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
Keywords: Euler−poinsot rigid body motion, conjugate locus on surfaces of revolution, Serret−Andoyer metric, spins dynamics
@article{COCV_2014__20_3_864_0,
author = {Bonnard, Bernard and Cots, Olivier and Pomet, Jean-Baptiste and Shcherbakova, Nataliya},
title = {Riemannian metrics on 2D-manifolds related to the Euler-Poinsot rigid body motion},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {20},
number = {3},
year = {2014},
pages = {864-893},
doi = {10.1051/cocv/2013087},
zbl = {1293.49040},
mrnumber = {3264227},
language = {en},
url = {http://www.numdam.org/item/COCV_2014__20_3_864_0}
}
Bonnard, Bernard; Cots, Olivier; Pomet, Jean-Baptiste; Shcherbakova, Nataliya. Riemannian metrics on 2D-manifolds related to the Euler-Poinsot rigid body motion. ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 3, pp. 864-893. doi : 10.1051/cocv/2013087. http://www.numdam.org/item/COCV_2014__20_3_864_0/
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