Riemannian metrics on 2D-manifolds related to the Euler-Poinsot rigid body motion. (English) Zbl 1293.49040

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

ESAIM, Control Optim. Calc. Var. 20, No. 3, 864-893 (2014).

Summary: 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 \(S^{2}\) associated to the dynamics of spin particles with Ising coupling is analyzed using both geometric techniques and numerical simulations.

MSC:

49K15	Optimality conditions for problems involving ordinary differential equations
49Q99	Manifolds and measure-geometric topics
53C20	Global Riemannian geometry, including pinching
70Q05	Control of mechanical systems
81Q93	Quantum control

Keywords:

Euler-Poinsot rigid body motion; conjugate locus; surfaces of revolution; Serret-Andoyer metric; spin dynamics