A generalization of the Helmholtz conditions for the existence of a first‐order Lagrangian using nonholonomic velocities
A generalization of the Helmholtz conditions for the existence of a first‐order Lagrangian using nonholonomic velocities
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DOI：
10.1002/zamm.200800114
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发表时间：
2009-01
期刊：
影响因子：
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通讯作者：
U. Jungnickel;G. Kielau;P. Maisser;A. Müller
中科院分区：
文献类型：
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作者：
U. Jungnickel;G. Kielau;P. Maisser;A. Müller
本文给出了一阶动力势（拉格朗日量）存在的亥姆霍兹条件的推广，在这种情况下，机械、电气或机电系统的运动方程是以非完整速度给出的。假定完整速度与非完整速度之间的变换是已知的。它显示了如何广义亥姆霍兹条件的同时存在的一阶拉格朗日和一阶耗散函数遵循广义亥姆霍兹条件的系统完整速度。此外，经典的Helmholtz条件在非耗散系统的情况下，可以直接推广到包含非完整速度的运动方程。在这两种情况下，使用Boltzmann-Hamel方程。通过两个例子证明了该方法：重型陀螺仪和三维旋转电机。
The paper presents a generalization of the Helmholtz conditions for the existence of a first‐order kinetic potential (Lagrangian) in that cases where the motion equations of mechanical, electrical, or electromechanical systems are given in terms of nonholonomic velocities. It is assumed that the transformation between holonomic and nonholonomic velocities is known. It is shown how the generalized Helmholtz conditions for the simultaneous existence of a first‐order Lagrangian and a first‐order dissipation function follow from the generalized Helmholtz conditions for systems with holonomic velocities. Moreover, the classical Helmholtz conditions in case of nondissipative systems can be directly generalized to motion equations which include nonholonomic velocities. In both cases the Boltzmann‐Hamel equations are used. The approaches are demonstrated by two examples: a heavy gyroscope and a 3‐dimensional rotating electrical machine.