\documentstyle[leqno,amssymb]{article}
%
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}{Definition}[section]
\newtheorem{remark}{Remark}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{proposition}{Proposition}[section]
\renewcommand{\Re}{\Bbb R}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\begin{document}
\title{Harmonic Legendre and Hamilton transformations}
\author{M.Trinidad P\'erez \and M.E. V\'azquez--Abal\thanks{Supported by
project XUGA 20702B96}}
\maketitle
\begin{abstract}
Lagrangians and Hamiltonians in (pseudo-)Riemannian manifolds
in\-du\-cing isometric, totally geodesic and harmonic Legendre and Hamilton
transformations are completely characterized. Physical examples of the
different
situations are given.
\end{abstract}
\section{Introduction}
The tangent bundle of a manifold $M$ reflect some
interesting properties of the base manifold and is frequently used
to study the manifold itself. When $M$ is equipped with a
(pseudo-)Riemannian metric $g$,
its complete lift, $g^C$, defines a pseudo-Riemannian metric
on $TM$. This metric has been used in the study of harmonic maps, showing
some nice properties. For example, a smooth map is harmonic (resp.
totally geodesic) if and only if its differential is a harmonic (resp. totally
geodesic) map between the tangent bundles with their corresponding
complete lift metrics \cite{MEVA}.
Any symmetric connection $D$ on $M$ induces a pseudo-Riemannian metric on
the cotangent bundle
$T^{*}M$. This induced metric, called the Riemannian extension $g_{D}$ of
$D$, presents some
interesting properties in the study of the curvature and has been used to
analyze the harmonicity of
one forms.
For a given Lagrangian $L$ on a pseudo-Riemannian manifold the induced
Le\-gen\-dre
transformation is a map:
$FL:(TM,g^C){\longrightarrow}(T^{*}M,g_{\nabla})$, where $g_{\nabla}$
denotes the Riemann
extension of the Levi-Civita connection ${\nabla}$ of $(M,g)$. Therefore,
it seemed to be interesting
to investigate these Lagrangians inducing harmonic maps from $(TM,g^C)$ to
$(T^{*}M,g_{\nabla})$.
The purpose of this paper is to characterize Lagrangians on a
pseudo-Rie\-man\-nian manifold where
asso\-cia\-ted Legendre transformations are harmonic (Theorem \ref{th:ha}),
totally geodesic (Theorem \ref{th:tx}) and
isometric (Theorem \ref{th:isol}) between the tangent and cotangent
bundle. It turns out that the musical isomorphism, viewed as the Legendre
transformation
induced by the Lagrangian $\frac{1}{2}\parallel{x}\parallel$ are isometries
and moreover, all
Lagrangians with isometric Legendre transformations are described as well
as those corresponding
to totally geodesic Legendre transformations.
Note here that physical examples of the different situations investigated
are those corresponding
to different mathematical models of the simple and sphe\-ri\-cal pendulum
and the Atwood´s machine.
Finally, the dual situation co\-rres\-pon\-ding to Hamiltonians and their
asso\-cia\-ted Hamilton
transformations is considered on the last section.
\section{Preliminaries}
In this section we collect some basic material that we will need further on.
\subsection{Harmonic maps}
Let $(M,g)$ and $(N,h)$ be Riemannian (or pseudo-Riemannian) manifolds
with dim$M$ = $m$ and dim$N$ = $n$ and let $f:(M,g)$ $\rightarrow$ $(N,h)$ be
a smooth map from $M$ to $N$. The Levi-Civita connections on $TM$ and $TN$
induce a connection in the bundle of one-forms on $M$ with values in
the pull-back bundle $f^{-1}(TN)$. The covariant derivative of the differential
$\nabla df$ is called the second fundamental form of $f$, and the section
${\tau}(f) = trace \nabla df$ is called the tension field of $f$.
$f$ is said to be {\it harmonic} if $\tau(f) = 0$, and {\it totally
geodesic} if
${\nabla}df = 0$. (See \cite{EL}
for more details and references.)
