}
Let $M$ be a riemannian manifold, $\calb $ its tangent bundle, and
$L$ the function which assigns to each vector one-half the square of
its length. The lagrangian vector field of $L$ is the geodesic flow.
Now let $\cala $ be an integrable subbundle, with $\psi :\cala
\rightarrow \calb $ the inclusion. The
dual comorphism is in this case a morphism, the restriction map from
$T^{*}M$ to $\cala ^{*}$. The pulled back lagrangian $K$ generates
the geodesic flow along the leaves of the foliation determined by
$\cala $; its trajectories map under $\psi $ to solutions of
Lagrange's equations for $L$ if and only if the leaves of the
foliation are totally geodesic in $M$. Two things have gone wrong here.
\begin{enumerate}
\item The energy functions $K$ and $L$, though they are $\psi
$-related, are not $\psi ^{*}$-related. In this case, the energy
functions essentially coincide with the lagrangians, but the pullback
of $E_{K}$ by $\psi ^{*}$ is the degenerate metric on $M$ which agrees
with the original metric on the foliation and which is null on the
orthogonal subbundle.
\item The Lagrange-Poisson structure
on $TM$ is the symplectic structure inherited from $T^{*}M$ by the
identification given by the metric, while the structure on $\cala
$ is that whose symplectic leaves are essentially the cotangent bundles along
the leaves of the foliation given by $\cala $. The inclusion mapping
$\psi $ is not a Poisson mapping for these structures. (Even a symplectic
embedding is not a Poisson mapping!)
\end{enumerate}
\end{ex}
Motivated by this example, we are led to correct the
previous ``Theorem'' by adding some hypotheses.
\begin{thm}
\label{thm-functorial}
Let $\psi :\cala \rightarrow \calb $ be a morphism of Lie algebroids,
and let $L$ be a regular lagrangian on $\calb $. If $\psi $ is an
isomorphism on each fibre,\footnote{\rm{In the terminology of
\cite{hi-ma:duality}, this makes $\psi $ an {\bf action morphism}.}}
then the image under $\psi $
of any solution of Lagrange's equations for $L\smalcirc \psi $ is a
solution of Lagrange's equations for $L$.
\end{thm}
\pf
The (quite strong) hypotheses in the statement of the theorem,
together with Lemma \ref{lemma-functorial legendre} imply
that the pair of maps $(FK,FL)$ takes the graph of $\psi $
diffeomorphically onto the
graph of $\psi ^{*}$. This implies that $\psi $ is a Poisson relation
for the Lagrange-Poisson structures on $\cala $ and $\calb $, so the
theorem follows from Proposition \ref{prop-poissonreduction} and the
uniqueness of solutions to the initial value problem.
\qed
It would be interesting to find weaker hypotheses under which the
conclusion of Theorem \ref{thm-functorial} is true; still, the theorem
is strong enough to yield a basic result on lagrangian reduction.
\begin{cor}
\label{cor-reduction}
Let $P\rightarrow M=P/G$ be a principal bundle. Then the quotient mapping
$\psi:TP\rightarrow TP/G$ is a morphism of Lie algebroids from the
tangent bundle of $P$ to the gauge algebroid. Consequently, if $K$ is
any $G$-invariant regular lagrangian on $TP$ and $L$ the reduced
lagrangian on $TP/G$, the solutions of Lagrange's equations for $K$
project to those for $L$.
\end{cor}
\pf
All that needs to be proven is that $\psi $ is a morphism of Lie
algebroids. The dual relation $\psi ^{*}:T^{*}P/G \rightarrow T^{*}P$
is just the inverse to the quotient mapping. On the other hand, as we
have already noted in Section \ref{subsec-algebroids}, the Poisson
structure on $T^{*}P/G$ considered as the dual of a gauge algebroid is
the ``Yang-Mills'' structure obtained when $T^{*}P/G$ is considered as
a quotient of $TP$. The mapping inverse to $\psi^{*} $ is therefore a
Poisson mapping, so its graph is coisotropic.
\qed
\begin{ex}
\label{ex-universal}
Let $\cala $ be the Lie algebroid of the Lie groupoid $\Gamma $ over $M$.
