The Reduced Euler-Lagrange Equations
The Reduced Euler-Lagrange Equations
复制标题
简化的欧拉-拉格朗日方程
DOI：
--
复制
发表时间：
1993
期刊：
影响因子：
--
通讯作者：
J. Scheurle
中科院分区：
文献类型：
--
作者：
J. Marsden;J. Scheurle
Marsden和Scheurle [1993]研究了拉格朗日约化，
的动量图约束-这里的意思是减少的标准
限制在动量映射水平集上的欧拉-拉格朗日系统。这
提供了一个拉格朗日平行减少辛流形。的
本文研究了哈密顿量泊松约化的拉格朗日平行性
系统.对于拉格朗日系统在水平集上的约化，
一个守恒量，一个关键的对象是Routhian，也就是Lagrangian
减去与动量固定值配对的机械连接
地图对于无约束系统，我们使用速度移动拉格朗日量，
在约束理论中扮演了Routhian的角色。汉密尔顿变分
欧拉-拉格朗日方程的一个基本原理可以分解为两组方程
它代表了一组具有陀螺强迫的欧拉-拉格朗日方程，
可以根据水平变化的连接的曲率来写，
并转化为欧拉-庞加莱方程的垂直变化。这
一组新的方程就是我们所说的简化欧拉-拉格朗日方程，
它包括Euler-Poincare和Hamel方程作为特殊情况。我们说明
该方法用于具有内部转子的刚体和用于颗粒
在磁场中移动
Marsden and Scheurle [1993] studied Lagrangian reduction in the context
of momentum map constraints—here meaning the reduction of the standard
Euler-Lagrange system restricted to a level set of a momentum map. This
provides a Lagrangian parallel to the reduction of symplectic manifolds. The
present paper studies the Lagrangian parallel of Poisson reduction for Hamiltonian
systems. For the reduction of a Lagrangian system on a level set of
a conserved quantity, a key object is the Routhian, which is the Lagrangian
minus the mechanical connection paired with the fixed value of the momentum
map. For unconstrained systems, we use a velocity shifted Lagrangian, which
plays the role of the Routhian in the constrained theory. Hamilton’s variational
principle for the Euler-Lagrange equations breaks up into two sets of equations
that represent a set of Euler-Lagrange equations with gyroscopic forcing that
can be written in terms of the curvature of the connection for horizontal variations,
and into the Euler-Poincar´e equations for the vertical variations. This
new set of equations is what we call the reduced Euler-Lagrange equations, and
it includes the Euler-Poincare and the Hamel equations as special cases. We illustrate
this methodology for a rigid body with internal rotors and for a particle
moving in a magnetic field.