Homogeneous variational problems: a minicourse

TL;DR

This paper explores homogeneous variational problems, especially Finsler geometry, focusing on extremal curves and submanifolds from a geometric perspective, highlighting their theoretical foundations and generalizations.

Contribution

It provides a geometric framework for understanding Finsler and related variational problems involving extremal submanifolds of arbitrary dimension.

Findings

Finsler geometry as a homogeneous variational problem

Extension of extremals from curves to submanifolds

Geometric interpretation of variational problems

Abstract

A Finsler geometry may be understood as a homogeneous variational problem, where the Finsler function is the Lagrangian. The extremals in Finsler geometry are curves, but in more general variational problems we might consider extremal submanifolds of dimension $m$. In this minicourse we discuss these problems from a geometric point of view.