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

Bernard Bonnard (1) , Olivier Cots (2) , Jean-Baptiste Pomet (2) , Nataliya Shcherbakova (3)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]
2 McTAO - Mathematics for Control, Transport and Applications
3 LGC - Laboratoire de Génie Chimique

The Euler-Poinsot rigid body motion is a standard mechanical system and is a model for left-invariant Riemannian metric 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 work 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.

Mots clés: Euler-Poinsot rigid body motion

Domaines: Optimisation et contrôle [math.OC]

Bernard Bonnard, Olivier Cots, Jean-Baptiste Pomet, Nataliya Shcherbakova. Riemannian metrics on 2D-manifolds related to the Euler-Poinsot rigid body motion. ESAIM: Control, Optimisation and Calculus of Variations, 2014, 20 (3), pp.864-893. ⟨10.1051/cocv/2013087⟩. ⟨hal-00918587v2⟩