旋度力的对偶变分原理
Dual Variational Principles for Curl Forces
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中文总结 AI 辅助
本文针对旋度力作用下的粒子动力学，提出一种对偶变分形式，通过引入对偶变量与辅助函数构建预对偶作用量，其欧拉-拉格朗日方程可恢复原始运动方程及初始条件，还引入了守恒的辅助对偶哈密顿量，经实例验证非保守旋度力动力学可具有变分描述。
中文摘要 AI 辅助
旋度力是依赖于位置、非保守且非耗散的力，通常无法由普通势能导出，因此其运动方程一般不遵循标准变分原理。本文针对旋度力作用下的粒子动力学，提出一种对偶变分形式：通过引入与位置、速度对偶的变量及辅助函数，构建预对偶作用量，其中运动方程作为约束；对原始变量的平稳性定义了对偶到原始的映射，将其代入预对偶作用量可得到完全由对偶变量表示的作用量，该对偶作用量的欧拉-拉格朗日方程可恢复原始运动方程及其给定初始条件。我们还引入了辅助对偶哈密顿量，其在平稳对偶轨迹上守恒，但不代表物理能量。该形式通过二维、三维的两种非线性旋度力场及经典齐格勒柱（Ziegler column）进行说明，这些例子表明，即使不存在普通势能或传统拉格朗日量，非保守旋度力动力学也可具有变分描述。
英文摘要
Curl forces are position-dependent, non-conservative, and non-dissipative forces that, in general, cannot be derived from an ordinary potential energy. Consequently, their equations of motion do not, in general, follow from a standard variational principle. In this paper, we present a dual variational formulation for particle dynamics under curl forces. By introducing variables dual to position and velocity and an auxiliary function, we construct a pre-dual action in which the equations of motion act as constraints. Stationarity with respect to the primal variables defines a dual-to-primal mapping, whose substitution into the pre-dual action gives an action expressed entirely in terms of the dual variables. The Euler--Lagrange equations of the dual action recover both the original equations of motion and their prescribed initial conditions. We also introduce an auxiliary dual Hamiltonian that is conserved along stationary dual trajectories, although it does not represent the physical energy. The formulation is illustrated using two nonlinear curl force fields in two and three dimensions and the classical Ziegler column. These examples demonstrate that non-conservative curl-force dynamics can admit variational descriptions even in the absence of an ordinary potential energy or a conventional Lagrangian.
