On the Dynamics of a Heavy Symmetric Ball that Rolls Without Sliding on a Uniformly Rotating Surface of Revolution
| Autor: | Marco Dalla Via, Francesco Fassò, Nicola Sansonetto |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2022 |
| Předmět: |
Moving energies
Nonholonomic mechanical systems with symmetry Integrable systems Hamiltonization Relative equilibria Quasi-velocities Nonlinear Sciences - Exactly Solvable and Integrable Systems 37J15 70F25 70G45 Applied Mathematics General Engineering FOS: Physical sciences Mathematical Physics (math-ph) Dynamical Systems (math.DS) Modeling and Simulation FOS: Mathematics Mathematics - Dynamical Systems Exactly Solvable and Integrable Systems (nlin.SI) Mathematical Physics |
| Popis: | We study the class of nonholonomic mechanical systems formed by a heavy symmetric ball that rolls without sliding on a surface of revolution, which is either at rest or rotates about its (vertical) figure axis with uniform angular velocity $$\Omega $$ Ω . The first studies of these systems go back over a century, but a comprehensive understanding of their dynamics is still missing. The system has an $$\mathrm {SO(3)}\times \mathrm {SO(2)}$$ SO ( 3 ) × SO ( 2 ) symmetry and reduces to four dimensions. We extend in various directions, particularly from the case $$\Omega =0$$ Ω = 0 to the case $$\Omega \not =0$$ Ω ≠ 0 , a number of previous results and give new results. In particular, we prove that the reduced system is Hamiltonizable even if $$\Omega \not =0$$ Ω ≠ 0 and, exploiting the recently introduced “moving energy,” we give sufficient conditions on the profile of the surface that ensure the periodicity of the reduced dynamics and hence the quasiperiodicity of the unreduced dynamics on tori of dimension up to three. Furthermore, we determine all the equilibria of the reduced system, which are classified in three distinct families, and determine their stability properties. In addition to this, we give a new form of the equations of motion of nonholonomic systems in quasi-velocities which, at variance from the well-known Hamel equations, use any set of quasi-velocities and explicitly contain the reaction forces. |
| Databáze: | OpenAIRE |
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