Now, let $U\subset M$ and $V\subset N$ be domains with coordinates
$(q^1,\dots,q^m)$ and
$(y^1,\dots,y^m)$ respectively, such that
$f(U)\subset V$. Locally, the map $f$ has the representation:
$y^a$ $=$ $f^a(q^1,\dots,q^m)$. Then the second fundamental form and the
tension field at
$q \in U$ can be locally expressed by the following:
\begin{equation}\label{sff}
\nabla(df)^a_{ij}=
\frac{\partial^2f^a}{\partial q^i\partial q^j}-
{}^g\Gamma^k_{ij}\frac{\partial f^a}{\partial q^k}+
{}^h\Gamma^a_{bc}(f)[\frac{\partial f^b}{\partial q^i}
\frac{\partial f^c}{\partial q^j}],
\end{equation}
\begin{equation}\label{tensi}
\tau(f)^a=g^{ij}(\nabla(df)^a_{ij}),
\end{equation}
for $i,j,k=1,\dots,m;\,a,b,c=1,\dots,n$.
\subsection{The complete lift metric on $TM$}
Let $(M,g)$ be a pseudo-Riemannian $m$-dimensional manifold with
Levi-Civita connection
${\nabla}$. Let $TM$ denote the tangent bundle of $M$ with projection
${\pi}_{TM}: TM {\rightarrow} M $. This $2m$-dimensional manifold can be
equipped with the
pseudo-Riemannian complete lift metric $g^C$, of signature $(m,m)$, defined by
\[\left\{\begin{array}{l}
g^C(X^H,Y^H)=g^C(X^V,Y^V)=0,\\
g^C(X^H,Y^V)=g^C(X^V,Y^H)=g(X,Y)^V.
\end{array}
\right.\]
Here, the horizontal and vertical lifts of tangent vector fields $X, Y$ on
$M$ refer to the
decomposition of the tangent space, $TM$, at every point in horizontal
vectors with respect to
${\nabla}$ and canonical vertical vectors. For vector fields $X, Y$ on $M$,
the function
$g(X,Y)^V$ on $M$ is the pull-back of $g(X,Y)$ along the projection
${\pi}_{TM}$.
We shall use the following notation: $\cal U$ $=$ $\{ U, (q^i), i=1,\dots,m
\}$
denotes a coordinate neighborhood on the $m$-dimensional manifold $M$, and
$T\cal U$ $=$
$\{{\pi}_{TM}^{-1}({\cal U}),\\ (q^1,\dots,q^m;\dot{q}^1,\dots, \dot{q}^m)\}$
denotes a coordinate neighborhood on $TM$ induced by $U$ and the projection
${\pi}_{TM}: TM {\rightarrow} M $. The local expression of $g^C$ is then
given by
\begin{equation}\label{eq:C}
g^C=\left(\begin{array}{cc} \dot{q}^k\frac{\partial g_{ij}} {\partial q^k}
& g_{ij}\\
\noalign{\medskip}
g_{ij} & 0
\end{array}\right),
\end{equation}
$i,j = 1, \dots, m$, with respect to the basis
$
\{\frac{\partial }{\partial q^1},\dots,\frac{\partial}{\partial q^m};
\frac{\partial}{\partial \dot{q}^1},\dots,\frac{\partial}{\partial
\dot{q}^m}\},
$
where $g_{ij}$ denote the local components of $g$
with respect to
$
\{\frac{\partial}{\partial q^1},\dots,\frac{\partial}{\partial q^m}\}$.
The Christoffel symbols,
$
{}^{TM}{\Gamma}^{\gamma}_{{\alpha}{\beta}},\;$
of the Levi-Civita connection of $g^C$ are given by \cite{YI}
\begin{equation}\label{eq:CTM}
{}^{TM}{\Gamma}^k=\left(\begin{array}{cc}
{\Gamma}^k_{ij} & 0\\
\noalign{\medskip}
0 & 0
\end{array}\right), \quad
{}^{TM}{\Gamma}^{\bar{k}}=\left(\begin{array}{cc}
\dot{q}^l\frac{\partial \Gamma^k_{ij}}{\partial q^l} & \Gamma^k_{ij}\\
\noalign{\medskip}
{\Gamma}^k_{ij} & 0
\end{array}\right),
\end{equation}
where ${\Gamma}^k_{ij}$ are the Christoffel symbols of ${\nabla}$ and
$
\bar{k} = k+m$ . (We are considering $\frac{\partial}{\partial
\dot{q}^{k}}$ as the
$\bar{k}=(k+m)$-th element from the basis of $T_{(q,\dot{q})}(TM)$.)
\subsection{ The Riemannian extension to $T^*M$}
Let $T^*M$ denote the cotangent bundle of $M$. We shall use the following
notation:
$\cal U$ $=$ $\{ U, (q^i), i=1,\dots,m \}$ denotes a coordinate
neighborhood on
the $m$-dimensional manifold $M$.
