A generalization of the Hamilton-Jacobi theory to arbitrary dynamical systems, including non-Hamiltonian ones, is considered. The generalized Hamilton-Jacobi theory is constructed as a theory of ensemble of identical systems moving in the configuration space and described by the continual equation of motion and the continuity equation. For Hamiltonian systems, the usual Hamilton-Jacobi equations naturally follow from this theory. The proposed formulation of the Hamilton-Jacobi theory, as the theory of ensemble, allows interpreting in a natural way the transition from quantum mechanics in the Schrödinger form to classical mechanics.

中文翻译：

---

哈密​​顿系统和非哈密顿系统的哈密顿-雅各比理论

考虑将汉密尔顿-雅各比理论推广到任意动力学系统，包括非汉密尔顿系统。广义汉密尔顿-雅各比理论被构造为同一系统在构型空间中运动的整体理论，并由运动的连续方程和连续性方程来描述。对于汉密尔顿系统，通常的汉密尔顿-雅各比方程式自然会遵循该理论。汉密尔顿-雅各比理论的拟议表述作为整体理论，可以自然地解释从薛定er形式的量子力学到古典力学的转变。