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\begin{document}
\title{ \vspace{-1.6cm}
\hfill {\normalsize DSMA-TS QM2 289} \\
\hfill {\normalsize FT/UCM/25-92} \\
~\\
On the inverse problem of Lagrangian supermechanics}
\author{L. A. Ibort$\dag$, G. Landi$\ddag ^*$,\\
J. Mar\'{\i}n-Solano$\dag$, G. Marmo$\S ^*$ }
\date{\today }
\maketitle
\medskip
\medskip
\parindent 0cm
\dag Dpto. de F\'{\i}sica Te\'orica. Universidad Complutense, 28040
Madrid, Spain.
\ddag Dpto. di Scienze Matematiche, Universit\`a di Trieste, Pl.
Europa 1, I-34100 Trieste, Italy.
\S Dpto. di Scienze Fisiche, Universit\`a di Napoli, Mostra
d'Oltremare, Pad. 19, I-80125 Napoli, Italy.
$^*$INFN, Sezione di Napoli, Mostra
d'Oltremare, Pad. 20, I-80125 Napoli, Italy.
\parindent .6cm
\medskip
\medskip
\begin{abstract} The
inverse problem for Lagrangian supermechanics is investigated. A
set of necessary and sufficient conditions for a system of second
order differential equations in superspace to derive from (a
possibly non regular) superlagrangian function are
given. The harmonic superoscillator is revisited and a family of
even and odd alternative superlagrangians are constructed for it.
Finally, we comment on the existence of recursion operators.
\end{abstract}
\medskip
PACS: 11.30.Pb.
\medskip
\newpage
\section{Introduction}
The inverse problem of the calculus of variations has attracted
a lot of attention over the years (for a recent review see
\cite{Mo90}). An elegant description of
necessary and sufficient conditions for a second order
differential equation to derive from a Lagrangian function were
given in \cite{Ba80} \cite{He82} \cite{Cr81}. These conditions
encode in a neat geometrical way the well-known Helmhotz
conditions for the existence of a local Lagrangian function. A
similar programme can be carried out in the setting of
supermechanics (or pseudomechanics). One of the difficulties
encountered in trying to formulate precisely the inverse
problem in this context, is to establish the correct notions of
second order differential equations and an intrinsic derivation
of Euler-Lagrange equations from a superlagrangian in such a way
that the ordinary conditions can be easily superized.
Such a geometrical foundation has been established recently
\cite{Ib92} and in this geometrical setting the inverse problem
of Lagrangian supermechanics acquires a structure similar to the
inverse problem in ordinary Lagrangian mechanics. One of the main
sources of interest for addressing the inverse problem lies in
the already well-known link between (super) integrability and the
existence of alternative Lagrangians \cite{Cr83}.
New features emerge in the
context of supermechanics \cite{La92} as we
will discuss later and the existence of alternative
superlagrangians is not always related to a recursion operator.
The simple example of the harmonic superoscillator is revisited.
Its linear supersymmetries and a whole family of alternative
superlagrangians, both even and odd, are described.
\section{The Inverse Problem of Lagrangian Supermechanics}
Let $L$ be a superlagrangian defined on the tangent
supermanifold of a superspace with local
supercoordinates $(q^a ,\theta^\alpha )$, $a=1,\cdots,n$ and
$\alpha =1,\cdots,m$.
The configuration superspace with local supercoordinates
$(q^a ,\theta^\alpha )$ can be thought to be a $n|m$
dimensional ${\cal G}$-supermanifold with superalgebra $B_L$, the
real Grassmann algebra with $L$ generators \cite{Ba91}. The
supermanifold is defined by ${\cal G}$-smooth transition
superfunctions between superdomains
\begin{eqnarray}\label{cocycle}
\acute{q}^a & = &\Phi^a (q,\theta ) \nonumber \\
\acute{\theta}^\alpha & = & \Psi^{\alpha} (q,\theta )
\end{eqnarray}
and the tangent supermanifold is the ${\cal G}$-supermanifold
with local supercoordinates $(q^a ,\dot{q}^a ,\theta^\alpha
,\dot{\theta}^\alpha )$ transforming by means of the tangent
cocycle
\begin{eqnarray}
\acute{q}^a & = & \Phi^a (q,\theta );~~~
\dot{q}^{\prime a} = \frac{\partial \Phi^a}{\partial q^b}
\dot{q}^b + \frac{\partial\Phi^a}{\partial\theta^\alpha}
\dot{\theta}^\alpha \nonumber\\
\acute{\theta}^\alpha & = & \Psi^{\alpha}(q,\theta );~~~
\dot{\theta}^{\prime \alpha} =
\frac{\partial\Psi^\alpha}{\partial q^a}\dot{q}^a +
\frac{\partial\Psi^\alpha}{\partial\theta^\beta}\dot{\theta}^\beta .
