Geodesic flows on real forms of complex semi-simple Lie groups of rigid body type
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RESEARCH
Geodesic flows on real forms of complex semi-simple Lie groups of rigid body type Tudor S. Ratiu1,2,3 and Daisuke Tarama4* * Correspondence:
[email protected] Department of Mathematical Sciences, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577, Japan Full list of author information is available at the end of the article 4
Abstract The geodesic flows are studied on real forms of complex semi-simple Lie groups with respect to a left-invariant (pseudo-)Riemannian metric of rigid body type. The Williamson types of the isolated relative equilibria on generic adjoint orbits are determined. Keywords: Geodesic flow, Real semi-simple Lie group, Free rigid body, Cartan subalgebra, Bi-Hamiltonian structure, Integrable system, Equilibrium, Williamson type, Lyapunov stability Mathematics Subject Classification: 34D20, 53D25, 70E15, 70E45
1 Introduction In theoretical mechanics, the motion of a free rigid body, i.e., a rigid body under no external forces, is one of the classical solvable problems. Its integrability and the stability of its equilibria are well known since the works of Euler, Poinsot, and Jacobi. In the 1970s, under the influence of the rapid development of infinite-dimensional integrable systems, such as the Korteweg–de Vries equation, free rigid body dynamics has been generalized, first to the higher-dimensional rotation groups SO(n) [9,22,24,31] and, later, to general semi-simple Lie groups [26,27]. Extensions to other Lie groups and symmetric spaces have also been considered [12]. From a differential geometric point of view, the motion of the above (generalized) free rigid bodies can be formulated as Hamiltonian systems on (co)tangent bundles to Lie groups for a left-invariant Hamiltonian. In addition, these Hamiltonian systems are equivalent to geodesic flows for certain left-invariant metrics on semi-simple Lie groups, called of rigid body type. They can be reduced to Lie–Poisson systems described by the Euler equation on the Lie algebra. Moreover, the complete integrability of the geodesic flow follows from that of the Euler equation. In the study of Hamiltonian dynamics, the analysis around equilibrium points is one of the most important issues. The equilibria of the Euler equation on generic adjoint orbits are relative equilibria of the geodesic flow. Thus, the dynamics around the relative equilibria of the geodesic flow is determined by the dynamical behavior of the Euler equation around its equilibria.
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T. S. Ratiu, D. Tarama Res Math Sci (2020)7:32
Bolsinov and Oshemkov [7] have given a systematic method of dealing with the complete integrability and the non-degeneracy of equilibria for bi-Hamiltonian systems on Poisson manifolds. A Hamiltonian system with respect to a Poisson bracket on a manifold is called bi-Hamiltonian, if it is also a Hamiltonian system with respect to another Poisson bracket on the same manifold which together with the original Poisson bracket generates a pencil o
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