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Normal forms and averaging in an acceleration problem in nonholonomic mechanics.

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This study explores nonholonomic systems with changing mass, revealing conditions for indefinite speed-up, analogous to Fermi acceleration. Specific mathematical conditions lead to velocity growth, with variable v exhibiting asymptotic behavior.

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Area of Science:

  • Physics
  • Mechanical Engineering
  • Applied Mathematics

Background:

  • Nonholonomic systems, such as the Chaplygin sleigh and Suslov system, are fundamental in mechanics.
  • Periodically varying mass distribution introduces complex dynamics.
  • Understanding indefinite acceleration phenomena is crucial for advanced mechanical designs.

Purpose of the Study:

  • To investigate the existence of modes for indefinite acceleration in nonholonomic systems with periodically varying mass.
  • To analyze the asymptotic behavior of velocities under specific dynamic conditions.
  • To identify regions within the phase space that exhibit accelerated motion.

Main Methods:

  • Analysis of a system of differential equations describing velocity dynamics.
  • Application of normal forms and averaging techniques.
  • Investigation of asymptotic behavior of system variables.

Main Results:

  • Demonstrated that indefinite acceleration (Fermi acceleration analog) is possible.
  • Proved that the variable v exhibits asymptotic behavior of t^(1/k) for k=1, 2, 3.
  • Identified specific regions in the phase space leading to observed speed-up.

Conclusions:

  • The study confirms the possibility of indefinite acceleration in specific nonholonomic systems.
  • The findings provide insights into the conditions governing accelerated motion in systems with time-varying parameters.
  • The research contributes to the theoretical understanding of complex dynamical systems.