\end{eqnarray}
In many particular instances it is enough to
consider configuration superspaces which are graded manifolds
$(Q,{\cal A}_Q )$ in the sense of Kostant \cite{Ko77}. In such case
the cocycle (\ref{cocycle}) defining the supermanifold can be
drastically simplified \cite{Ba79} and takes the simpler form
\begin{eqnarray} \acute{q}^a & = &\phi^a (q) \nonumber \\
\acute{\theta}^\alpha & = & \psi^{\alpha}_\beta (q)\theta^\beta
\end{eqnarray}
and the tangent supermanifold $(TQ,T{\cal A}_Q )$ is
again a graded supermanifold defined by the tangent cocycle \cite{Ib92}:
\begin{eqnarray}
\acute{q}^a & = & \phi^a (q);~~~
\dot{q}^{\prime a} = \frac{\partial\phi^a}{\partial q^b}\dot{q}^b\nonumber \\
\acute{\theta}^\alpha & = & \psi^{\alpha}_\beta
(q)\theta^\beta;~~~ \dot{\theta}^{\prime\alpha} =
\psi^{\alpha}_\beta (q) \dot{\theta}^\beta +
\frac{\partial\psi^\alpha_\beta}{\partial q^a}\dot{q}^a \theta^{\beta} .
\end{eqnarray}
\medskip
It is a simple, but tedious, computation to check that the
$(1,1)$-supertensor field defined in local supercoordinates by
\begin{equation}
S=dq^a \otimes\frac{\partial}{\partial\dot{q}^a}
+d\theta^{\alpha}\otimes\frac{\partial}{\partial\dot{\theta}^\alpha}
\end{equation}
is well-defined. We can use it to define the analogous to the
Cartan $1$-superform as
\begin{equation}\label{thetaL}
\Theta_L =S\circ dL= dq^a \frac{\partial
L}{\partial\dot{q}^a} + d\theta^\alpha \frac{\partial
L}{\partial\dot{\theta}^\alpha}
= \frac{\partial
L}{\partial\dot{q}^a}dq^a + (-1)^{\vert L\vert} \frac{\partial
L}{\partial\dot{\theta}^\alpha}d\theta^\alpha ,
\end{equation}
and the Cartan $2$-superform
\begin{equation}\label{omegaL}
\Omega_L =-d\Theta_L
\end{equation}
The superenergy $E_L$ is defined as the superfunction
\begin{equation}\label{energyL}
E_L =\Delta
(L)-L=\dot{q}^a\frac{\partial
L}{\partial\dot{q}^a}+\dot{\theta}^{\alpha}\frac{\partial
L}{\partial\dot{\theta}^\alpha}-L
\end{equation}
where $\Delta$ denotes the Liouville superfield defined by $\Delta =
\dot{q}^a\frac{\partial}{\partial\dot{q}^a} +
\dot{\theta}^{\alpha}
\frac{\partial}{\partial\dot{\theta}^{\alpha}}$.
\medskip
The superlagrangian $L$ is said to be nondegenerate or regular
if the supermatrix defined by the superform $\Omega_L$ in the
basis $(dq^a ,d\dot{q}^a ,d\theta^{\alpha},
d\dot{\theta}^{\alpha})$ is invertible. This is equivalent to
say that the weak kernel of $\Omega_L$, defined as the set of
supervector fields $Z$ such that $\epsilon (\Omega_L (Z,X))=0$
for any $X$, is zero. By $\epsilon$ we mean that we take the
ordinary part of the superfunction $\Omega_L (Z,X)$. If $L$ is
regular, it is simple to show that the dynamical equation
\begin{equation}
i_{\Gamma}\Omega_L =dE_L
\end{equation}
has a unique solution $\Gamma$ which happens to be a super
second order differential equation (superSODE) \cite{Ib92}, i.e.
$S(\Gamma ) = \Delta$, or locally
\begin{equation}
\Gamma =\dot{q}^a\frac{\partial}{\partial
q^a} + \dot{\theta}^\alpha\frac{\partial}{\partial \theta^\alpha}
+f^a\frac{\partial}{\partial \dot{q}^a}
+f^\alpha\frac{\partial}{\partial\dot{\theta}^\alpha}
\end{equation}
where $f^a$, $f^\alpha$, even and odd superfunctions
respectively, are the generalized superforces.
Locally the flow defined by $\Gamma$ can be written as
the set of Newton's like equations
\begin{equation}
\ddot{q}^a = f^a(q,\dot{q},\theta,\dot{\theta});~~~~~~
\ddot{\theta}^\alpha = f^\alpha (q,\dot{q},\theta,\dot{\theta}).
