连续和离散Clebsch变分原理
Continuous and discrete Clebsch variational principles
摘要：
Clebsch方法提供了一种统一的方法来推导连续和离散动力系统的变分原理，其中向量空间的元素用于控制李群\emph{via}的余切丛上的动力学，即速度映射。本文证明了一个约化定理，即李群上的典型变量可以消去，当且仅当速度映射是李代数作用，从而得到向量空间变量的Euler-Poincaré（EP）方程.在这种情况下，从李群上的典型变量到向量空间的映射是使用菱形算子定义的标准动量映射。我们将Clebsch方法应用于旋转刚体和不可压缩Euler方程的例子中。一路上，我们解释了如何奇异解的EP方程的同胚群（EPDiff）出现的动量映射的Clebsch方法。在有限维李群的情况下，将Clebsch变分原理离散化以产生动力系统的变分积分器。我们得到一个离散映射，从李群的余切丛上的变量可以被消除，以产生一个离散的EP方程的向量空间的元素。作为一个例子，我们给出了一个积分器的旋转刚体。我们还简要地讨论了如何离散无限维Clebsch系统，从而产生保守的数值方法的流体动力学。
英文摘要：
The Clebsch method provides a unifying approach for deriving variational principles for continuous and discrete dynamical systems where elements of a vector space are used to control dynamics on the cotangent bundle of a Lie group \emph{via} a velocity map. This paper proves a reduction theorem which states that the canonical variables on the Lie group can be eliminated, if and only if the velocity map is a Lie algebra action, thereby producing the Euler-Poincaré (EP) equation for the vector space variables. In this case, the map from the canonical variables on the Lie group to the vector space is the standard momentum map defined using the diamond operator. We apply the Clebsch method in examples of the rotating rigid body and the incompressible Euler equations. Along the way, we explain how singular solutions of the EP equation for the diffeomorphism group (EPDiff) arise as momentum maps in the Clebsch approach. In the case of finite dimensional Lie groups, the Clebsch variational principle is discretised to produce a variational integrator for the dynamical system. We obtain a discrete map from which the variables on the cotangent bundle of a Lie group may be eliminated to produce a discrete EP equation for elements of the vector space. We give an integrator for the rotating rigid body as an example. We also briefly discuss how to discretise infinite-dimensional Clebsch systems, so as to produce conservative numerical methods for fluid dynamics.
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