Hamiltonian Systems with a Functional Parameter in the Form of a Potential

Abstract

Let $T=T(h)$ be a real-analytic at $0\in \mathbb{R}$ function, $T(0)=\pi$. Let $H(x,y) = x^2+y^2 + {\cal O} _4(x,y)$ be a real-analytic at $0\in \mathbb{R} ^2$ even function. We prove that there exists a real-analytic $\varphi = \varphi (x) = {\cal O} _4(x)$ even function such that any solution of the Hamiltonian equations

$$\dot x = \partial_y(H + \varphi ), \quad \dot y = - \partial_x(H + \varphi )$$

near the origin on the energy level $h>0$ has the period $T(h)$. We discuss motivations and possible generalizations of this result.