$T^*\cal U$ $=$
$\{{\pi}_{T^*M}^{-1}({\cal U}), (q^1,\dots,q^m;p^1,\dots, p^m)\}$
denotes a coordinate neighborhood on $TM$ induced by $U$ and the projection
${\pi}_{T^*M}: T^*M {\rightarrow} M $.
For a given symmetric
connection $D$ on $M$, the cotangent bundle, $T^*M$, can be equipped with a
pseudo-Rie\-man\-nian
metric of signature $(m,m)$, called the {\it Rie\-man\-nian extension} of
$D$, and denoted by
$g_D$.
This pseudo-Riemannian metric is characterized by
\[\left\{\begin{array}{l}\label{eq:rie1}
g_D(X^C,Y^C)=-\gamma (D_XY + D_YX),\\
g_{D}({\omega}^{V},X^{C})=({\omega}(X))^{V},\quad
g_{D}({\omega}^{V},{\theta}^{V})=0,
\end{array}
\right.\]
{\noindent}where $X^C$, $Y^C$ are the complete lifts to $T^*M$ of vector
fields $X$,
$Y$ on $M$. For any vector field $Z$ on $M$, $(Z$ $=$
$Z^i\frac{\partial}{\partial q^i})$,
$\gamma Z$ is the function on $T^*M$ defined by
$\gamma Z$ $=$ $p_iZ^i$, and ${\omega}^{V}, {\theta}^{V}$ are the vertical
lifts to $T^*M$ of 1-forms ${\omega},{\theta}$ on $M$. $({\omega}(X))^{V}$ is
the vertical lift to $T^*M$ of the function ${\omega}(X)$ on $M$.\\
With respect to the basis
$
\{\frac{\partial }{\partial q^1},\dots,\frac{\partial}{\partial q^m};
\frac{\partial}{\partial p_1},\dots,\frac{\partial}{\partial p_m}\},
$
the local components of the Riemannian extension and the Christoffel symbols
of the Levi-Civita connection of $g_D$,
$
{}^{T^*M}{\Gamma}^{\gamma}_{{\alpha}{\beta}},\;
$,
are given by \cite{YI}
\begin {equation}\label{eq:Ex}
(g_{D})=\left(\begin{array}{cc}
-2p_k^{D}{\Gamma}_{ij}^k & \delta_{ij}\\
\noalign{\medskip}
\delta_{ij} & 0
\end{array}\right),
\end{equation} and
\begin{equation}\label{eq:CT*M}
{}^{{T^{\ast}}M}\Gamma^k=\left(\begin{array}{cc}
\Gamma^k_{ij} & 0\\
0 & 0
\end{array}\right),\quad
{}^{{T^{\ast}}M}\Gamma^{\bar{k}}=\left(\begin{array}{cc}
p_l\,{}^{k} \, A^l_{ij} & -\Gamma^i_{kj}\\
-{\Gamma}^j_{ik} & 0
\end{array}\right),
\end{equation}
{\noindent}respectively, where $\Gamma^k_{ij}$ are the Christoffel symbols of
$\nabla$, $\bar{k}=k+m$ (where as before we consider
$\frac{\partial}{\partial p_k}$
as the $\bar{k}=(k+m)$-th element from the basis of
$T_{(q,p)}(T^{\ast}M))$, and
$$
{}^kA^l_{ij} = \frac{\partial {\Gamma}^l_{ij}}{\partial q^k}-
\frac{\partial {\Gamma}^l_{ik}}{\partial q^j}-
\frac{\partial {\Gamma}^l_{kj}}{\partial q^i}+
2{\Gamma}^l_{kt}{\Gamma}^t_{ij},\;\ i,j,k,l,t=1,\dots,m.
$$
For a given (pseudo-)Riemannian manifold $(M,g)$, let $g_{\nabla}$ denote
the Riemannian extension of the Levi-Civita connection. (See \cite{YI} for
more details.)
\section{Legendre transformation}
In the context of mechanics, a differentiable map $L : TM {\rightarrow}
{\Re}$ from a manifold $M$ to $\Re$
is called a {\it Lagrangian}. The {\it transformation of Legendre}, $FL:TM
\longrightarrow T^{\ast}M $, is
the fibre derivative of
the Lagrangian $L$,
\begin{eqnarray*}
X_{q} \mapsto DL_q({X}_{q}) \in {\cal L}(T_qM,{\Re}) = T^{\ast}_qM,
\end{eqnarray*}
where $L_q$ denotes the restriction of $L$ to the fibre over $q \in M$.
