Using the non-standard geometric structure proposed by Loos, we present a coordinate-free formulation of the theory for time-dependent finite-dimensional mechanical systems with n degrees of freedom. The state space containing the system’s information on time, position and velocity is defined as a (2n+1)-dimensional affine bundle over an (n+1)-dimensional generalized space-time. The main goal is to present a geometric postulate that characterizes a second-order vector field whose integral curves describe the motions of a time-dependent finite-dimensional mechanical system. The core objects of the postulate are differential two-forms on the state space, called action forms, which are in a bijective relation with second-order vector fields. The requirements for a differential two-form to be an action form allow for a coordinate-free definition of non-potential forces, which may depend on time, position and velocity. Finally, we show that not only Lagrange’s equations but also Hamilton’s equations follow directly as mere coordinate representations of the same coordinate-free postulate.

中文翻译：

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瞬态有限维机械系统的几何描述

使用 Loos 提出的非标准几何结构，我们提出了具有 n 个自由度的瞬态有限维机械系统的理论的无坐标公式。包含系统时间、位置和速度信息的状态空间被定义为 (n+1) 维广义时空上的 (2n+1) 维仿射丛。主要目标是提出一个几何假设来表征二阶矢量场，其积分曲线描述了与时间相关的有限维机械系统的运动。该公设的核心对象是状态空间上的微分二元形式，称为动作形式，与二阶向量场呈双射关系。将微分二形式作为动作形式的要求允许非势力的无坐标定义，这可能取决于时间、位置和速度。最后，我们表明，不仅拉格朗日方程，而且哈密顿方程也直接遵循相同的无坐标假设的坐标表示。