标题： 泊松几何在约束系统中的应用 显示英文标题

标题： Poisson Geometry in Constrained Systems

摘要： 约束哈密顿系统属于预辛几何的范畴。 然而，我们表明泊松几何在此背景下也是有用的。 当约束形成一个闭合代数时，该系统有两个与之相关的自然泊松流形，它们相对于原始的、无约束的相空间形成辛对偶。 我们提供了充分条件，以便可以将约束系统的约化相空间识别为其中之一中的辛叶。 在第二类情况下，通过这些方法，原始的约束系统可以等效地重新表述为扩展相空间中的阿贝尔第一类系统。 受一般第二类约束系统狄拉克括号与原始无约束相空间之间关系的启发，我们探讨了这样一个问题：一个正则泊松流 manifold 是否允许其叶在辛流 manifold 中进行辛嵌入。 这是必要且充分的条件是泊松张量的特征形式类在第三相对上同调中的某个元素消失。 显示英文摘要

摘要： Constrained Hamiltonian systems fall into the realm of presymplectic geometry. We show, however, that also Poisson geometry is of use in this context. For the case that the constraints form a closed algebra, there are two natural Poisson manifolds associated to the system, forming a symplectic dual pair with respect to the original, unconstrained phase space. We provide sufficient conditions so that the reduced phase space of the constrained system may be identified with a symplectic leaf in one of those. In the second class case the original constrained system may be reformulated equivalently as an abelian first class system in an extended phase space by these methods. Inspired by the relation of the Dirac bracket of a general second class constrained system to the original unconstrained phase space, we address the question of whether a regular Poisson manifold permits a leafwise symplectic embedding into a symplectic manifold. Necessary and sufficient for this is the vanishing of the characteristic form-class of the Poisson tensor, a certain element of the third relative cohomology.