李群上的三次多项式：哈密顿系统的约化
Cubic polynomials on Lie groups: reduction of the Hamiltonian system
摘要：
本文从Hamilton的观点及其对称性出发，分析了紧李群上三次多项式的最优控制问题。该问题的动力学由李群及其李代数的半直积的余切丛上的标准辛形式的前辛形式描述。利用这些控制几何工具，这里开发的哈密顿方法和已知的变分之一之间的关系进行了分析。在明确了左平凡化系统后，我们利用Marsden-Weinstein约化技术去除了哈密顿系统的对称性。鉴于约化动力学，我们能够保证，通过Lie-Cartan定理，存在相当数量的独立积分的对合运动。
英文摘要：
This paper analyzes the optimal control problem of cubic polynomials on compact Lie groups from a Hamiltonian point of view and its symmetries. The dynamics of the problem is described by a presymplectic formalism associated with the canonical symplectic form on the cotangent bundle of the semidirect product of the Lie group and its Lie algebra. Using these control geometric tools, the relation between the Hamiltonian approach developed here and the known variational one is analyzed. After making explicit the left trivialized system, we use the technique of Marsden-Weinstein reduction to remove the symmetries of the Hamiltonian system. In view of the reduced dynamics, we are able to guarantee, by means of the Lie-Cartan theorem, the existence of a considerable number of independent integrals of motion in involution.
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