Stochastic Averaging for Quasi-Integrable Hamiltonian Systems With Variable Mass

Abstract

Variable-mass systems become more and more important with the explosive development of micro- and nanotechnologies, and it is crucial to evaluate the influence of mass disturbances on system random responses. This manuscript generalizes the stochastic averaging technique from quasi-integrable Hamiltonian systems to stochastic variable-mass systems. The Hamiltonian equations for variable-mass systems are firstly derived in classical mechanics formulation and are approximately replaced by the associated conservative Hamiltonian equations with disturbances in each equation. The averaged Itô equations with respect to the integrals of motion as slowly variable processes are derived through the stochastic averaging technique. Solving the associated Fokker–Plank–Kolmogorov equation yields the joint probability densities of the integrals of motion. A representative variable-mass oscillator is worked out to demonstrate the application and effectiveness of the generalized stochastic averaging technique; also, the sensitivity of random responses to pivotal system parameters is illustrated.