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Extended Hamilton–Jacobi Theory, Symmetries and Integrability by Quadratures
Mathematics ( IF 2.3 ) Pub Date : 2021-06-11 , DOI: 10.3390/math9121357 Sergio Grillo , Juan Carlos Marrero , Edith Padrón
Mathematics ( IF 2.3 ) Pub Date : 2021-06-11 , DOI: 10.3390/math9121357 Sergio Grillo , Juan Carlos Marrero , Edith Padrón
In this paper, we study the extended Hamilton–Jacobi Theory in the context of dynamical systems with symmetries. Given an action of a Lie group G on a manifold M and a G-invariant vector field X on M, we construct complete solutions of the Hamilton–Jacobi equation (HJE) related to X (and a given fibration on M). We do that along each open subset , such that has a manifold structure and , the restriction to U of the canonical projection , is a surjective submersion. If is not vertical with respect to , we show that such complete solutions solve the reconstruction equations related to and G, i.e., the equations that enable us to write the integral curves of in terms of those of its projection on . On the other hand, if is vertical, we show that such complete solutions can be used to construct (around some points of U) the integral curves of up to quadratures. To do that, we give, for some elements of the Lie algebra of G, an explicit expression up to quadratures of the exponential curve , different to that appearing in the literature for matrix Lie groups. In the case of compact and of semisimple Lie groups, we show that such expression of is valid for all inside an open dense subset of .
中文翻译:
通过正交扩展 Hamilton-Jacobi 理论、对称性和可积性
在本文中,我们在具有对称性的动力系统的背景下研究了扩展的哈密顿-雅可比理论。给定一个李群的动作ģ上的歧管中号和ģ -invariant矢量场X上中号,我们构建汉密尔顿-雅可比等式(HJE)相关的完整的解决方案X(和在给定的纤维化中号)。我们沿着每个开放的子集这样做 ,这样 具有流形结构和 ,规范投影对U的限制 , 是一个满射浸没。如果 不是垂直的 ,我们表明这样的完全解解决了与相关的重建方程 和G,即使我们能够写出积分曲线的方程 就其对 . 另一方面,如果 是垂直的,我们表明这种完整的解决方案可用于构造(围绕U 的某些点)的积分曲线 直到正交。为了做到这一点,我们给出,对于某些元素 李代数的 的G,一个显式表达式,直到指数曲线的正交 ,与文献中出现的矩阵李群不同。在紧李群和半单李群的情况下,我们证明了 对所有人都有效 在一个开放的密集子集内 .
更新日期:2021-06-11
中文翻译:
通过正交扩展 Hamilton-Jacobi 理论、对称性和可积性
在本文中,我们在具有对称性的动力系统的背景下研究了扩展的哈密顿-雅可比理论。给定一个李群的动作ģ上的歧管中号和ģ -invariant矢量场X上中号,我们构建汉密尔顿-雅可比等式(HJE)相关的完整的解决方案X(和在给定的纤维化中号)。我们沿着每个开放的子集这样做