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

Bonnard, Bernard; Cots, Olivier; Pomet, Jean-Baptiste; Shcherbakova, Nataliya

ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 3, pp. 864-893.

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.

Classification : 49K15, 53C20, 70Q05, 81Q93
Mots clés : Euler−poinsot rigid body motion, conjugate locus on surfaces of revolution, Serret−Andoyer metric, spins dynamics

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