In this talk, I will first introduce Bernad-Oliviera dos Santos's symplectic description of Mane critical value, Aubry set and Mane set, and Arnaud's graphicality theorem of  invariant Lagrangian submanifold under the flow of Tonelli Hamiltonians. Then I will explain construction of the graph selector of exact Lagrangian submanifold  and its extension to the class of Lipschitz-exact Lagrangian submanifolds. Finally I will explain generalization of above mentioned results in Aubry-Mather theory to this class of Lipschitz-exact Lagrangian submanifolds. The main ingredient of the constructions is the Floer homology theory in symplectic topology. This talk is based on the joint work with Amorim and Oliviera dos Santos.