| Hamiltonian system theory is both classical and modern research area, which is studied by different methods. In this paper, we use variational method because Hamiltonian system has a variational structure, solutions of Hamiltonian system will be obtained as critical points of the corresponding functionals. During the past twenty years , there has been a great deal of progress for calculus of variations to find periodic, homoclinic, heteroclinic orbits for Hamiltonian systems.In this paper, it will be discussed the existences of multiple T -periodic solutions q = q (t) of a class of ordinary differential equations:where L denotes the Lagrangian functionand V(t,q) is a real function on Rn+1. (L1) is called Lagrangian systems. Ifαij(q)≡ 1, then (L1) is the usual second order hamiltonian systems.The present note is devoted to the study of problem(L1) when V is subquadratic at infinity, i.e.Our main results areTheorem1. Assume that functions αij(q) (i, j=1,　… , n) satisfy (A) there exists μ > 0 such thatμ1|ξ|2≤(α(q)ξ,ξ)≥μ2|ξ|2, for q∈Rn, ξ∈RNV(t,q) is independent of t, denoted by V(q), and satisfy for M, r1,r2,ε>0 where φ1, φ2 ∈ C (R+, R+) are two functions satisfying the following(i) φ1(x)/x →0 as x→ + ∞ and φ1 is mcreasing.(ii) φ2(x)→+∞ as x→+∞;(iii) φ2(x)/φ11+ε(x) → +∞ as |x| →∞(iv) φ11+ε(x) is Lipschitz continuous.And also suppose that α (q) = (αij(q)) and V(q) satisfy (V4) there exists β∈(0,2) such that α'(q)q + βα(q)≥0 for q∈Rn; (V5)V(0) = 0 is the minimum of V and V'(q)≠0 for any q≠0. (V6)αij and V are twice differentiable at the origin and V"(0) has allpositive eigenvalues.For any k∈N,k≠0, let T(k) = 2π[(k2+1)v/λ]1/2, where v is thelargest eigenvalue of {αij(0)}, λ the first eigenvalue of the matrix V"(0),Then for any T > T(k), problem(L1) possesses at least distinctkn T - periodic solutions.The multiple T -periodic solutions of problem (L1) are found ascritical points of the action functional:Theorem2. Assume that (A) and (V1)～(V4) hold, and g∈L2(R,Rn)is a T - periodic function, then for T > 0, the forced Lagrangian systempossesses at least one nontrivial T - periodic solution.In the paper, we apply the variational method to prove the above results.
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