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

Abstract

Consider the ensemble of Gaussian random potentials {VL(q)}L=1∞ on the d-dimensional torus where, essentially, VL(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→∞, the dynamical system associated with the natural Hamiltonian function defined by this random potential, HL:=12|p|2+VL(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.