论文标题
回顾有关雅各比和莱布尼兹身份的几何汉密尔顿 - 雅各比理论
Reviewing the Geometric Hamilton-Jacobi Theory concerning Jacobi and Leibniz identities
论文作者
论文摘要
在这项调查中，我们从不同几何背景中的几何学角度回顾了古典汉密尔顿雅各比理论。我们为汉密尔顿雅各布方程提出了一个特定特征的不同几何结构：它们是否同时满足雅各比和莱布尼兹的身份，或者至少在他们满足其中之一时。 在这方面，我们将依赖时间和耗散物理系统作为满足雅各比身份而不是莱布尼茨身份的系统。此外，我们将接触进化论汉密尔顿雅各比理论视为定期接触几何形状的分裂，实际上满足了莱布尼兹规则而不是雅各比。 此外，我们还包括一个新的结果，这是汉密尔顿 - 雅各比方程的汉密尔顿载体场作为众所周知的汉密尔顿·雅各比（Hamilton Jacobi）在符号歧管上的概括，在零保串因子的情况下可以检索到它。 The interest of a geometric Hamilton Jacobi equation is the primordial observation that if a Hamiltonian vector field can be projected into a configuration manifold by means of a 1-form dW, then the integral curves of the projected vector field can be transformed into integral curves of the Hamiltonian vector field provided that W is a solution of the Hamilton-Jacobi equation.从几何学上讲，汉密尔顿雅各比方程的解决方案扮演着某个捆绑包的拉格朗日式submanifold的角色。在不同的几何场景中利用这些特征，我们根据其动态所满足的基本身份为多个物理系统提出了一种几何理论。图为不同的示例以反映所提供的结果，这是所有新的，除了重新评估先前被认为的示例的结果。
In this survey, we review the classical Hamilton Jacobi theory from a geometric point of view in different geometric backgrounds. We propose a Hamilton Jacobi equation for different geometric structures attending to one particular characterization: whether they fulfill the Jacobi and Leibniz identities simultaneously, or if at least they satisfy one of them. In this regard, we review the case of time dependent and dissipative physical systems as systems that fulfill the Jacobi identity but not the Leibnitz identity. Furthermore, we review the contact evolution Hamilton Jacobi theory as a split off the regular contact geometry, and that actually satisfies the Leibniz rule instead of Jacobi. Furthermore, we include a novel result, which is the Hamilton-Jacobi equation for conformal Hamiltonian vector fields as a generalization of the well known Hamilton Jacobi on a symplectic manifold, that is retrieved in the case of a zero conformal factor. The interest of a geometric Hamilton Jacobi equation is the primordial observation that if a Hamiltonian vector field can be projected into a configuration manifold by means of a 1-form dW, then the integral curves of the projected vector field can be transformed into integral curves of the Hamiltonian vector field provided that W is a solution of the Hamilton-Jacobi equation. Geometrically, the solution of the Hamilton Jacobi equation plays the role of a Lagrangian submanifold of a certain bundle. Exploiting these features in different geometric scenarios we propose a geometric theory for multiple physical systems depending on the fundamental identities that their dynamic satisfies. Different examples are pictured to reflect the results provided, being all of them new, except for one that is reassessment of a previously considered example.
