To describe time-dependent finite-dimensional mechanical systems, their generalized space-time is modeled as a Galilean manifold. On this basis, we present a geometric mechanical theory that unifies Lagrangian and Hamiltonian mechanics. Moreover, a general definition of force is given, such that the theory is capable of treating nonpotential forces acting on a mechanical system. Within this theory, we elaborate the interconnections between classical equations known from analytical mechanics such as the principle of virtual work, Lagrange's equations of the second kind, Hamilton's equations, Lagrange's central equation, Hamel's generalized central equation as well as Hamilton's principle.

中文翻译：

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伽利略流形上的虚功原理和哈密顿原理

为了描述与时间相关的有限维机械系统，它们的广义时空被建模为伽利略流形。在此基础上，我们提出了统一拉格朗日力学和哈密顿力学的几何力学理论。此外，给出了力的一般定义，使得该理论能够处理作用在机械系统上的非势力。在这个理论中，我们详细阐述了分析力学中已知的经典方程之间的相互联系，例如虚功原理、第二类拉格朗日方程、哈密顿方程、拉格朗日中心方程、哈默尔广义中心方程以及哈密顿原理。