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
$$
\frac{dv}{d\tau}=f_2(\tau)u^2+f_1(\tau)u+f_0(\tau), \,\, \frac{du}{d\tau}=uv+g(\tau),
$$
where the coefficients are periodic functions of time $\tau$ 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.
This paper continues a series of studies [1–5] of the
dynamics of nonholonomic systems with vary
ing mass distribution due to the prescribed pe
riodic motion of some structural components (ro
tor, point masses etc.). Depending on the choice
of the law of variation of mass distribution, such
systems generally exhibit a large variety of be
havior, both regular and chaotic. In addition,
it turns out that nonholonomic systems are one
of the simplest mathematical models of mechan
ical systems exhibiting a phenomenon known as
unbounded speedup, which is due to redistribu
tion of internal masses. In this paper, for a cer
tain class of nonholonomic systems (including the
Chaplygin sleigh and the Suslov system) we find a
criterion which must be satisfied by the periodic
variation of mass distribution for the existence of
speedingup trajectories.
Citation: 
Bizyaev I. A., Bolotin S. V., Mamaev I. S., Normal forms and averaging in an acceleration problem in nonholonomic mechanics, Chaos, 2021, vol. 31, 013132, 16 pp. 
DOI: 
10.1063/5.0030889 
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