AI 中文总结
本文重构牛顿力学的时空几何，将惯性原理从定理导出，提出经典自旋的新对应关系，证明相关数学结果并作出公理与猜想，构建不含引力的非退化时空动力学框架。
AI 中文摘要
牛顿力学通常建立在具有固定几何结构的时空之上。本文我们构建该几何结构，仅在描述需要时引入传播子和双线性形式。在这种重构中，惯性原理作为定理出现，而非独立的公理。惯性运动对双线性形式的尺度以及传播子定义的挠率不敏感，这种不敏感性是引力介入前“自由落体的普适性”的几何前驱。牛顿第二定律是沿曲线的两种导数之间的相容性条件。力超出质量与降标加速度乘积的部分为一个精确项，且该定律在任意挠率下均成立。要求无外力运动为沿弧长参数化曲线的惯性运动，将挠率限制为完全反对称的。剩余的挠率是一个三形式，我们假设粒子沿其世界线携带该三形式。该三形式与速度的收缩会湮灭该速度，因此自旋粒子力学的补充条件恒成立，且我们将该收缩提议为经典自旋。该代数在相对论自旋流体动力学中较为常见；据我们所知，该对应关系尚未在其他文献中提出。由于惯性运动既不确定质量也不确定自旋，预测内容在于我们提出的输运定律，该定律使自旋以仅由加速度决定、无耦合常数的速率相对于平行移动的框架反向旋转。只有当弧长参数被识别为时钟时，该速率才成为实验室数值，我们暂不对此作出限定。该动力学要求双线性形式非退化，因此该时空并非牛顿-嘉当的伽利略时空，且引力不在该构造范围内。我们证明了所有数学结果，提出了三个构成性公理和一个物理猜想。
英文摘要
Newtonian mechanics is usually written on a spacetime of fixed geometry. Here we build that geometry, introducing a propagator and a bilinear form only as the description demands. Within this reconstruction the Principle of Inertia emerges as a theorem rather than an independent postulate. Inertial motion is blind to the scale of the form and to the torsion the propagator defines, and that blindness is a geometric precursor of the \emph{universality of free fall}, before gravity enters. Newton's Second Law is a compatibility between two derivatives along a curve. The force exceeds the mass times the lowered acceleration by an exact term, and the law survives arbitrary torsion. Requiring force-free motion to be inertial along curves parametrised by arc length restricts the torsion to be totally antisymmetric. The remaining torsion is a three-form which we postulate the particle carries along its worldline. Its contraction with the velocity annihilates that velocity, so the supplementary condition of spinning-particle mechanics holds identically, and we propose that contraction as a classical spin. The algebra is familiar from relativistic spin hydrodynamics; the assignment, to the best of our knowledge, has not been proposed elsewhere. Since inertial motion resolves neither the mass nor the spin, the predictive content lies in the transport law we postulate, which turns the spin against a parallel-transported frame at a rate fixed by the acceleration alone and by no coupling constant. That rate becomes a laboratory number only once the arc-length parameter is identified with a clock, which we leave open. The dynamics asks non-degeneracy of the form, so this spacetime is not Newton--Cartan's Galilean one, and gravitation lies outside the construction. We prove every mathematical result, make three constitutive postulates and one physical conjecture.
