We present a systematic treatment of line bundle geometry and Jacobi manifolds with an application to geometric mechanics that has not been noted in the literature. We precisely identify categories that generalize the ordinary categories of smooth manifolds and vector bundles to account for a lack of choice of a preferred unit, which in standard differential geometry is always given by the global constant function [Formula: see text]. This is what we call the “unit-free” approach. After giving a characterization of local Lie brackets via their symbol maps, we apply our novel categorical language to review Jacobi manifolds and related notions such as Lichnerowicz brackets and Jacobi algebroids. The main advantage of our approach is that Jacobi geometry is recovered as the direct unit-free generalization of Poisson geometry, with all the familiar notions translating in a straightforward manner. We then apply this formalism to the question of whether there is a unit-free generalization of Hamiltonian mechanics. We identify the basic categorical structure of ordinary Hamiltonian mechanics to argue that it is indeed possible to find a unit-free analogue. This paper serves as a prelude to the investigation of dimensioned structures, an attempt at a general mathematical framework for the formal treatment of physical quantities and dimensional analysis.

中文翻译：

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雅可比几何和哈密顿力学：无单位方法

我们提出了对线束几何和雅可比流形的系统处理，并将其应用于几何力学，这在文献中没有提到。我们精确地确定了概括光滑流形和向量丛的普通类别的类别，以解释缺乏首选单位的选择，在标准微分几何中，首选单位总是由全局常数函数给出[公式：见正文]。这就是我们所说的“无单元”方法。在通过符号映射给出局部李括号的特征之后，我们应用我们新颖的分类语言来回顾雅可比流形和相关概念，例如 Lichnerowicz 括号和雅可比代数。我们方法的主要优点是雅可比几何被恢复为泊松几何的直接无单位推广，所有熟悉的概念都以直截了当的方式翻译。然后，我们将这种形式主义应用于哈密顿力学是否存在无单位推广的问题。我们确定了普通哈密顿力学的基本分类结构，以证明确实有可能找到无单位的类似物。本文作为维度结构研究的前奏，是对物理量和维度分析的形式化处理的一般数学框架的尝试。