The operations of right translation define a bundle mapping $\psi
:\ker T\beta \rightarrow \cala $ which covers the target mapping
$\beta :\Gamma \rightarrow M$ and which is an isomorphism on
fibres. In fact, $\psi $ is a morphism of Lie algebroids, since the
dual comorphism $\psi ^{*}:\cala ^{*}\rightarrow (\ker T\beta )^{*}
$ is the composition (as relations) of the inverse of the restriction
map $T^{*}\Gamma \rightarrow (\ker T\beta )^{*} $ and the
Poisson map $\tilde{\alpha }:T^{*}\Gamma \rightarrow \cala ^{*}$ which
is the source map of $T^{*}\Gamma $ as a symplectic groupoid.
We can now apply Theorem \ref{thm-functorial} to conclude that, for
any regular lagrangian $L$ on $\cala $, the solutions of Lagrange's
equations are the image under $\psi $ of the solutions of Lagrange's
equations for the pulled back lagrangian on the tangent bundle to the
foliation of $\Gamma $ by its $\beta $-fibres.
\end{ex}
Finally, there is another corrected version of Fake Theorem
\ref{fakethm-functorial}, in which the conclusion is weakened.
\begin{thm}
\label{thm-weakened}
Let $\psi :\cala \rightarrow \calb $ be a morphism of Lie algebroids,
and let $L$ be a lagrangian on $\calb $. If the image $\psi\smalcirc
\sigma $ of a admissible path $\sigma$ in $\cala $ is a solution of
Lagrange's equations for $L$, then $\sigma $ itself is a solution of
Lagrange's equations for $L\smalcirc \psi.$
\end{thm}
\pf
It is convenient to identify the Lie algebroids with their duals via
the Legendre maps. The path $(\sigma (t), \psi (\sigma (t)))$ lies
in the graph of $\psi $; by Lemma \ref{lemma-functorial
legendre} it therefore lies in $\psi ^{*}$ as well. On the other
hand, if we choose a $t_{0}$ in the domain of $\sigma $, we can
consider the solution $\tau (t)$ of Lagrange's equations for
$L\smalcirc \psi $ with $\tau (t_{0})=\sigma (t_{0})$. The curve
$(\tau (t),\psi (\sigma (t))$ then lies in $\psi ^{*}$ by Proposition
\ref{prop-poissonreduction}.
Our proof will be complete if we can show that $\sigma =\tau $. To
see this, we note that $\sigma $ and $\tau $ are both admissible paths
with the same image under the relation $\psi ^{*}$. Note that the
typical element of
$\psi ^{*}$ is determined (see Equation \ref{eq-psistar}) by $m$ and
$\omega $. It follows that the typical element of $T\psi ^{*}$ is
determined by its ``components'' $\delta m$ and $\delta \omega $. If
this element consists of a pair of admissible tangent vectors, the
component $\delta m$ is in turn determined by $m$ and $\omega $.
Thus, there is a
unique second order differential equation on $\cala $ whose image
under $T\psi ^{*}$ is the lagrangian vector field of $L$; since $\sigma
$ and $\tau $ are solution curves of this equation with a common
point, they must be equal.
\qed
For instance, in Example \ref{ex-fake}, a geodesic in $M$ which
happens to lie in a leaf of the foliation is geodesic for the induced
metric on the leaf. Or, if one has any mapping $M\rightarrow N$ and a
riemannian metric on $N$, then if $M$ is given the pulled back
(possibly degenerate) metric, any path in $M$ which projects to a geodesic in
$N$ is a ``geodesic'' for the pulled back metric. On the other hand,
Theorem \ref{thm-weakened} does not encompass the fact that, for a
riemannian submersion $M\rightarrow N$, a {\em horizontal} path in $M$
is a geodesic if its image in $N$ is a geodesic. It would be nice to
understand this fact in the context of Lie algebroids.
\subsection{Lie groupoid morphisms and lagrangian reduction}
\label{subsec-groupoid morphisms}
The definition of groupoid morphism is a straightforward
generalization of that of a group homomorphism. In the language of
category theory, it is just a functor.
If $\Gamma $ and $\Delta $ are groupoids over $M$ and $N$
respectively, a {\bf groupoid morphism} from $\Gamma $ to $\Delta $
consists of maps $\Psi :\Gamma \rightarrow \Delta $ and
$\underline{\Psi}:M\rightarrow N$ which are
compatible with the source and target maps of the two groupoids,
such that $\Psi (g)\Psi (h)=\Psi (gh)$ whenever $(g,h) \in \Gamma
_{2}$.
