## Institute for Mathematical Sciences

## Preprint ims96-1b

** P. Boyland and C. Gole**
* Dynamical Stability in Lagrangian Systems*

Abstract: This paper surveys various results concerning stability for the dynamics of Lagrangian (or Hamiltonian) systems on
compact manifolds. The main, positive results state,
roughly, that if the configuration manifold carries a
hyperbolic metric, \ie a metric of constant negative
curvature, then the dynamics of the geodesic flow persists
in the Euler-Lagrange flows of a large class of
time-periodic Lagrangian systems. This class contains all
time-periodic mechanical systems on such manifolds. Many of
the results on Lagrangian systems also hold for twist maps
on the cotangent bundle of hyperbolic manifolds.
We also present a new stability result for autonomous
Lagrangian systems on the two torus which shows, among other
things, that there are minimizers of all rotation
directions. However, in contrast to the previously known
\cite{hedlund} case of just a metric, the result allows the
possibility of gaps in the speed spectrum of minimizers.
Our negative result is an example of an autonomous
mechanical Lagrangian system on the two-torus in which this
gap actually occurs. The same system also gives us an
example of a Euler-Lagrange minimizer which is not a Jacobi
minimizer on its energy level.

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