Note that $L$ is not
neccesarily an homomorphism of fibred spaces, however it is a
differentiable map that preserves the fibres.
The Lagrangian $L$ is said to be {\it regular} if $FL$ is a regular map at
every point,
and regular Lagrangians
correspond to those whose transformations of Legendre are local
diffeomorphisms from $TM$ to $T^*M$.
If $FL$ is a diffeomorphism, then $L$ is a {\it hyperregular
Lagrangian}. (See \cite{RJ} for more details.)
From now on let $(M,g)$ be a (pseudo-)Riemannian manifold. In this case the
trans\-for\-ma\-tion of
Legendre can be considered as a map between the pseudo-Riemannian manifolds:
$$
FL: (TM,g^C) {\rightarrow} (T^{\ast}M,g_{\nabla}).
$$
A special example of a Legendre transformation is the musical isomorphism
${\flat}: TM \rightarrow T^{\ast}M$, which is the diffeomorphism defined by
${\flat}(X) = g(X,.)$
Next we will characterize the Lagrangians for which $FL$ is an isometry,
totally geodesic or
harmonic.
\begin{theorem}\label{th:isol}
The transformation of Legendre
$FL : (TM,g^C) \rightarrow\- (T^{\ast}M,\-g_{\nabla})$ associated to a
regular Lagrangian $L$, is a local isometry if and only if:
$$
L(q,\dot{q}) = \frac{1}{2}g(\dot{q},\dot{q}) + \varphi(q),\ \dot{q} \in T_qM,
$$
{\noindent}where
${\varphi} : M \rightarrow {\Re}$
is a differentiable map.
\end{theorem}
\noindent{\normalsize\it Proof.} For a given Lagrangian, the associated
Legendre transformation
$FL$ is a local isometry if and
only if
\begin{equation}\label{eq:iso}
(FL)^*g_{\nabla} = g^C.
\end{equation}
{\noindent}Considering the coordinate neighborhoods ${\cal U}$, $T{\cal U}$
and $T^{\ast}{\cal U}$
described in the previous section and the local expression of the Legendre
transformation,
$FL(q^i,\dot{q}^i) = (q^i, \frac{\partial L}{\partial \dot{q}^i})$, the
isometry condition (\ref{eq:iso})
amounts to the following system of equations:
\begin{equation}\label{eq:l}
\left\{ \begin{array}{l}
\displaystyle{\frac{\partial^2 L}{\partial \dot{q}^s \partial \dot{q}^r} =
g_{rs}}\\
\noalign{\medskip}
\displaystyle{\frac{\partial^2 L}{\partial q^r \partial \dot{q}^s}
+\frac{\partial^2 L}{\partial \dot{q}^r \partial q^s}
-2{\Gamma}^k_{sr}
\frac{\partial L}{\partial \dot{q}^k} = \dot{q}^t
\frac{\partial g_{rs}}{\partial q^t}}.
\end{array}
\right.
\end{equation}
{\noindent}Now, it immediately follows that $L(q,\dot{q}) =
\frac{1}{2}g(\dot{q},\dot{q}) + {\varphi}(q)$
satisfies the equations (\ref{eq:l}), and thus it induces a locally
isometric Legendre transformation, for any arbitrary
function ${\varphi}$ on $M$.
Conversely, suppose that $FL$ is a local isometry. Then from the first
condition in (\ref{eq:l}) it follows that:
$$
L(q,\dot{q})=\frac{1}{2}g_{rs}\dot{q}^r\dot{q}^s + {\phi}_r\dot{q}^r
+{\varphi}(q)\\,
$$
where ${\phi}_r$ defines a 1-form on $M$ and ${\varphi}$ is an arbitrary
function on $M$
Now, a straightforward calculation from the second equation in
(\ref{eq:l}) shows that ${\phi}_r$
vanishes and the desired result is obtained.
$\hfill\Box$
\vspace{2ex}
As an inmediate application of the previous theorem, we obtain
\vspace{2ex}
\begin{corollary} The musical isomorphism ${\flat}: X \rightarrow
{\flat}(X) = g(X,.)$
is an isometry beetwen
$(TM,g^C)$ and $(T^{\ast}M,\-g_{\nabla})$.