\end{equation}
It is an immediate consequence that $\Gamma$ satisfies the
equation
\begin{equation}
{\cal L}_{\Gamma}\Theta_L = dL,
\end{equation}
and then, the dynamical equations above are equivalent to the
Euler-Lagrange superequations for $L$
\begin{equation}
\frac{d}{dt}\left(\frac{\partial
L}{\partial\dot{q}^a}\right) =\frac{\partial L}{\partial q^a}
~;~~\frac{d}{dt}\left(\frac{\partial L}
{\partial\dot{\theta}^\alpha}\right) =\frac{\partial L}{\partial
\theta^\alpha} . \end{equation}
\bigskip
\noindent
{\bf Statement of the inverse problem.}
\noindent
Given a superSODE
$\Gamma$, under which conditions does there exists a
superlagrangian function $L$ such that $\Gamma$ is a solution of
the dynamical equation defined by $L$, namely
$i_{\Gamma}\Omega_L =dE_L$ ?
\bigskip
\noindent
\prop 1.
A necessary and sufficient set of conditions assuring the
existence of a (local) superlagrangian for the superSODE $\Gamma$
are:
\medskip
i) There exists a closed $2$-superform $\Omega$.
ii) ${\cal L}_\Gamma \Omega = 0$.
iii) $i_{V_1 \wedge V_2} \Omega = 0$ for all $V_1$, $V_2$ vertical
supervector fields.
\medskip
A vertical supervector field is a supervector field $V$ that is
written locally as
$V = k^a\frac{\partial}{\partial\dot{q}^a} +
k^\alpha\frac{\partial}{\partial\dot{\theta}^\alpha}$.
\medskip
The necessity of conditions i) and ii) is obvious from the
construction of $\Omega_L$. Computing explicitely $\Omega_L$ in
local supercoordinates it is also a simple exercise to check
condition iii).
\medskip
Let us check now directly the sufficiency of conditions
i), ii), iii) obtaining a local superlagrangian function $L$ for
$\Gamma$. Because of condition i) and applying the graded
Poincare's lemma
\cite{Ko77} there exists a $1$-superform $\Phi$ such that
$$\Omega =d\Phi .$$
Writting down $\Phi$ in local supercoordinates we have:
$$\Phi =dq^a M_a (q,\dot{q},\theta ,\dot{\theta}) + d\dot{q}^a
M_{\dot{a}}(q,\dot{q},\theta ,\dot{\theta}) + d\theta^\alpha N_{\alpha}
(q,\dot{q},\theta ,\dot{\theta}) + d\dot{\theta}^\alpha
N_{\dot{\alpha}} (q,\dot{q},\theta,\dot{\theta}). $$
Denoting by $\dot{\Phi}=d\dot{q}^a M_{\dot{a}} + d\dot{\theta}^{\alpha}
N_{\dot{\alpha}}$, we have that because
of iii),
$$ i_{V_1 \wedge V_2} d\dot{\Phi} = i_{V_1 \wedge V_2} d\Phi =
i_{V_1 \wedge V_2} \Omega = 0$$
for all supervertical $V_1$, $V_2$. Then, there exists a
superfunction $f(q,\dot{q},\theta ,\dot{\theta})$ such that
$$i_{V} df = i_{V} \dot{\Phi}$$
for all $V$ supervertical, or locally,
$$\frac{\partial f}{\partial\dot{q}^a} = N_{\dot{a}};~~~~
\frac{\partial f}{\partial\dot{\theta}^\alpha}
= M_{\dot{\alpha}}.$$
Then defining the $1$-superform
$$\Theta =\Phi -df$$
it is clear that $i_V \Theta = 0$ for all $V$ supervertical,
hence, in local supercoordinates $\Theta$ has the expression
$$\Theta = dq^a A_a +d\theta^\alpha B_{\alpha}$$
where $A_a$ is an even superfunction and $B_\alpha$ is odd if
$\Omega$ is even and conversely if $\Omega$ is odd. It is clear
that $d\Theta = \Omega$.
\medskip
Using now the invariance condition ii) we obtain
$${\cal L}_\Gamma d\Theta =d{\cal L}_\Gamma \Theta = 0$$
or, in other words, ${\cal L}_\Gamma \Theta$ is a closed
$1$-superform. Using again Poincare's lemma, we can affirm that
there exists a superfunction $L$ such that
$${\cal L}_\Gamma \Theta = dL$$
and computing the previous Lie derivative in local supercoordinates
we obtain the following identities
\begin{eqnarray*}
A_a & = & \frac{\partial L}{\partial\dot{q}^a};~~~~~
B_\alpha = \frac{\partial L}{\partial\dot{\theta}^\alpha} \\
\dot{q}^a\frac{\partial A_b}{\partial q^a} & + &
\dot{\theta}^\alpha\frac{\partial
A_b}{\partial\theta^\alpha} + f^a\frac{\partial
A_b}{\partial\dot{q}^a}+f^{\alpha}\frac{\partial
A_b}{\partial\dot{\theta}^\alpha} = \frac{\partial
L}{\partial q^b} \\
\dot{q}^a\frac{\partial B_\beta}{\partial q^a} & + &
\dot{\theta}^\alpha\frac{\partial
B_\beta}{\partial\theta^\alpha} + f^a\frac{\partial
B_\beta}{\partial\dot{q}^a}+f^{\alpha}\frac{\partial
B_\beta}{\partial\dot{\theta}^\alpha} = \frac{\partial
L}{\partial \theta^\beta}.
