The fundamental Lepage form in variational theory for submanifolds

TL;DR

This paper develops a geometric variational framework for submanifolds using the fundamental Lepage form, extending classical mechanics concepts to Grassmann fibrations and deriving key equations like Euler-Lagrange and Noether's theorem.

Contribution

It introduces a global variational geometry approach on Grassmann fibrations utilizing the fundamental Lepage form, generalizing classical mechanics tools for submanifold analysis.

Findings

Derived the first infinitesimal variation formula.

Established Euler-Lagrange equations for extremal submanifolds.

Proved Noether's theorem for invariant functionals.

Abstract

A setting for global variational geometry on Grassmann fibrations is presented. The integral variational functionals for finite dimensional immersed submanifolds are studied by means of the fundamental Lepage equivalent of a homogeneous Lagrangian, which can be regarded as a generalization of the well-known Hilbert form in the classical mechanics. Prolongations of immersions, diffeomorphisms and vector fields to the Grassmann fibrations are introduced as geometric tools for the variations of immersions. The first infinitesimal variation formula together with its consequences, the Euler-Lagrange equations for extremal submanifolds and the Noether theorem for invariant variational functionals are proved. The theory is illustrated on the variational functional for minimal submanifolds.