# Dynamical Stability in Lagrangian Systems

@article{Boyland1999DynamicalSI, title={Dynamical Stability in Lagrangian Systems}, author={Philip Boyland and Christopher Gol'e}, journal={arXiv: Dynamical Systems}, year={1999}, pages={90-114} }

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 least as complicated as those of the geodesic flow of the hyperbolic metric. The second part of the paper presents original results on autonomous Lagrangian systems on the two torus including a precise description of Mather’s beta function. Examples are given of mechanical systems with non-strictly…

## One Citation

Complicated dynamics from simple topological hypotheses

- Biology, MathematicsPhilosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
- 2001

A diverse range of more sophisticated examples of dynamical system for which quite simple topological hypotheses imply very complicated behaviour are produced and their relevance to physical applications is explored.

## References

SHOWING 1-10 OF 37 REFERENCES

Lagrangian systems on hyperbolic manifolds

- Mathematics
- 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…

Periodic orbits for Hamiltonian systems in cotangent bundles

- Mathematics
- 1991

We prove the existence of at least cl(M) periodic orbits for certain time-dependent Hamiltonian systems on the cotangent bundle of an arbitrary compact manifold M. These Hamiltonians are not…

Mather Sets for Twist Maps and Geodesics on Tori

- Physics
- 1988

The title refers to a theory which is based on independent research in three different fields—differential geometry, dynamical systems and solid state physics—and which has attracted growing interest…

Generic properties and problems of minimizing measures of Lagrangian systems

- Mathematics
- 1996

It is proved here that minimizing measures of a Lagrangian flow are invariant and the Lagrangian is cohomologous to a constant on the support of their ergodic components. Moreover, it is shown that…

Mather sets for plane Hamiltonian systems

Recently Mather (Ma) developed a theory for area preserving monotone twist maps of an annulus, the main result of which is the generalization of the notion of an invariant curve of such maps. An easy…

Symplectic maps, variational principles, and transport

- Physics
- 1992

Symplectic maps are the discrete-time analog of Hamiltonian motion. They arise in many applications including accelerator, chemical, condensed-matter, plasma, and fluid physics. Twist maps correspond…

Action minimizing invariant measures for positive definite Lagrangian systems

- Mathematics
- 1991

In recent years, several authors have studied "minimal" orbits of Hamiltonian systems in two degrees of freedom and of area preserving monotone twist diffeomorphisms. Here, "minimal" means action…

Consequences of contractible geodesics on surfaces

- Mathematics
- 1998

The geodesic flow of any Riemannian metric on a geodesically convex surface of negative Euler characteristic is shown to be semi-equivalent to that of any hyperbolic metric on a homeomorphic surface…

Topological methods in surface dynamics

- Mathematics
- 1994

Abstract This paper surveys applications of low-dimensional topology to the study of the dynamics of iterated homeomorphisms on surfaces. A unifying theme in the paper is the analysis and application…

Birkhoff periodic orbits for small perturbations of completely integrable Hamiltonian systems with convex Hamiltonians

- Mathematics
- 1987

Pour des systemes a n degres de liberte, on montre l'existence d'au moins n orbites periodiques distinctes avec un vecteur frequence rationnel admissible proche du tore correspondant du systeme non…