\end{eqnarray*}
Because of eqn. (\ref{thetaL}) the first two equations clearly
imply that $\Theta =\Theta_L$ and in consequence (\ref{omegaL})
$\Omega =\Omega_L$. Expanding again the invariance condition we
get $i_\Gamma d\Theta_L +di_\Gamma \Theta_L =dL$, this is
$i_\Gamma \Omega_L =d(i_\Gamma \Theta_L -L)$, but
$i_\Gamma \Theta_L = i_\Gamma (S \circ dL) = i_\Delta (dL) = \Delta(L)$,
hence $i_\Gamma \Omega_L =dE_L$ because of (\ref{energyL}).
\bigskip
\noindent
{\bf Remarks.}
1) The 2-superform $\Omega$ can be even or odd. In each
situation we get an even or odd superlagrangian respectively.
\medskip
2) It is important to realize that with respect to the ordinary
assumptions in \cite{Ba80} \cite{Cr81},\cite{He82},
$\Omega$ does not have to
be regular, i.e. $L$ needs not to be a regular superlagrangian. If
this were not the case the weak kernel of $\Omega$ would impose
algebraic conditions on $\Gamma$, and the strong kernel of
$\Omega$ (defined as the set of supervector fields $Z$ such that
$\Omega (Z,X)=0$, $\forall X$) defines an invariant Lie
superalgebra, the gauge superalgebra of $\Gamma$ and $L$. Under
such circumstances, the tangent supermanifold and the dynamical
equation can be reduced quotienting out the gauge degrees of
freedom defined by the strong kernel of $\Omega_L$.
\medskip
3) The existence of alternative superlagrangians
allows to construct a mixed $(1,1)$ tensor field which could play the
role of recursion operator. In fact, if the superlagrangians have
opposite parity this is not the case and they cannot be used in the
analysis of complete integrability. Their existence will rather
imply the existence of a supersymmetry.
\medskip
4) A similar inverse problem can be formulated for
superequations of motion which are first order in the
velocities. This is usually the situation of superparticles and
Supersymmetric (Quantum) Classical Mechanics \cite{Ba83}.
The inverse problem for such class of systems has been
discussed in the setting of ordinary geometry in \cite{Ib91}
and the results there can be easily translated into the graded
realm following the ideas above.
\section{The Superoscillator Revisited}
We will discuss in this section a family of alternative
superlagrangians for the superoscillator. Let $\R^{n\mid n}$
be the configuration superspace with local supercoordinates
$(q^a ,\theta^a )$, $a=1,\dots ,n$. The tangent supermanifold is
easily seen to be $\R^{2n\mid 2n}$ with local supercoordinates
$(q^a ,\dot{q}^a ,\theta^a ,\dot{\theta}^a )$. The equations of
motion of the superoscillator are simply
\begin{equation}
\ddot{q}^a =-q^a~;~~\ddot{\theta}^a =-\theta^a,
\end{equation}
corresponding to the superSODE
\begin{equation}\label{superosci}
\Gamma =\dot{q}^a\frac{\partial}{\partial q^a} -
q^a\frac{\partial}{\partial\dot{q}^a} +
\dot{\theta}^a\frac{\partial}{\partial \theta^a} -
\theta^a\frac{\partial}{\partial\dot{\theta}^a}.
\end{equation}
\bigskip
\noindent
{\bf The linear supersymmetry algebra of the superoscillator.}
\noindent
Let $M(m|n, B_L)$ be the Lie superalgebra of supermatrices
with entries in the superalgebra $B_L$. A supermatrix $A$ will
have the block structure
$$A = \left( \begin{array}{c|c}
A_{++} & A_{+-} \\ \hline
A_{-+} & A_{--}
\end{array} \right) .$$
\noindent
Any $A\in M(m|n, B_L)$, has associated a linear supervector
field $X_A$ on $\R^{m|n}$.
If we use collective notations for the coordinates,
namely $z^i = (x^a, \theta^b)$, the supervector field $X_A$
will be given by
\begin{equation}
X_A = z^j A_j~^i \frac{\partial}{\partial z^i}.
\end{equation}
Clearly, $[X_A ,X_B]=X_{[A,B]}$, where $[A,B]$ denotes the
supercommutator in the Lie superalgebra $M(m|n, B_L)$.
Similarly, if $F$ is another supermatrix,
we can define a $2$-form $\Omega_F$ as
\begin{equation}
\Omega_F = d z^i \wedge d z^j F_{ji}.
\end{equation}
In order to have a 2-form the matrix $F$ should be
skewsymmetric, i.e., $F^t = - F$ or, in components,
$F_{ij} = - (-1)^{\vert i \vert \vert j \vert}F_{ji}$
(to be precise, in the previous expression, will survive only the part
of $F$ with these properties), irrespective of the degree of $F$.