\end{corollary}
\noindent{\normalsize\bf Proof.} The musical isomorphism ${\flat}: X
\rightarrow {\flat}(X) = g(X,.)$
is the transformation of Legendre associated to the Lagrangian
$L(q,\dot{q})=\frac{1}{2}g_{rs}\dot{q}^r\dot{q}^s$.
$\hfill\Box$
\vspace{2ex}
Note that this corollary is a generalization of the totally geodesic
character of ${\flat}$, which was stablished
in \cite{G}.
\vspace{2ex}
A {\it mathematical model} $(M,g_{T})$ for a system of particles moving
in ${\Re}^3$ with holonomic constraints consist on the following
ingredients: A differentiable manifold $M \hookrightarrow
{\Re}^3$ as {\it configuration space}, the tangent bundle $TM$, {\it space
of the velocities},
and a Riemannian metric $g_T$ in $M$, {\it metric of kinetic energy}.
\vspace{2ex}
\begin{corollary}\label{th:mec}
Let $(M,g_T)$ be a mathematical model for a system of particles moving in
${\Re}^3$ with holonomic constraints and let $V : M \rightarrow {\Re}$ be a
potential
function defined on $M$. Then, the kinetic energy Lagrangian $L = T-(V
\circ {\pi}_{TM})$
have isometric Legendre transformation.
\end{corollary}
\noindent{\normalsize\it Proof.} Note that the kinetic energy Lagrangian is
a hyperregular
Lagrangian. Then the result follows from Theorem \ref{th:isol} and the
definition of kinetic energy.
$\hfill\Box$
\vspace{2ex}
Next we will illustrate this result by showing examples motivated by some
mathematical models.
\vspace{2ex}
{\it Example 3.1}
{\noindent}Let us consider a mechanical system of a {\it Spherical
Pendulum}, that is,
a particle of mass $m$ restricted to move under the effect of gravity on
a sphere without friction. The constraints can be obtained, for example,
putting the
particle at one end of a light bar while the other end of the bar
remains fixed.
The sphere on which the particle moves is $S^2$, and consider the immersion
given by:
$$
i:(q^1,q^2) \in S^2 \hookrightarrow (\sin q^1 \cos q^2, \sin q^1 \sin q^2,
\cos q^1) \in {\Re}^3,\\
$$
The metric of the kinetic energy in ${\Re}^3$, in terms of the
identity chart of ${\Re}^3$ is
$\tilde{g}_T = m[(dx)^2+(dy)^2+(dz)^2]$
and its pull-back gives us a metric in $S^2$:
$
(i^* \tilde{g}_T) = g_T = m[(dq^1)^2 + {\sin}^2 q^1 (dq^2)^2].$
The kinetic and potential energies are given by
$
V(q) = {\bold g} \,{\cos} \, q^1
$
and
$
T(q,\dot{q}) = \frac{1}{2}m[(\dot{q}^1)^2 + {\sin}^2 q^1 (\dot{q}_2)^2],
$
respectively,
{\noindent}where ${\bold g}$ denotes the constant of gravitation. Next,
the kinetic energy Lagrangian, becomes
\begin{equation}\label{eq:lt}
L_T(q,\dot{q}) = T-(V \circ {\pi}_{TM}) (q,\dot{q})= \frac{1}{2}
m[(\dot{q}^1)^2 + {\sin}^2 q^1 (\dot{q}_2)^2] -
{\bold g} \cos q^1.
\end{equation}
\vspace{2ex}
{\it Example 3.2}
{\noindent}Consider now the classic {\it Simple Pendulum}. Here $M$ is
$S^1$ and each point is
determined by a coordinate $q$. The metric of the kinetic energy on $S^1$ is
$
g_T = i^*\tilde{g_T} = ml^2dq \otimes dq,
$
and the kinetic energy Lagrangian becomes
\begin{eqnarray*}
L_T(q,\dot{q}) = [T-(V \circ {\pi}_{TM})] (q,\dot{q})
= \frac{1}{2}ml^2(\dot{q})^2 + m {\bold g} l \, \cos q.
\end{eqnarray*}
\vspace{2ex}
{\it Example 3.3}
{\noindent}The {\it Atwood's Machine} consists of two masses $\mu_1$ and
$\mu_2$ joined by a string which passes through
a pulley without friction.
We consider ${\Re}^3$ with the $z$ axis "outwards". The configuration space
is $M = \{(-a,q,0,a,l-{\pi}a-q,0)
\in {\Re}^6 ; 0