The class of integral variational functionals for paths in smooth manifolds, whose extremals are (nonparameterized) sets, is considered in this study. Recently, it was shown that for the functionals depending on tangent vectors, this property follows from any of the following two equivalent conditions: (a) the Lagrange function, defined on the tangent bundle, is positively homogeneous in the components of tangent vectors and (b) the Lepage differential form of the Lagrange function is projectable onto the Grassmann fibrations of rank 1 and order 1 (projective bundle); the classical Hilbert form was rediscovered this way as the projection of the Lepage form. In this paper, we extend these results to variational functionals of any order. Projectability conditions onto Grassmann fibrations of any order are found. The case of projective bundles is then studied in full generality. The proofs are based on the Euler–Zermelo conditions and the properties of higher-order Grassmann fibrations of rank 1. As an application, equations for set solutions in Riemannian geometry are derived.

中文翻译：

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变分原理：可投影到格拉斯曼纤维

本研究考虑了光滑流形中路径的积分变分泛函类，其极值是（非参数化）集合。最近，研究表明，对于依赖于切向量的泛函，该性质遵循以下两个等价条件中的任何一个：（a）定义在切丛上的拉格朗日函数在切向量的分量中是正齐次的，并且（ b) 拉格朗日函数的 Lepage 微分形式可投影到 1 阶和 1 阶 Grassmann fibrations（投影丛）上；以这种方式重新发现了经典的希尔伯特形式，作为 Lepage 形式的投影。在本文中，我们将这些结果扩展到任何阶的变分泛函。找到了任何阶 Grassmann 纤维的可投影性条件。然后全面研究射影丛的情况。证明基于 Euler-Zermelo 条件和 1 阶高阶 Grassmann 纤维化的性质。作为应用，导出了黎曼几何中的集合解方程。