Fermi-like acceleration and power-law energy growth in nonholonomic systems

TL;DR

This paper investigates energy growth in a nonholonomic system, specifically a Chaplygin sleigh with time-varying mass, demonstrating conditions for unbounded acceleration and effects of viscous friction.

Contribution

It provides a detailed analysis of Fermi-like acceleration in a nonholonomic system with parametric excitation, revealing conditions for unbounded energy growth and the impact of friction.

Findings

Existence of trajectories with unbounded energy growth and asymptotic velocity $ au^{1/3}$.

Unbounded acceleration disappears when viscous friction is introduced.

Trajectories tend to a limit cycle under viscous friction.

Abstract

This paper is concerned with a nonholonomic system with parametric excitation - the Chaplygin sleigh with time-varying mass distribution. A detailed analysis is made of the problem of the existence of regimes with unbounded growth of energy (an analogue of Fermi's acceleration) in the case where excitation is achieved by means of a rotor with variable angular momentum. The existence of trajectories for which the translational velocity of the sleigh increases indefinitely and has the asymptotics $\tau^{\frac{1}{3}}$ is proved. In addition, it is shown that, when viscous friction with a nondegenerate Rayleigh function is added, unbounded speed-up disappears and the trajectories of the reduced system asymptotically tend to a limit cycle.