$DIFF M$ DOES NOT ACT TRANSITIVELY ON THE
SOLUTIONS OF  AN INVARIANT VARIATIONAL PROBLEM ON $FM$

J. Mu\~{n}oz Masqu\'{e}, M.E. Rosado  Mar\'{\i}a

1991 Math. Subject Classification: 58E30, 57R25, 58D19, 58G35.

Key words: Invariant Lagrangians, Euler--Lagrange morphism.

Abstract: Let $\pi :FM \rightarrow M$ be the bundle of linear frames
of a $C^{\infty }$ $m-$manifold $M$, and let
${\cal L}:J^1(FM)\rightarrow {\Bbb R}$ be an invariant
under diffeomorphisms function. We set $E({\cal L})=\{ j^2_xs\in J^2(FM) ;
\;{\cal E}({\cal L})(j^2_xs)=0\}$, where ${\cal E}({\cal L})$ is
the Euler-Lagrange operator associated to the Lagrangian density
${\cal L}\theta ^1\wedge ... \wedge \theta ^m,\;\;\theta =
(\theta ^1,...,\theta ^m)$ being the canonical form on $FM$.
Let Diff$_xM$ be the group of germs of diffeomorphisms $\phi :(M,x)\rightarrow
(M,x)$ and let ${\cal S} _x $ be the set of germs of sections of $\pi $,
defined around $x\in M$, which are extremals of the above density.
The following result is proved. If (i) $M$ and $\cal{L}$ are
real analytic, (ii) $E({\cal L})\rightarrow M$ is a fibered
subbundle of $J^2(FM)$, and (iii) $E({\cal L})$ is formally integrable, then:
1) Under the natural action of the diffeomorphisms on linear frames,
Diff$_xM$ leaves invariant ${\cal S}_x$.
2) The action of Diff$_xM$ on ${\cal S}_x$ is non-transitive.