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.
Mots clés : 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}, pages = {864--893}, publisher = {EDP-Sciences}, volume = {20}, number = {3}, year = {2014}, doi = {10.1051/cocv/2013087}, mrnumber = {3264227}, zbl = {1293.49040}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2013087/} }
TY - JOUR AU - Bonnard, Bernard AU - Cots, Olivier AU - Pomet, Jean-Baptiste AU - Shcherbakova, Nataliya TI - Riemannian metrics on 2D-manifolds related to the Euler-Poinsot rigid body motion JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 864 EP - 893 VL - 20 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2013087/ DO - 10.1051/cocv/2013087 LA - en ID - COCV_2014__20_3_864_0 ER -
%0 Journal Article %A Bonnard, Bernard %A Cots, Olivier %A Pomet, Jean-Baptiste %A Shcherbakova, Nataliya %T Riemannian metrics on 2D-manifolds related to the Euler-Poinsot rigid body motion %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 864-893 %V 20 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2013087/ %R 10.1051/cocv/2013087 %G en %F 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, Tome 20 (2014) no. 3, pp. 864-893. doi : 10.1051/cocv/2013087. http://www.numdam.org/articles/10.1051/cocv/2013087/
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