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

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)