The $\Omega_F$ will be even or odd if $F$ is even or odd
respectively.
\bigskip
Since $F$ is a constant matrix, $\Omega_F$ is closed and the action of
the supervector field
$X_A$ is easly obtained. It turns out to be
\begin{eqnarray}
{\cal L}_{X_A} \Omega_F & & = (-1)^{\vert A \vert \vert i \vert}
d z^i \wedge d z^j \{ (AF) _{ji} -
(-1)^{\vert i \vert \vert j \vert} (AF)_{ij} \} \\ \nonumber
& & =
(-1)^{\vert A \vert \vert i \vert}
d z^i \wedge d z^j \{ (AF) - (AF)^t \}_{ji}~.
\end{eqnarray}
In particular, ${\cal L}_{X_A} \Omega_F$ can be associated to a
supermatrix iff $A$ is even. If this is the case, then
\begin{equation}
{\cal L}_{X_A} \Omega_F = 2 \Omega_{(AF)^{sa}}~, ~~~~~
(AF)^{sa} = \frac{1}{2}((AF) - (AF)^t)~. \label{@@}
\end{equation}
\bigskip
For the analysis of the harmonic superoscillator we shall consider the
configuration superspace $\R^{n\mid n}$ as a graded supermanifold.
This assumption will also implies that the only constant numbers are
ordinary (i.e. real or complex) numbers.
Consider then the supermatrix $J$ in $M(2n|2n, \R)$ given by
$$ J = \left( \begin{array}{c|c}
J_0 & 0 \\ \hline 0 & J_0
\end{array}\right) $$
where $J_0$ is the $2n\times 2n$ symplectic matrix
$$ J_0 = \left( \begin{array}{c|c}
0 & I \\ \hline -I & 0 \end{array}\right)~. $$
\noindent
Clearly the superSODE $\Gamma$ defining the harmonic
superoscillator dynamics, eq. (\ref{superosci}), is simply
\begin{equation} \Gamma =X_J
\end{equation}
and then, a linear supervector field $X_A$ is a symmetry of
$\Gamma$ iff
$$0 = {\cal L}_{X_A}\Gamma =[X_A ,X_J ]=X_{[A,J]}$$
i.e., iff $[A,J]=0$.
In other words, the
linear symmetry supergroup for $\Gamma$ is the supergroup of
linear transformations $Gl(2n|2n, \R)$.
\bigskip
\noindent
{\bf Linear Inverse problem.}
\noindent
Let $\Omega_F$ be a 2-superform
associated with the skewsymmetric supermatrix $F$ in
$M(2n|2n, \R)$.
The first condition in the inverse problem, namely
that $\Omega_F$ is closed, is automatically satisfied.
>From (\ref{@@}), the invariance condition
${\cal L}_\Gamma\Omega_F = 0$ implies that $(JF)^{sa} = 0$
and this is equivalent to
$F$ being in the Lie superalgebra of the Lie supergroup
$Gl(2n|2n, \R)\cap Osp(2n|2n)$, and having the block
structure
$$ F = \left( \begin{array}{c|c}
\begin{array}{c|c}
F_{++}^d & F_{++}^a \\ \hline
-F_{++}^a & F_{++}^d
\end{array} & \begin{array}{c|c}
F_{+-}^d & F_{+-}^a \\ \hline
-F_{+-}^a & F_{+-}^d
\end{array} \\ \hline
\begin{array}{c|c}
-(F_{+-}^d)^t & (F_{+-}^a)^t \\ \hline
-(F_{+-}^a)^t & (F_{+-}^d)^t
\end{array} & \begin{array}{c|c}
F_{--}^d & F_{--}^a \\ \hline
-F_{--}^a & -F_{--}^d
\end{array} \end{array} \right) $$
\noindent
with $F_{++}^a$ a symmetric matrix of ordinary numbers and
$F_{--}^a$ a skwesymmetric matrix of ordinary numbers.
