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 $U\subseteq M$, such that $\pi \left(U\right)$ has a manifold structure and ${\pi }_{\left|U\right.}:U\to \pi \left(U\right)$, the restriction to U of the canonical projection $\pi :M\to M/G$, is a surjective submersion. If $X_{\left|U\right.}$ is not vertical with respect to ${\pi }_{\left|U\right.}$, we show that such complete solutions solve the reconstruction equations related to $X_{\left|U\right.}$ and G, i.e., the equations that enable us to write the integral curves of $X_{\left|U\right.}$ in terms of those of its projection on $\pi \left(U\right)$. On the other hand, if $X_{\left|U\right.}$ is vertical, we show that such complete solutions can be used to construct (around some points of U) the integral curves of $X_{\left|U\right.}$ up to quadratures. To do that, we give, for some elements $\xi$ of the Lie algebra $\mathfrak{g}$ of G, an explicit expression up to quadratures of the exponential curve $exp\left(\xi t\right)$, 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 $exp\left(\xi t\right)$ is valid for all $\xi$ inside an open dense subset of $\mathfrak{g}$.

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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$你\subseteq 米$，这样 $\pi \left(你\right)$ 具有流形结构和 ${\pi }_{\left|你\right.}：你\to \pi \left(你\right)$,规范投影对U的限制$\pi ：米\to 米/G$, 是一个满射浸没。如果$X_{\left|你\right.}$ 不是垂直的 ${\pi }_{\left|你\right.}$，我们表明这样的完全解解决了与相关的重建方程 $X_{\left|你\right.}$和G，即使我们能够写出积分曲线的方程$X_{\left|你\right.}$ 就其对 $\pi \left(你\right)$. 另一方面，如果$X_{\left|你\right.}$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$X_{\left|你\right.}$直到正交。为了做到这一点，我们给出，对于某些元素$\xi$ 李代数的 $\mathfrak{G}$的G，一个显式表达式，直到指数曲线的正交$经验值\left(\xi 吨\right)$，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$经验值\left(\xi 吨\right)$ 对所有人都有效 $\xi$ 在一个开放的密集子集内 $\mathfrak{G}$.