Note that it follows already from the compatibility of source and
target maps that $(\Psi (g),\Psi (h))\in \Delta _{2}$ when $(g,h)\in
\Gamma _{2}$; we will denote the resulting map from $\Gamma _{2}$ to
$\Delta _{2}$ by $\Psi _{2}$. Compatibility with multiplication means
that the following diagram is commutative.
%\begin{equation}
%\label{diagram}
%\begin{array}{ccc}
%\Gamma_{2} &\stackrel{\Psi _{2}}{\longrightarrow}&\Delta _{2} \\
%m_{\Gamma } \downarrow \phantom{m_{\Gamma }}& &
%\phantom{m_{\Delta }}\downarrow m_{\Delta }\\
%\Gamma &\stackrel{\Psi }{\longrightarrow}&\Delta
%\end{array}
%\end{equation}
\begin{equation}
\begin{CD}
\Gamma_{2} @>\Psi _{2}>> \Delta _{2} \\
@Vm_{\Gamma }VV @Vm_{\Delta }VV \\
\Gamma @>\Psi>> \Delta
\end{CD}
\end{equation}
For Lie groupoids, one requires of course that the maps be differentiable.
It now follows that, if $\Psi _{2}(g,h)$ is a critical point of
$L_{2}$ along the fibres of $m_{\Delta }$, $(g,h)$ must be critical for
$L_{2} \smalcirc \Psi _{2}$ (which equals $(L\smalcirc \Psi )_{2}$ )
along the fibres of $m_{\Gamma }$; i.e. $\Psi _{2}^{-1}\Sigma _{L}
\subset \Sigma _{L\smalcirc \Psi }. $ We have thus proven the
following discrete analogue of Theorem \ref{thm-weakened}.
\begin{thm}
\label{thm-groupoidweakened}
Let $\Psi :\Gamma \rightarrow \Delta $ be a morphism of Lie groupoids,
and let $L$ be a lagrangian on $\Delta $. If the image of an
admissible sequence
in $\Gamma $ is a solution of Lagrange's equations for $L$, then that
sequence is a solution of Lagrange's equations for $L\smalcirc \Psi $.
\end{thm}
There is also a discrete analogue of Theorem \ref{thm-functorial}.
Its proof rests on the ideas preceding Theorem
\ref{thm-groupoidweakened} and the fact that an isomorphism on $\alpha
$-fibres gives rise to an isomorphism on fibres of $m$.
\begin{thm}
\label{thm-groupoid functorial}
Let $\Psi :\Gamma \rightarrow \Delta $ be a morphism of Lie groupoids
and let $L$ be a lagrangian on $\Delta $. If $\Psi $ is a
local diffeomorphism when restricted to $\alpha $-fibres (or,
equivalently, to $\beta$-fibres), then the image under $\Psi $ of any
solution of Lagrange's equations for $L\smalcirc \Psi $ is a solution
of Lagrange's equations for $L$.
\end{thm}
\begin{ex}
\label{ex-group reduction}
If $G$ is a group, the map $\Psi :(g,h)\mapsto gh^{-1}$ is a morphism
from the pair groupoid $G\times G$ to the group $G$. A lagrangian
$K$ on $G\times G$ is the pullback of a lagrangian $L$ on $G$ if and
only if it is invariant under the diagonal action of $G$ on $G\times
G$ by right translations. When this is the case, the image under
$\Psi $ of the solutions of Lagrange's equations for
$K(g,h)=L(gh^{-1})$ are the solutions of Lagrange's equations for $L$.
This is exactly the discrete analogue of the passage to the Euler
equations for a right invariant lagrangian system on $TG$; it is a
basic construction in \cite{mo-ve:discrete}.
\end{ex}
\begin{ex}
\label{ex-gauge}
Let $P$ be a principal $G$-bundle over $M$, $\Gamma =(P\times P)/G$
its gauge groupoid. The quotient map from the pair groupoid
$P\times P$ to $\Gamma $ is a Lie groupoid morphism which is a
diffeomorphism on $\alpha $-fibres, so there is a correspondence
between $G$-invariant lagrangian systems on $P\times P$ and lagrangian
systems on $\Gamma $. A particular example of this setup occurs in
\cite{mo-ve:discrete}. There, $P$ is the Stiefel manifold of
$n$-frames in $R^{n+N}$, and $G$ is $O(n)$. Invariant systems on
$P\times P$ can be reduced to systems on a gauge groupoid over $P/G$,
which is a Grassmann manifold.
