标题： 高阶拉格朗日形式化在格拉斯曼流形上 显示英文标题

标题： Higher-Order Lagrangian Formalism on Grassmann Manifolds

摘要： 考虑任意非纤维化流形上的拉格朗日形式。 这种一般情况的动力学描述基于（高阶）格拉斯曼流形的概念，即正则速度流形对微分群作用的商空间。 在此背景下，我们引入拉格朗日形式的基本概念，包括拉格朗日形式、欧拉-拉格朗日形式和亥姆霍兹-松因形式。 这些对象成对出现，即我们有定义在正则速度流形上的齐次对象和定义在格拉斯曼流形上的非齐次对象。 我们将建立齐次对象与其非齐次对应物之间的联系。 最终，我们将得出结论：变分平凡的拉格朗日表达式和局部变分微分方程的一般表达式与纤维化情形相同。 显示英文摘要

摘要： The Lagrangian formalism on a arbitrary non-fibrating manifold is considered. The kinematical description of this generic situation is based on the concept of (higher-order) Grassmann manifolds which is the factorization of the regular velocity manifold to the action of the differential group. Here we introduce in this context the basic concepts of the Lagrangian formalism as Lagrange, Euler-Lagrange and Helmholtz-Sonin forms. These objects come in pairs, namely we have homogeneous objects (defined on the regular velocity manifold) and non-homogeneous objects (defined on the Grassmann manifold). We will establish the connection between the homogeneous objects and their non-homogeneous counterparts. As a result we will conclude that the generic expressions for a variationally trivial Lagrangian and for a locally variational differential equation remain the same as in the fibrating case.