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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.

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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