71,698 research outputs found
Hamiltonisation, measure preservation and first integrals of the multi-dimensional rubber Routh sphere
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 Garc\'ia-Naranjo (arXiv: 1805:06393)
and Garc\'ia-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.Comment: 23 pages, 1 figure. Submitted to the special issue of Theor. Appl.
Mech. in honour of Chaplygin's 150th anniversar
Non-existence of an invariant measure for a homogeneous ellipsoid rolling on the plane
It is known that the reduced equations for an axially symmetric homogeneous
ellipsoid that rolls without slipping on the plane possess a smooth invariant
measure. We show that such an invariant measure does not exist in the case when
all of the semi-axes of the ellipsoid have different length.Comment: v2: Minor changes after journal review. This text uses the theory
developed in arXiv:1304.1788 for the specific example of a homogeneous
ellipsoid rolling on the plan
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