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

Consider the ensemble of Gaussian random potentials {VL(q)}infinity 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 -> infinity, 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. (c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY -NC license (http:// creativecommons .org /licenses /by -nc /4 .0/).