受控拉格朗日量与具有对称破缺非完整约束的欧拉-庞加莱方程的镇定
Controlled Lagrangians and Stabilization of Euler--Poincaré Equations with Symmetry Breaking Nonholonomic Constraints
AI总结：
本文将受控拉格朗日量方法推广到带平流参数的非完整欧拉-庞加莱方程，通过引入依赖平流参数的准速度消去拉格朗日乘子并导出匹配条件，最终为摆式冰鞋模型设计了镇定滑动平衡的控制律。
摘要：
我们将受控拉格朗日量方法推广到带有平流参数的非完整欧拉-庞加莱方程，特别是那些李群上不仅被势能力而且被非完整约束破坏对称性的力学系统。我们引入了依赖平流参数的准速度，以系统地消去非完整欧拉-庞加莱方程中的拉格朗日乘子。准速度有助于对这些系统应用受控拉格朗日量方法，并导出与 Bloch、Leonard 和 Marsden 针对标准完整欧拉-庞加莱方程所得到的匹配条件相似的匹配条件。我们的启发性例子是我们所称的摆式冰鞋，这是 Gzenda 和 Putkaradze 提出的一个花样滑冰运动员的简单模型。我们证明，在某些条件下，摆式冰鞋的直立旋转是稳定的，而直立滑动平衡总是不稳定的。利用匹配条件，我们推导出一个控制律来镇定滑动平衡。
英文摘要：
We extend the method of Controlled Lagrangians to nonholonomic Euler--Poincaré equations with advected parameters, specifically to those mechanical systems on Lie groups whose symmetry is broken not only by a potential force but also by nonholonomic constraints. We introduce advected-parameter-dependent quasivelocities in order to systematically eliminate the Lagrange multipliers in the nonholonomic Euler--Poincaré equations. The quasivelocities facilitate the method of Controlled Lagrangians for these systems, and lead to matching conditions that are similar to those by Bloch, Leonard, and Marsden for the standard holonomic Euler--Poincaré equation. Our motivating example is what we call the pendulum skate, a simple model of a figure skater developed by Gzenda and Putkaradze. We show that the upright spinning of the pendulum skate is stable under certain conditions, whereas the upright sliding equilibrium is always unstable. Using the matching condition, we derive a control law to stabilize the sliding equilibrium.
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