论文标题
动量映射和降低接触汉密尔顿系统
Momentum mapping and reduction in contact Hamiltonian systems
论文作者
论文摘要
由于出现了符号几何形状，在上个世纪，力学的几何处理经历了巨大的发展。在这种情况下，哈密顿系统中对称性的压力自然导致存在保守数量。这种运动的积分是通过众所周知的动量映射来描述的。此外，如果系统在某个组的作用下是不变的，则可以通过称为还原的过程来简化运动方程。在接触几何的框架中，也可以考虑此过程，这是一个近期的研究领域。事实证明，这种几何形状在与热力学，控制理论或神经几阶的区域不同。我们对接触几何形状的了解要比符合性几何学要小得多，因此称为Symphectificacion的过程非常有用，因为它允许在符号框架中研究接触问题。 文本的目的是研究还原和符合性过程之间的通勤关系。为此，我们首先介绍了符号和接触几何形状的基础，并在两种情况下都展示了如何通过动量图进行减少。众所周知的共同体还原定理在描述中至关重要。最后，详细分析了符号化过程，并研究了其与象征性和接触减少的关系。
Due to the emergence of symplectic geometry, the geometric treatment of mechanics underwent a great development during the last century. In this scenario the pressence of symmetries in Hamiltonian systems leads naturally to the existence of conserved quantities. This integrals of motions are described by the well-known momentum mapping. Furthermore, the equations of motion can be simplified by a process known as reduction, if the system is invariant under the action of a certain group. This process can also be considered in the framework of contact geometry, a much more recent field of study. This kind of geometry has proven to be valuable in areas as different as thermodynamics, control theory, or neurogeometry. Our knowledge about contact geometry is much smaller than that of symplectic, and so a process known as symplectificacion is extremely useful, since it allows to study contact problems in the symplectic frame. The aim of the text is to study the commutativity relations between the processes of reduction and symplectification. To do so, we first introduce the basis of symplectic and contact geometry and we show how to perform the reduction via the momentum map in both scenarios. The well-known coisotropic reduction theorem will be crucial in the description. Finally, the symplectification process is analyzed in detail, and its relation with symplectic and contact reduction is studied.
