arXiv 1708.014692017-08-04math-phmath.MP

李群为位形空间的力学系统的多辛几何和协变形式：应用于Reissner梁

Multisymplectic geometry and covariant formalism for mechanical systems with a Lie group as configuration space: application to the Reissner beam

Joël Bensoam, Florie-Anne Baugé

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摘要：

许多物理上重要的力学系统可以用李群$G$作为位形空间来描述。根据著名的Noether定理，李群的基本对称性可以大大降低问题的复杂性。然而，这些简化技术，不关心一般问题（波，场论）使用，可能会导致不舒服的无限维空间。作为替代方案，\emph{covariant}公式允许通过增加独立变量的数量来考虑有限维配置空间。但归约所需的几何元素，适应于承认李群作为配置空间的协变问题的特殊性，在文献中很难理解（据我们所知，有些甚至是缺失的）。为了填补这一空白，本文对E.在这个特殊的"协变李群"上下文中。因此，这是这项工作的主要兴趣，Poincaré-Cartan和多辛形式获得的主$G$丛。它允许制定的\emph{Euler-Poincaré equations}的运动，并导致一个诺特的当前形式定义的对偶李代数。

英文摘要：

Many physically important mechanical systems may be described with a Lie group $G$ as configuration space. According to the well-known Noether's theorem, underlying symmetries of the Lie group may be used to considerably reduce the complexity of the problems. However, these reduction techniques, used without care for general problems (waves, field theory), may lead to uncomfortable infinite dimensional spaces. As an alternative, the \emph{covariant} formulation allows to consider a finite dimensional configuration space by increasing the number of independent variables. But the geometric elements needed for reduction, adapted to the specificity of covariant problems which admit Lie groups as configuration space, are difficult to apprehend in the literature (some are even missing to our knowledge). To fill this gap, this article reconsiders the historical geometric construction made by E. Cartan in this particular "covariant Lie group" context. Thus, and it is the main interest of this work, the Poincaré-Cartan and multi-symplectic forms are obtained for a principal $G$ bundle. It allows to formulate the \emph{Euler-Poincaré equations} of motion and leads to a Noether's current form defined in the dual Lie algebra.

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