We study some properties of multidimensional Hamiltonian systems in the adiabatic limit. Using the properties of the Poincare-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.
Maurice A. de Gosson. "The Adiabatic Limit for Multidimentsional Hamiltonian Systems." J. Geom. Symmetry Phys. 4 19 - 43, 2005. https://doi.org/10.7546/jgsp-4-2005-19-43