The vertical condition iii) in proposition 1,
implies that the coefficients of
$d\dot{q}\wedge d\dot{q}$, $d\dot{q}\wedge d\dot{\theta}$ and
$d\dot{\theta}\wedge d\dot{\theta}$ in $\Omega_F$ must vanish,
thus $F$ has the form
$$ F = \left( \begin{array}{c|c}
\begin{array}{c|c}
0 & F_{++}^a \\ \hline
-F_{++}^a & 0
\end{array} & \begin{array}{c|c}
0 & F_{+-}^a \\ \hline
-F_{+-}^a & 0
\end{array} \\ \hline
\begin{array}{c|c}
0 & (F_{+-}^a)^t \\ \hline
-(F_{+-}^a)^t & 0
\end{array} & \begin{array}{c|c}
0 & F_{--}^a \\ \hline
-F_{--}^a & 0
\end{array} \end{array} \right) = F_{even} + F_{odd}~,$$
with
$$ F_{even} = \left( \begin{array}{c|c}
\begin{array}{c|c}
0 & F_{++}^a \\ \hline
-F_{++}^a & 0
\end{array} &
0 \\ \hline
0 &
\begin{array}{c|c}
0 & F_{--}^a \\ \hline
-F_{--}^a & 0
\end{array} \end{array} \right)~
$$ and
$$
F_{odd} = \left( \begin{array}{c|c}
0 &
\begin{array}{c|c}
0 & F_{+-}^a \\ \hline
-F_{+-}^a & 0
\end{array} \\ \hline
\begin{array}{c|c}
0 & (F_{+-}^a)^t \\ \hline
-(F_{+-}^a)^t & 0
\end{array} &
0
\end{array} \right)~. $$
\bigskip
\noindent
{\bf Alternative superlagrangians.}
\noindent
In both cases what we get
is that the product $FJ$ defines a supermetric $G$ on the
superspace $\R^{2n|2n}$, even or odd if $F$ is even or odd
respectively and the superlagrangians from which $\Omega_F$
derives are given by
\begin{equation}\label{hoel}
L_{even} = \frac{1}{2} ((F_{++}^a)_{ab}\dot{q}^a\dot{q}^b +
(F_{--}^a)_{ab}\dot{\theta}^a\dot{\theta}^b -
(F_{++}^a)_{ab}q^aq^b - (F_{--}^a)_{ab}\theta^a\theta^b)
\end{equation}
or
\begin{equation}\label{hool}
L_{odd} = (F_{+-}^a)_{ab}\dot{q}^a\dot{\theta}^b
- (F_{+-}^a)_{ab}q^a\theta^b
\end{equation}
\bigskip
Any choice of an admissible Lagrangian will break the group
$Gl(2n|2n,\R)$ to a subgroup that depends on $L_F$.
This implies that the association of symmetries and constants of the
motion depends on $L_F$ and moreover, upon quantization, we get
different quantum mechanical systems depending on the Lagrangian we
choose. This fact is evident without the need of heavy computations.
Indeed, according to which Lagrangian we use, we can select a
subgroup of symmetries that contains a compact or a non compact even
part and upon quantization we shall have a discrete or a continuous
spectrum respectively.
\section{Alternative Lagrangians and recursion operators}
What we would like to show now is that the existence of alternative
superlagrangians of opposite parity for the same superSODE $\Gamma$
does not allow to construct
recursion operators and cannot be used for the complete integrability
of $\Gamma$. If the alternative Lagrangians are of the same parity one
has the usual kind of analysis.
Before we do that we rewiev the analysis done with ordinary (i.e. non
graded) Lagrangian dynamical systems \cite{Cr83}
by stressing the conditions that
should (and in same cases cannot) be superized.
So we work on an ordinary tangent bundle
$TQ$; the following objects can (and some have already been
\cite{Ib92})
superized. There are two natural
lifting procedure for vector fields from $Q$ to $TQ$,
namely the tangent and the vertical lifting. If $X = X^a(q)
\pd{},{q^a} \in \vectq $, its {\it tangent lift} $X^T$ and its
{\it vertical lift} $X^V$ are the elements in $\vecttq$ given by
\beq
X^T = X^a(q) \pd{},{q^a} + \dot{q}^b \pd{X^a},{\dot{q}^b}
\pd{},{\dot{q}^a}~,~~~
X^V = X^a(q) \pd{},{\dot{q}^a}~. \label{AB}
\eeq
A third lifting procedure is associated with any SODE
$\Gamma$: the {\it horizontal lift} of $X \in
\vectq$ is the vector field $X^H \in \vecttq$ given by
\beq
X^H = X^a H_a~,
~~~ H_a =: ( \pd{},{q^a} )^H = \pd{},{q^a} + \frac{1}{2}
(\pd{\Gamma^b},{\dot{q}^a})
\pd{},{\dot{q}^b}~, ~~~ a \in \{1,\dots,n \}. \label{AD}
\eeq
The horizontal lift provides a connection on $TQ$ and, for any point
$(q,\dot{q})$ of $TQ$, the set of horizontal lifts of vectors from $Q$
forms a
subspace of $T_{(q,\dot{q})}(TQ)$ which is complementary to the subspace
spanned by vertical lifts. This decomposition of the tangent spaces
to $TQ$ allows also to define an almost complex structure ${\cal J}$
on $TQ$ by setting
\beq
{\cal J}(\xi ^H) = \xi ^V~,~~~~~
{\cal J}(\xi ^V) = - \xi ^H~,~~~~~ \forall ~\xi \in
T_{(q,\dot{q})}(TQ)~. \label{AE}
\eeq
\bigskip
If $L$ is a regular Lagrangian on $TQ$,
its Cartan $2$-form
$\Omega_{L}$ besides being invariant under $\Gamma$
has additional properties related to the
horizontal distribution generated by $\Gamma$. The most relevant one
is the possibility of defining a
pseudo-hermitian metric $g$ on $TQ$ by
\beq
g(X,Y) = \Omega_{L}(X, {\cal J}Y)~, ~~~\forall~ X, Y \in \vecttq~. \label{AM}
\eeq
Then, horizontal and vertical subspaces are ortogonal with respect to
$g$ and $g(\xi^V, \eta^V) = g(\xi^H, \eta^H)$.