\end{ex}
\section{Variational principles}
\label{sec-variational}
At this point in our exposition, it is easier to treat the
case of groupoids before that of algebroids.
\subsection{The case of groupoids}
\label{subsec-groupoid variational}
It is quite simple to show that the solutions of Lagrange's equations
are the extremals for a variational principle. To characterize the solutions
among the admissible sequences defined on a given interval $[a,b]$, we
consider the functional $\call (g_{.})=\sum _{j=a}^{b}L(g_{j})$. The
example of a group, in which all sequences are admissible, shows that
we need to impose a supplementary condition. The case $b=a+1$ already
suggests the condition in the following theorem.
\begin{thm}
\label{thm-variational groupoid}
Given integers $a < b$ and an element $g$ of the groupoid $\Gamma
$, let $\cals_{g} $ denote the set of admissible sequences
$g_{a},\ldots, g_{b}$ with values in $\Gamma $ such that $g_{a}\cdots
g_{b}=g$. Then $\cals_{g} $ is a submanifold of $\Gamma ^{b-a+1}$,
and the critical points of the functional $\call (g_{.})=\sum
_{j=a}^{b}L(g_{j})$ on $\cals_{g} $ are precisely those elements of
$\cals_{g} $ which satisfy Lagrange's equations.
\end{thm}
\pf
We begin by observing that $\cals _{g}$ can be identified via the map
$$(g_{a},\ldots g_{b})\mapsto
(g_{a}\cdots g_{b},g_{a+1}\cdots g_{b},\ldots, g_{b-1}g_{b},g_{b},\beta
(g_{b}))$$
with the set of sequences $(h_{a},\ldots h_{b+1})$ in
$\calf ^{\beta }(g)$ which satisfy the boundary conditions
$h_{a}=g$ and $h_{b+1}=\beta (g)$. (Compare Remark
\ref{rmk-diffeo}.) Thus, $\cals _{g}$ is essentially a product
manifold on which $(h_{a},\ldots ,h_{b-1})$ can be taken as
``independent variables''. Setting to zero the derivative of $\call
$ with respect to each of these variables gives precisely Lagrange's
equations.
\qed
\begin{ex}
\label{ex-variational pair}
When $\Gamma $ is a pair groupoid $M\times M$, the functional $\call $
can be written in terms of sequences in $M$ as $\sum
_{j}L(x_{j-1},x_{j})$. The boundary condition is that the endpoints
of the sequence should be fixed. This example occurs already in
\cite{mo-ve:discrete}.
\end{ex}
\subsection{The case of Lie algebroids}
\label{subsec-variational algebroids}
Given a lagrangian $L:\cala \rightarrow \reals $, it is natural to
consider the functional $ {\script L}:\sigma \mapsto \int L(\sigma
(t)dt $ restricted to admissible curves. For instance, if $\cala
=TM$, this is the usual restriction to curves in $TM$ which are
tangent lifts of curves in $M$.
The example of the tangent bundle shows that we must impose a boundary
condition before we can identify the critical points of ${\script L}$
with solutions of Lagrange's equations. The most appropriate boundary
condition in this case is a pair of points in $M$, which can be
considered as an element of the pair groupoid $M\times M$. Another
example is furnished by a Lie algebra $\frakg $. Since any path in
$\frakg $ is admissible, the critical points of ${\script L}$ are just
the paths with values in the critical point set of $L$. The fact that
these are ``wrong'' goes back at least to Lin \cite{li:hydrodynamics}, who
introduced what are now known as ``Lin constraints''. If we think of
the lagrangian on $\frakg $ as being reduced from a
translation invariant lagrangian on $TG$, then the natural boundary
values are points in $G$. By translation invariance, this pair of
points can be reduced to its quotient, a single point in $G$.
