Second Order Hamiltonian Systems

Abstract

We consider the system

$$\displaystyle -\ddot x(t)=\; B(t)x(t) +\nabla _xV(t,x(t)), $$

where

$$\displaystyle x(t)=(x_1(t),\cdots ,x_n(t)) $$

is a map from I = [0, T] to $\mathbb R^n$ such that each component x _j(t) is a periodic function in H ¹ with period T, and the function V (t, x) = V (t, x ₁, ⋯ , x _n) is continuous from $\mathbb R^{n+1}$ to $\mathbb R$ with

$$\displaystyle \nabla _xV(t,x)=(\partial V/\partial x_1,\cdots ,\partial V/ \partial x_n) \in C(\mathbb R^{n+1},\mathbb R^n). $$

For each $x \in \mathbb R^n,$ the function V (t, x) is periodic in t with period T.