Infinitely many periodic solution orbits of autonomous Lagrangian systems on tori

We prove that for any τ>0 the autonomous Lagrangian systemddt Lẋ(x,ẋ)−Lx(x,ẋ)=0on the standard n-dimensional torus possesses infinitely many geometrically distinct contractible periodic solution orbits whose periods are integer multiples of τ, provided that the Lagrangian function satisfies L(x,p)=12Ap•p+V(x), where A is a positive definite symmetric real matrix, and V is C3 and 1-periodic in all of its variables