# Lagrangian systems on hyperbolic manifolds

@article{Boyland1996LagrangianSO, title={Lagrangian systems on hyperbolic manifolds}, author={Philip Boyland and Christopher Gol'e}, journal={arXiv: Dynamical Systems}, year={1996} }

This paper gives two results that show that the dynamics of a time-periodic Lagrangian system on a hyperbolic manifold are at least as complicated as the geodesic flow of a hyperbolic metric. Given a hyperbolic geodesic in the Poincar\'e ball, Theorem A asserts that there are minimizers of the lift of the Lagrangian system that are a bounded distance away and have a variety of approximate speeds. Theorem B gives the existence of a collection of compact invariant sets of the Euler-Lagrange flow…

## 11 Citations

Dynamical Stability in Lagrangian Systems

- Mathematics
- 1999

The first part of this paper surveys results on time-periodic Lagrangian systems on a hyperbolic manifolds. Results of the authors show that the dynamics of such systems are, in a precise sense, at…

New dynamical invariants on hyperbolic manifolds

- Mathematics
- 2000

The rotation measure is an asymptotic dynamical invariant assigned to a typical point of a flow in a fiber bundle over a hyperbolic manifold. The total mass of the rotation measure is the average…

Geometry of Hamiltonian Dynamics with Conformal Eisenhart Metric

- Computer Science, MathematicsInt. J. Math. Math. Sci.
- 2011

The geometry of the Hamiltonian dynamics with a conformal metric is characterized and the instability of the associated geodesic spreads is shown.

Minimal measures for Euler–Lagrange flows on finite covering spaces

- Mathematics
- 2016

In this paper we study the minimal measures for positive definite Lagrangian systems on compact manifolds. We are particularly interested in manifolds with more complicated fundamental groups.…

Melnikov Potential for Exact Symplectic Maps

- Mathematics
- 1997

Abstract:The splitting of separatrices of hyperbolic fixed points for exact symplectic maps of n degrees of freedom is considered. The non-degenerate critical points of a real-valued function (called…

Homotopy Classes for Stable Periodic and Chaotic¶Patterns in Fourth-Order Hamiltonian Systems

- Mathematics
- 2000

Abstract: We investigate periodic and chaotic solutions of Hamiltonian systems in ℝ4 which arise in the study of stationary solutions of a class of bistable evolution equations. Under very mild…

On Aubry sets and Mather’s action functional

- Mathematics
- 2001

We study Lagrangian systems on a closed manifoldM. We link the differentiability of Mather’sβ-function with the topological complexity of the complement of the Aubry set. As a consequence, whenM is a…

Homoclinic orbits of twist maps and billiardsAmadeu

- 1998

The splitting of separatrices for hyperbolic xed points of twist maps with d degrees of freedom is studied through a real-valued function , called the Melnikov potential. Its non-degenerate critical…

Erratum to “On Aubry sets and Mather’s action functional”

- Mathematics
- 2015

We study Lagrangian systems on a closed manifold M . We link the differentiability of Mather’s β -function with the topological complexity of the complement of the Aubry set. As a consequence, when…

Minimal measures on surfaces of higher genus

- Mathematics
- 2009

Abstract We study the minimal measures for positive definite autonomous Lagrangian systems defined on the tangent bundles of compact surfaces with genus greater than one. We present some results on…

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