Title: On the integrability of Tonelli Hamiltonians


Abstract:

In this talk I shall discuss a weaker version of Liouville's theorem on 
the integrability of Hamiltonian systems. 
I shall show that in the case of Tonelli Hamiltonians the involution 
hypothesis on the integrals of motion can be completely dropped and 
still interesting information on the dynamics of the system can be 
deduced. 
Moreover, in the case of the n-dimensional torus this weaker notion of 
integrability implies the classical one (i.e., integrability the sense 
of Liouville). 
The main idea consists in relating the existence of independent 
integrals of motion of a Tonelli Hamiltonian to the "size" of its 
Mather and Aubry sets.