Multisymplectic structures and the variational bicomplex
Multisymplectic structures and the variational bicomplex
复制标题
多重辛结构和变分双复形
DOI：
10.1017/s0305004109990259
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发表时间：
2009
影响因子：
0.8
通讯作者：
J. Lawson
中科院分区：
文献类型：
--
作者：
T. Bridges;P. Hydon;J. Lawson
多辛性和变分双复形是两个独立发展的学科。我们的主要观察是，从变分双复的角度重新分析多辛系统不仅是自然的，而且还产生了新的基本思想的多辛哈密顿偏微分方程。变分双复体提供了一个自然的分级的微分形式，根据其基地和纤维组件，这种结构产生了一个新的关系几何基础，协变多辛偏微分方程和辛性的保护。我们的配方也提出了一个新的观点，诺特理论的多辛系统，导致多动量映射的定义，我们适用于给出一个坐标自由描述的多辛相对平衡。我们的主要例子是一类多辛系统的全外代数丛黎曼流形。
Abstract Multisymplecticity and the variational bicomplex are two subjects which have developed independently. Our main observation is that re-analysis of multisymplectic systems from the view of the variational bicomplex not only is natural but also generates new fundamental ideas about multisymplectic Hamiltonian PDEs. The variational bicomplex provides a natural grading of differential forms according to their base and fibre components, and this structure generates a new relation between the geometry of the base, covariant multisymplectic PDEs and the conservation of symplecticity. Our formulation also suggests a new view of Noether theory for multisymplectic systems, leading to a definition of multimomentum maps that we apply to give a coordinate-free description of multisymplectic relative equilibria. Our principal example is the class of multisymplectic systems on the total exterior algebra bundle over a Riemannian manifold.