Abstract: The periodic solutions of some Lagrange systems with suitable conditions are studied through the least action principle theorems in variational methods.The author presents some applications of the least principle on Lagrange systems.The author introduces the results obtained by the least action principle and the study of the transformation of the periodic solutions of Lagrange systems to the solutions of corresponding Euler equation.The periodic solutions of some Lagrange systems are studied with the coercivity conditions.

Key words: Lagrange systems, periodic solutions, least principle