The integrability of positively definite Lagrangian systems via variational criterion: Mechanical systems

We study the behavior of Mather's α-function for positive definite mechanical systems of arbitrary degree of freedom, give the estimate of the size of the flat part of the α-functions near 0, and then the criterion of the integrability of the systems with real analytic potential by the strict convexity of α-functions. Some example are also discussed.