The adiabatic limit for multidimensional Hamiltonian systems. (English) Zbl 1132.37318

de Gosson, Maurice A.

J. Geom. Symmetry Phys. 4, 19-43 (2005).

Summary: We study some properties of multidimensional Hamiltonian systems in the adiabatic limit. Using the properties of the Poincaré-Cartan invariant we show that in the integrable case conservation of action requires conditions on the frequencies together with conservation of the product of energy and period. In the ergodic case the most general conserved quantity is not volume but rather symplectic capacity; we prove that even in this case there are periodic orbits whose actions are conserved.

MSC:

37J99	Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
70H11	Adiabatic invariants for problems in Hamiltonian and Lagrangian mechanics
70H12	Periodic and almost periodic solutions for problems in Hamiltonian and Lagrangian mechanics
37J45	Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)

Keywords:

Poincaré-Cartan invariant; integrable case; symplectic capacity; periodic orbits