On the accuracy of conservation of adiabatic invariants in slow-fast systems

TL;DR

This paper investigates the precision of adiabatic invariants in slow-fast Hamiltonian systems, providing estimates for their conservation accuracy using complex analysis and canonical transformations.

Contribution

It introduces a method combining iso-energetic reduction and complex canonical transformations to estimate adiabatic invariant conservation in analytic systems.

Findings

Exponential smallness of the difference between limiting values

Application of complexified phase space methods

Estimation of conservation accuracy for adiabatic invariants

Abstract

Let the adiabatic invariant of action variable in slow-fast Hamiltonian system with two degrees of freedom have two limiting values along the trajectories as time tends to infinity. The difference of two limits is exponentially small in analytic systems. An iso-energetic reduction and canonical transformations are applied to transform the slow-fast systems to form of systems depending on slowly varying parameters in a complexified phase space. On the basis of this method an estimate for the accuracy of conservation of adiabatic invariant is given for such systems.