Geometrical approach for description of the mixed state in multi-well potentials

Abstract

We use so-called geometrical approach in description of transition from regular motion to chaotic in Hamiltonian systems with potential energy surface that has several local minima. Distinctive feature of such systems is coexistence of different types of dynamics (regular or chaotic) in different wells at the same energy Mixed state reveals unique opportunities in research of quantum manifestations of classical stochasticity. Application of traditional criteria for transition to chaos (resonance overlap criterion, negative curvature criterion and stochastic layer destruction criterion) is inefficient in case of potentials with complex topology. Geometrical approach allows considering only configuration space but not phase space when investigating stability. Trajectories are viewed as geodesics of configuration space equipped with suitable metric. In this approach all information about chaos and regularity consists in potential function. The aim of this work is to determine what details of geometry of potential lead to chaos in Hamiltonian systems using geometrical approach. Numerical calculations are executed for potentials that are relevant with lowest umbilical catastrophes.