To formulate the boundary conditions in general, we need to lift paths
in $\cala $ to paths in a Lie groupoid $\Gamma $ whose Lie algebroid is
$\cala $. Given a path $\zeta (t)$ in $\Gamma $ for which $\beta
\smalcirc \zeta $ is constant, we may form the right-translated tangent
vectors $\sigma (t) = Tr_{\zeta (t)}^{-1} \zeta '(t)$, which
determine an admissible path in $\cala $. Conversely, one can show
(by integrating an appropriate time-dependent vector field), that
every admissible path $\sigma :[a,b]\rightarrow \cala $ arises in this
way from a unique
path $\zeta $ in $\Gamma $ for which $\zeta (a)$ is the identity
element $\pr _{\cala }(\sigma (a))$. We call $\zeta$ the {\bf development} of
$\sigma $ in $\Gamma $.
The following theorem is the continuous analogue of Theorem
\ref{thm-variational groupoid}. It is probably related to
Montgomery's theorem \cite{mo:isoholonomic} that the shortest paths
with prescribed holonomy come from solutions of the equations of
motion of a particle in a Yang-Mills field, but we have not been able
to find a direct connection.
\begin{thm}
\label{thm-variational}
Let $L$ be a regular lagrangian on the Lie algebroid $\cala $, and
let $g$ be an element of a Lie groupoid $\Gamma$ whose Lie algebroid
is $\cala $. The critical points of the functional $\sigma \mapsto
\int L(\sigma (t))dt$ on the space of admissible paths whose
development begins at $\beta (g)$ and ends at $g$ are precisely those
elements of that space which satisfy Lagrange's equations.
\end{thm}
\pf
The statement of the theorem is well known to be true if $\cala $ is a
tangent bundle, and the same argument, leaf by leaf, shows that it is
true if $\cala $ is the tangent bundle to a foliation. We can reduce
the general case to this one by using reduction.
Consider the morphism of Lie algebroids $\Psi
:\ker T\beta \rightarrow \cala $ discussed in Example
\ref{ex-universal}. The tangent lifts of admissible paths in $\Gamma
$ starting at the identity section project onto the admissible
paths in $\cala $. This shows that the extremals of the variational
problems for $L$ on $\cala $ and $L\smalcirc \psi $ on $\ker T\beta $
correspond under $\psi$. On the other hand, we saw in
Example \ref{ex-universal} that the solutions to Lagrange's equations
also correspond under $\psi$. Since the solutions and the extremals
coincide for $\ker T\beta $, they must coincide for $\cala $ as well.
\qed
It would be nice to have a proof of Theorem \ref{thm-variational}
like the very simple proof of its discrete analogue, in which a
variational problem for sequences is localized to a variational problem
for pairs. The continuous version of such a proof\footnote{Hints for
this continuous version may lie in Euler's original
derivation of the ``Euler-Lagrange equations'' as described, for
instance, in Section 2.2 of \cite{go:history}.} should somehow
consider the values of an admissible path in $\cala $ as ``independent
variables'' and should lead to Lagrange's equations as the constrained
critical point equations for a function S on the affine
$T_{\operatorname{adm}}\cala $ of $T\cala $ consisting of the
admissible tangent vectors. The values of the
lagrangian vector field should then be the critical points of $S$
along the fibres of the bundle projection $T_{\operatorname{adm}}\cala
\rightarrow \cala .$
We know {\em a posteriori} that such a function $S$ exists, at least
when $L$ is regular. In this case, we can take $S$ to be the
affine-quadratic function whose restriction to each fibre of
$T_{\operatorname{adm}\cala }$ has a critical point at the value of
the lagrangian vector field and whose (constant) hessian equals the
fibre-hessian of $L$ at the basepoint in $\cala $. For instance, when
$\cala =TM$, we have in coordinates
$$ S(q,\dot{q},\ddot{q})=\half \frac{\del ^{2}L}{\del
\dot{q}^{2}}\ddot{q}^{2}+\frac{\del E}{\del q}\ddot{q}-\half
(\frac{\del E}{\del q})^{2}/\frac{\del ^{2}L}{\del \dot{q}^{2}}
, $$ where $E=\dot{q}\frac{\del L}{\del \dot{q}}-L.$
Setting to zero the derivative of this function with respect to
$\ddot{q}$ gives Lagrange's {\em equation} (rather than its solutions).
The constant term must be included in order for $S$ to be independent
of the choice of coordinates.
What would be of real interest would be to have an intrinsic
construction of $S$ as a limit of the function $L(x,y)+L(y,z)$ in the
discrete case, as the triple $(x,y,z)$ ``approaches'' an admissible
tangent vector.
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\end{document}