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 EulerLagrange flows of a large class of timeperiodic Lagrangian systems. This class contains all timeperiodic 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 twotorus in which this gap actually occurs. The same system also gives us an example of a EulerLagrange minimizer which is not a Jacobi minimizer on its energy level.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 January 1996
 arXiv:
 arXiv:math/9601213
 Bibcode:
 1996math......1213B
 Keywords:

 Mathematics  Dynamical Systems