\bigskip
Let us suppose now that the field $\Gamma$ is the common
Euler-Lagrange field associated with two regular Lagrangians
$L_1$ and $L_2$ which are not trivially related by addition of a
total time derivative or by multiplication by a constant.
The associated Cartan 2-forms $\Omega_{L_1}$ and $\Omega_{L_2}$, which
both have the properties described before, can be used to define a
type $(1,1)$ tensor field $T$ on $TQ$ by the property
$i_{X} \Omega_{L_2} = i_{{T(X)}} \Omega_{L_1}$, namely
\beq
T = \Omega_{L_2} \circ \Omega_{L_1}^{-1}~, \label{AN}
\eeq
where we use the same symbols for the 2-forms and the associated
endomorphisms of $\vecttq$ and $\formtq$.
The tensor $T$ is compatible with
$(\Omega_L, {\cal J}, g)$ and as
a consequence it preserves the
direct sum decomposition of the tangent spaces to $TQ$, and maps
vertical vectors to vertical vectors and horizontal ones to horizontal
ones. In addition, $T$ is the direct sum of two linear non-singular
transformations of the vertical and the horizontal subspaces, which
are identical in the sense that they are given by the same matrix in a
chosen basis of vertical and horizontal vectors.
Since $T$ is simmetrical with respect to the metric $g$, its
eigenvalues at each points are real, and, acting identically on the
vertical and horizontal subspaces, each eigenvalue is at least
doubly degenerate with a vertical and a horizontal eigenvector.
If the eigenvalues of $T$ are as little degenerate as possible, namely
doubly degenerate, they determine functions $\lambda_a~, a \in
\{1, \dots, n \}$, on $TQ$,
whose values at each point are just the values of
the eigenvalues of $T$ at that point.
These functions are constants of
the motion for $\Gamma$ as follows from the
invariance of $T$ under $\Gamma$. They will give
its complete integrability if there are in involution.
In order to show the latter, one needs to impose an
additional condition on $T$, namely that its {\it Nijenhuis tensor}
$\NT$, defined for all $X,Y\in\vecttq$ by
\beq
\NT(X,Y) = T^{2} [X,Y] + [T X,T Y]
- T [T X,Y] - T [X,T Y]~, \label{AC}
\eeq
vanishes. This is equivalent to the fact that the two Poisson
structures associated with $\Omega_{L_1}$ and $\Omega_{L_2}$ are
compatible; their sum is still a Poisson structure.
An immediate consequence of the condition $\NT = 0$ is the fact
that, if $X_a$ is any
eigenvector field corresponding to the eigenfunction $\lambda_a$, then
${\cal L}_{X_b} \lambda_{a} = 0$, whenever $a \not= b$;
this also expresses the
vanishing of the Poisson bracket of any two eigenfunctions $\lambda_a,
\lambda_b$.
\medskip
An equivalent way of using a (1-1) tensor field $T$
with vanishing Nijenhuis torsion is to generate a
sequence of constants of motion in involution by successive
application of $T$. This is (locally) the content of the
\prop 2.
If the Nijenhuis tensor $\NT$ vanishes and $d(T dF) = 0$
then $d(T^k dF) = 0$ for all $k>1$.
\proof
Let $\alpha$ be any 1-form. Some simple algebra gives
\beq
i_{{X \wedge Y}} d(T^2 \alpha) =
i_{\{X \wedge TY + TX \wedge Y\}} d(T \alpha)
- i_{\{TX \wedge TY\}} d\alpha
- i_{{\NT (X, Y)}} \alpha~. \label{EB}
\eeq
If both $\alpha$ and $T \alpha$ are closed,
than $T^2 \alpha$ is closed if and only if
$\NT = 0$.
\bigskip
With $T$ as in (\ref{AN}), one has $T dE_{L_1} = dE_{L_2}$ and
$T^k dE_L$ is a function of the eigenfunctions of $T$.
\bigskip
Let us analize now the graded situation. It turns out that one cannot
define a (super) Nijenhuis torsion.
