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