Symplectic twist maps without conjugate points

Symplectic Twist Maps

Optical Hamiltonians and Symplectic Twist Maps

This paper concentrates on optical Hamiltonian systems of T T, i.e. those for which Hpp is a positive definite matrix, and their relationship with symplectic twist maps. We present theorems of decomposition by symplectic twist maps and existence of periodic orbits for these systems. The novelty of these results resides in the fact that no explicit asymptotic condition is imposed on the system. ...

REGULAR ARTICLES Twist singularities for symplectic maps

Near a nonresonant, elliptic fixed point, a symplectic map can be transformed into Birkhoff normal form. In these coordinates, the dynamics is represented entirely by the Lagrangian ‘‘frequency map’’ that gives the rotation number as a function of the action. The twist matrix, given by the Jacobian of the rotation number, describes the anharmonicity in the system. When the twist is singular the...

Convex Hamiltonians without Conjugate Points

Introduction.

Normal Forms for Symplectic Maps with Twist Singularities

We derive a normal form for a near-integrable, four-dimensional symplectic map with a fold or cusp singularity in its frequency mapping. The normal form is obtained for when the frequency is near a resonance and the mapping is approximately given by the time-T mapping of a two-degree-of freedom Hamiltonian flow. Consequently there is an energy-like invariant. The fold Hamiltonian is similar to ...