The stability of the equilibrium position of a Hamiltonian system of ordinary differential equations in the general elliptic case

Abstract

1. Let p = q = 0 be a fixed point of the system

$$ \label{1} \dot{q} = \frac{\partial H}{\partial p}, \dot{p} = -\frac{\partial H}{\partial p}, $$

((1))

where H(p, q, t) is an analytic function of p, q, t and periodic in t with period 2π. A case is called elliptic if the equilibrium position is stable in the first (linear) approximation. Then, as was shown by Birkhoff [1], by a proper choice of the variables p, q, t the Hamiltonian assumes the form

$$ \label{2} H = \lambda r + c_{2}r^{2} + ... + c_{n}r^{n} + \tilde{H}(p, q, t), $$

((2))

where $2r = p^{2} + q^{2}, \tilde{H} = 0(r^{n+1})$ is an analytic function of p, q, t, n ≥ 2 and arbitrary. We call a case a general elliptic case if among the constants c _l (2 ≤ l < ∞) is different from zero.