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Riemannian metrics on 2D-manifolds related to the Euler−Poinsot rigid body motion∗

Published online by Cambridge University Press:  10 June 2014

Bernard Bonnard ,
Olivier Cots ,
Jean-Baptiste Pomet  and
Nataliya Shcherbakova
Bernard Bonnard
Affiliation:
Institut de Mathématiques de Bourgogne, 9 avenue Savary, 21078 Dijon, France. bernard.bonnard@u-bourgogne.fr
Olivier Cots
Affiliation:
INRIA Sophia-Antipolis Méditerranée, 2004, route des Lucioles, 06902 Sophia Antipolis, France; olivier.cots@inria.fr
Jean-Baptiste Pomet
Affiliation:
INRIA Sophia-Antipolis Méditerranée, 2004, route des Lucioles, 06902 Sophia Antipolis, France; olivier.cots@inria.fr
Nataliya Shcherbakova
Affiliation:
Université de Toulouse, INPT, UPS, Laboratoire de Génie Chimique, 4 allée Emile Monso, 31432 Toulouse, France; nataliya.shcherbakova@inp-toulouse.fr
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Abstract

The Euler−Poinsot rigid body motion is a standard mechanical system and it is a model for left-invariant Riemannian metrics on SO(3). In this article using the Serret−Andoyer variables we parameterize the solutions and compute the Jacobi fields in relation with the conjugate locus evaluation. Moreover, the metric can be restricted to a 2D-surface, and the conjugate points of this metric are evaluated using recent works on surfaces of revolution. Another related 2D-metric on S2 associated to the dynamics of spin particles with Ising coupling is analysed using both geometric techniques and numerical simulations.

Keywords

Euler−poinsot rigid body motionconjugate locus on surfaces of revolutionSerret−Andoyer metricspins dynamics
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 3 , July 2014 , pp. 864 - 893
Copyright
© EDP Sciences, SMAI, 2014

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