Higher-Order Lagrangian Formalism on Grassmann Manifolds

TL;DR

This paper develops a higher-order Lagrangian formalism on Grassmann manifolds, generalizing classical variational calculus to non-fibrating manifolds using the concept of differential group actions.

Contribution

It introduces the basic Lagrangian concepts on Grassmann manifolds and establishes their connection to traditional formulations, extending variational calculus beyond fibrating manifolds.

Findings

Homogeneous and non-homogeneous Lagrangian objects are connected.

Expressions for variationally trivial Lagrangians are consistent with fibrating cases.

The formalism applies to arbitrary non-fibrating manifolds.

Abstract

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.