Non-integrability and chaos for natural Hamiltonian systems with a random potential

Abstract

Consider the ensemble of Gaussian random potentials $\{V^L(q){\}}_{L=1}^{\infty }$ on the $d$-dimensional torus where, essentially, $V^L(q)$ is a real-valued trigonometric polynomial of degree $L$ whose coefficients are independent standard normal variables. Our main result ensures that, with a probability tending to 1 as $L\to \infty$, the dynamical system associated with the natural Hamiltonian function defined by this random potential, $H^L:=\frac{1}{2}|p{|}^2+V^L(q)$, exhibits a number of chaotic regions which coexist with a positive-volume set of invariant tori. In particular, these systems are typically neither integrable with non-degenerate first integrals nor ergodic. An analogous result for random natural Hamiltonian systems defined on the cotangent bundle of an arbitrary compact Riemannian manifold is presented too.