Quantitative Destruction and Persistence of Lagrangian Torus in Hamiltonian Systems

For an integrable Hamiltonian systems with $d$ degrees of freedom ($d\geq 2$), we consider quantitatively the existence and non-existence of the flow-invariant Lagrangian torus with given frequency under the perturbation beyond the scope of the classical KAM method in the $C^r$ topology. As applications, the non-existence result gives a partial answer to an open problem on non-existence of invariant circles by Mather from 1988. The existence result sheds a light on another open problem on the existence of invariant circles with lower regularity by Mather from 1998.