从局部能量守恒出发的粒子动力学
Particle Mechanics from Local Energy Conservation
AI总结：
本文提出了一种基于局部能量守恒的粒子动力学框架，通过推导力与动能的关系，统一了保守和非保守动力学，并展示了牛顿和相对论力学作为对称性选择的实现。
摘要：
我们发展了一种粒子动力学的表述方法，其中力与动能之间的函数关系直接从局部保守机械能$E$导出，而不是通过牛顿第二定律或变分原理提出。从瞬时条件$\dot{E}=0$出发，将其作为粒子轨迹上的点约束条件，我们得到一个广义的力定律，不假定特定的动能函数、动量-速度关系或运动方程。所得到的惯性响应自然分解为一个与加速度平行的分量，负责动能的变化，以及一个横向分量，它在改变运动方向的同时保持能量。施加旋转等价性约束限制了力定律的几何结构，而惯性参考系之间的相对性原理进一步限制了其允许的实现。在严格的一维运动中，惯性系等价性意味着总力在惯性提升下不变；结合局部能量守恒，这唯一确定了动能和动量的函数形式。伽利略不变性选择牛顿表达式，而洛伦兹不变性则产生相对论的表达式。该框架在单个粒子层面统一了保守和非保守（但非耗散）动力学，并明确了能量基和变分力学表述在动态上等价的精确条件。牛顿力学和相对论力学因此作为对称性选择的实现，出现了共同的能量守恒力-能结构。
英文摘要：
We develop a formulation of particle mechanics in which the functional relation between force and kinetic energy is derived directly from local conservation mechanical energy $E$, rather than postulated through Newton's second law or a variational principle. Starting from the instantaneous condition $\dot{E}=0$, imposed as a pointwise constraint along a particle trajectory, we obtain a generalized force law that does not assume a specific kinetic-energy function, momentum-velocity relation, or equation of motion. The resulting inertial response naturally decomposes into a component parallel to the acceleration, responsible for changes in kinetic energy, and a transverse component that preserves energy while altering the direction of motion. Imposing rotational equivariance constrains the geometric structure of the force law, while the relativity principle between inertial reference frames further restricts its admissible realizations. In strictly one-dimensional motion, inertial-frame equivalence implies invariance of the total force under inertial boosts; together with local energy conservation, this uniquely fixes the functional form of kinetic energy and momentum. Galilean invariance selects the Newtonian expressions, whereas Lorentz invariance yields the relativistic ones. The framework unifies conservative and non-conservative (yet non-dissipative) dynamics at the single-particle level and clarifies the precise conditions under which energy-based and variational formulations of mechanics are dynamically equivalent. Newtonian and relativistic mechanics thus emerge as symmetry-selected realizations of a common energy-conserving force-energy structure.
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