Hamiltonisation, measure preservation and first integrals of the multi-dimensional rubber Routh sphere
arXiv:1901.11092 · doi:10.2298/TAM190130004G
Abstract
We consider the multi-dimensional generalisation of the problem of a sphere, with axi-symmetric mass distribution, that rolls without slipping or spinning over a plane. Using recent results from García-Naranjo (arXiv: 1805:06393) and García-Naranjo and Marrero (arXiv: 1812.01422), we show that the reduced equations of motion possess an invariant measure and may be represented in Hamiltonian form by Chaplygin's reducing multiplier method. We also prove a general result on the existence of first integrals for certain Hamiltonisable Chaplygin systems with internal symmetries that is used to determine conserved quantities of the problem.
arXiv comment23 pages, 1 figure. Submitted to the special issue of Theor. Appl. Mech. in honour of Chaplygin's 150th anniversary
References in corpus (9)
- Nonholonomic LR systems as Generalized Chaplygin systems with an Invariant Measure and Geodesic Flows on Homogeneous Spaces
- Hamiltonization and Integrability of the Chaplygin Sphere in R^n
- Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization
- Some multidimensional integrable cases of nonholonomic rigid body dynamics
- Gauge momenta as Casimir functions of nonholonomic systems
- Rolling balls over spheres in R^n
- Generalisation of Chaplygin's Reducing Multiplier Theorem with an application to multi-dimensional nonholonomic dynamics
- Integrability and dynamics of the n-dimensional symmetric Veselova top
- A Geometric Characterization of Certain First Integrals for Nonholonomic Systems with Symmetries
Cited by in corpus (6)
- Gyroscopic Chaplygin systems and integrable magnetic flows on spheres
- Generalisation of Chaplygin's Reducing Multiplier Theorem with an application to multi-dimensional nonholonomic dynamics
- Integrable systems in cosymplectic geometry
- Demchenko's nonholonomic case of gyroscopic ball rolling without sliding over a sphere after his 1923 Belgrade doctoral thesis
- Spherical and planar ball bearings -- nonholonomic systems with invariant measures
- Spherical and Planar Ball Bearings -- a Study of Integrable Cases