Katarzyna Grabowska, Janusz Grabowski, Marek Kuś and Giuseppe Marmo
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引用次数: 0
摘要
如果 η 是流形 M 上的接触形式,使得里布向量场的轨道在 M 上形成简单折射,那么 M 上的前折射 2 形式会在商流形上诱导出交映结构 ω 。我们称之为交映流形的接触化。首先,我们提出了连通李群某些共轭轨道接触化的明确几何构造。我们的构造是对众所周知的复投影空间Ⅳ的接触化的进一步概括,Ⅳ是Ⅳ中的单位球,并在Ⅳ上配备了Liouville 1-form的限制。其次,我们描述了在 Marsden-Weinstein-Meyer 对称还原过程中获得接触化的构造过程,并指出了紧凑接触化存在的几何障碍。第三,我们证明了接触化为紧凑交映流形上哈密尔顿系统的拉格朗日描述提供了一个很好的几何工具,而在紧凑交映流形上交映形式永远不承认 "矢量势"。
Contactifications: a Lagrangian description of compact Hamiltonian systems *
If η is a contact form on a manifold M such that the orbits of the Reeb vector field form a simple foliation on M, then the presymplectic 2-form on M induces a symplectic structure ω on the quotient manifold . We call a contactification of the symplectic manifold . First, we present an explicit geometric construction of contactifications of some coadjoint orbits of connected Lie groups. Our construction is a far going generalization of the well-known contactification of the complex projective space , being the unit sphere in , and equipped with the restriction of the Liouville 1-form on . Second, we describe a constructive procedure for obtaining contactification in the process of the Marsden–Weinstein–Meyer symplectic reduction and indicate geometric obstructions for the existence of compact contactifications. Third, we show that contactifications provide a nice geometrical tool for a Lagrangian description of Hamiltonian systems on compact symplectic manifolds , on which symplectic forms never admit a ‘vector potential’.
期刊介绍:
Publishing 50 issues a year, Journal of Physics A: Mathematical and Theoretical is a major journal of theoretical physics reporting research on the mathematical structures that describe fundamental processes of the physical world and on the analytical, computational and numerical methods for exploring these structures.