Let us suppose $T$ is a graded $(1,1)$ tensor field which is homogeneous of
parity $\vert T \vert$. Then, if $\alpha$ is any 1-form,
the closest possible expression to (\ref{EB}) reads
\beqar
i_{{X \wedge Y}} d(T^2 \alpha) &=&
i_{ \{(-1)^{\vert T \vert\vert Y \vert} X \wedge TY
+ (-1)^{\vert T \vert (\vert X \vert+\vert Y \vert) } TX \wedge Y\} }
d(T \alpha)
\nonumber \\
&&~~- (-1)^{\vert T \vert (\vert X \vert+\vert T \vert)}
i_{{TX \wedge TY}}
d\alpha - (-1)^{\vert T \vert}~ i_{\GNT (X,Y)} \alpha
\nonumber \\
&&~~+ (-1)^{\vert T \vert\vert X \vert} [ 1 - (-1)^{\vert T \vert}]
{\cal L}_{TX} ( i_{{TY}}\alpha)~.
\label{EC}
\eeqar
Where $\GNT$ is defined by
\beqar
\GNT(X,Y) &=:& T^{2} [X,Y] + (-1)^{\vert T \vert\vert X \vert}[T X,T Y]
- T [T X,Y] \nonumber
\\
&&~~ - (-1)^{\vert T \vert\vert X \vert} T [X,T Y]~. \label{BM}
\eeqar
We see from (\ref{EC}), that for an $(1,1)$ odd tensor a $(2,1)$ tensor
corresponding to its
torsion (super-Nijenhuis torsion) can
be defined only when $\vert T \vert = 0$.
This is a consequence of the fact that the map $\GNT$ defined in
(\ref{BM}) is
linear over superfunctions (and graded antisymmetric)
if and only if $\vert T \vert = 0$.
The Lagrangians (\ref{hoel}) and (\ref{hool}) for the harmonic
oscillator provide examples of numerical $(1,1)$ tensors that cannot
be recursion operators. As an explicit example we consider a harmonic
oscillator on $\R^{4|4}$ with an even and an odd Lagrangian given
respectively by
\beqar
&&L_1 = \frac{1}{2} [(\dot{q}^1)^2 + (\dot{q}^2)^2 -
(q^1)^2 - (q^2)^2 ]
+ i(\dot{\theta}^1\dot{\theta}^2 - \theta^1\theta^2) ~, \label{BN}\\
&&L_2 = \dot{q}^1\dot{\theta}^1 + \dot{q}^2\dot{\theta}^2
-{q}^1{\theta}^1 - {q}^2{\theta}^2~. \label{BO}
\eeqar
The Cartan $2$-forms and the energy functions are respectively
\beqar
&&\Omega_1 = d q^{1} \wedge d \dot{q}^1
d q^{2} \wedge d \dot{q}^2 + i d \theta^1 \wedge d \dot{\theta}^1
- i d \theta^2 \wedge d \dot{\theta}^2~, \label{BP}\\
&&\Omega_2 = d q^{1} \wedge d \dot{\theta}^1
d q^{1} \wedge d \dot{\theta}^1 +
d \theta^1 \wedge d \dot{q}^1 +
d \theta^2 \wedge d \dot{q}^2~; \label{BQ}\\
&&E_1 = \frac{1}{2} [(\dot{q}^1)^2 + (\dot{q}^2)^2 +
(q^1)^2 + (q^2)^2 ]
+ i(\dot{\theta}^1\dot{\theta}^2 + \theta^1\theta^2) ~, \label{BR}\\
&&E_2 = \dot{q}^1\dot{\theta}^1 + \dot{q}^2\dot{\theta}^2
+{q}^1{\theta}^1 + {q}^2{\theta}^2~. \label{BS}
\eeqar
The associated $(1,1)$ odd tensor field is given by
\beq
T = \Omega_2 \circ (\Omega_1)^{-1} =
\left( \begin{array}{c|c}
0 &
\col 0,-i,0,0 \col i,0,0,0
\col 0,0,0,-i \col 0,0,i,0 \\ \hline
\col 1,0,0,0 \col 0,1,0,0
\col 0,0,1,0 \col 0,0,0,1 &
0
\end{array} \right)~, \label{BT}
\eeq
and, is spite of the fact that $T$ is numerical, one finds that
\beqar
&& T d E_1 = d E_2 \nonumber \\
&& T^2 d E_1 = i((d q^1) q^2 - (d q^2) q^1 + (d\dot{q}^1) \dot{q}^2
+ (d\dot{q}^2) \dot{q}^1) \nonumber \\
&& ~~~~~~~~~~+ (d \theta^1) \theta^1
+(d \theta^2) \theta^2 + (d \dot{\theta}^1) \dot{\theta}^1
+ (d \dot{\theta}^2) \dot{\theta}^2~, \nonumber \\
&& d T^2 d E_1 \not= 0~.
\label{BU}
\eeqar
\bigskip
{\bf Acknowledgements.} GL thanks U. Bruzzo for a discussion.
JMS would like to thank the financial
support provided for a Grant awarded by the UCM. This work
was possible by partial financial support by CICYT under
programme PS89/0013.
GL and GM are partially supported by the Italian Ministero dell'
Universit\`a e della Ricerca Scientifica e Tecnologica.
\bigskip
\bigskip
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\end{document}