非保守Lagrange系统的广义Helmholtz条件
Generalized Helmholtz conditions for non-conservative Lagrangian systems
摘要：
本文给出了以半基本1-形式表示的广义Helmholtz条件，该条件刻画了给定的二阶常微分方程对任意非保守力与Lagrange方程等价的情形。对于耗散力或陀螺力的特殊情况，当用乘子矩阵表示时，这些条件可简化为[18]中得到的条件。当所涉及的几何结构是均匀的纤维坐标，我们展示了如何可以进一步简化广义亥姆霍兹条件。我们提供的例子中提出的广义亥姆霍兹条件，表示在一个半基本的1-形式，可以集成和相应的拉格朗日和拉格朗日方程可以找到。
英文摘要：
In this paper we provide generalized Helmholtz conditions, in terms of a semi-basic 1-form, which characterize when a given system of second order ordinary differential equations is equivalent to the Lagrange equations, for some given arbitrary non-conservative forces. For the particular cases of dissipative or gyroscopic forces, these conditions, when expressed in terms of a multiplier matrix, reduce to those obtained in [18]. When the involved geometric structures are homogeneous with respect to the fibre coordinates, we show how one can further simplify the generalized Helmholtz conditions. We provide examples where the proposed generalized Helmholtz conditions, expressed in terms of a semi-basic 1-form, can be integrated and the corresponding Lagrangian and Lagrange equations can be found.
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