2021/05/05 by Sergio Grillo, Juan Carlos Marrero, Grillo, Sergio +3
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2105.02130
openalex publication_date 2021/05/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we study the extended Hamilton-Jacobi Theory in the context of\ndynamical systems with symmetries. Given an action of a Lie group G on a\nmanifold M and a G-invariant vector field X on M, we construct complete\nsolutions of the Hamilton-Jacobi equation (HJE) related to X (and a given\nfibration on M). We do that along each open subset U\⊆ M such that\n\π\(U\) has a manifold structure and\n\π\|U\.:U\→\π\(U\), the restriction to U of\nthe canonical projection \π:M\→ M/G, is a surjective submersion. If\nX\|U\. is not vertical with respect to \π\|U\., we\nshow that such complete solutions solve the "reconstruction equations" related\nto X\|U\. and G, i.e., the equations that enable us to write\nthe integral curves of X\|U\. in terms of those of its projection\non \π\(U\). On the other hand, if X\|U\. is vertical,\nwe show that such complete solutions can be used to construct (around some\npoints of U) the integral curves of X\|U\. up to quadratures.\nTo do that we give, for some elements \ξ of the Lie algebra mathfrakg\nof G, an explicit expression up to quadratures of the exponential curve\n\exp\(\ξ ,t\), different to that appearing in the literature for\nmatrix Lie groups. In the case of compact and of semisimple Lie groups, we show\nthat such expression of \exp\(\ξ ,t\) is valid for all \ξ inside\nan open dense subset of mathfrakg.\n