Normal forms and averaging in an acceleration problem in nonholonomic mechanics

This paper investigates nonholonomic systems (the Chaplygin sleigh and the Suslov system) with periodically varying mass distribution. In these examples, the behavior of velocities is described by a system of the form d v d τ = f 2 ( τ ) u 2 + f 1 ( τ ) u + f 0 ( τ ) , d u d τ = − u v + g ( τ ), where the coefficients are periodic functions of time τ with the same period. A detailed analysis is made of the problem of the existence of modes of motion for which the system speeds up indefinitely (an analog of Fermi’s acceleration). It is proved that, depending on the choice of coefficients, variable v has the asymptotics t 1 k , k = 1 , 2 , 3. In addition, we show regions of the phase space for which the system, when the trajectories are started from them, is observed to speed up. The proof uses normal forms and averaging in a slightly unusual form since unusual form averaging is performed over a variable that is not fast.