An abstract theorem on the existence of periodic motions of non-autonomous Lagrange systems

In this paper, we established an abstract theorem on the existence of periodic solutions for Lagrange systems under Tonelli framework (strictly convex and superlinear, completeness of phase flow) which is based on the principle of least action in geometry and dynamics. An example of the application of the abstract theorem is given and the existence of periodic solutions for a class of time-periodic Lagrange systems will be proved.