Euler-Poincare-Suslov方程的几何与可积性
Geometry and integrability of Euler-Poincare-Suslov equations
摘要：
我们考虑紧连通李群上左不变度量和左不变不可积分布的非完整测地线流。将测地线流方程化为相应李代数上的Euler-Poincare-Suslov方程。泊松和辛结构给出了可积哈密顿系统的各种代数构造。另一方面，非完整系统不是哈密顿系统，非完整系统的积分方法还很不成熟。在本文中，使用链的子代数，我们给出的结构，导致一个大的第一积分集和可积的情况下的Euler-Poincare-Suslov方程。进一步，我们给出了非完整测地线流的例子，这些非完整测地线流可以看作是可积次黎曼测地线流的一个限制。
英文摘要：
We consider nonholonomic geodesic flows of left-invariant metrics and left-invariant nonintegrable distributions on compact connected Lie groups. The equations of geodesic flows are reduced to the Euler-Poincare-Suslov equations on the corresponding Lie algebras. The Poisson and symplectic structures give raise to various algebraic constructions of the integrable Hamiltonian systems. On the other hand, nonholonomic systems are not Hamiltonian and the integration methods for nonholonomic systems are much less developed. In this paper, using chains of subalgebras, we give constructions that lead to a large set of first integrals and to integrable cases of the Euler-Poincare-Suslov equations. Further, we give examples of nonholonomic geodesic flows that can be seen as a restrictions of integrable sub-Riemannian geodesic flows.
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