Invariant Variation Problems

TL;DR

This work develops a rigorous link between variational invariants and Lie group symmetries, unifying calculus of variations with group-theoretic methods. It shows that invariance under finite, infinite, and mixed continuous transformation groups imposes divergence relations and identities among Lagrange expressions, with converses established, thereby connecting symmetry to conservation-like laws. The analysis extends to energy relationships and Hilbertian assertions about the modification or failure of traditional conservation laws in generalized relativity contexts. Through illustrative examples, the paper situates these results within a broad framework of invariance, conservation, and group structure, offering a spectrum from standard Noether-type results to generalized, function-based symmetries. Overall, it provides a comprehensive bridge between variational calculus and Lie group theory with implications for physics and geometry.

Abstract

The problems in variation here concerned are such as to admit a continuous group (in Lie's sense); the conclusions that emerge from the corresponding differential equations find their most general expression in the theorems formulated in Section 1 and proved in following sections. Concerning these differential equations that arise from problems of variation, far more precise statements can be made than about arbitrary differential equations admitting of a group, which are the subject of Lie's researches. What is to follow, therefore, represents a combination of the methods of the formal calculus of variations with those of Lie's group theory. For special groups and problems in variation, this combination of methods is not new; I may cite Hamel and Herglotz for special finite groups, Lorentz and his pupils (for instance Fokker), Weyl and Klein for special infinite groups. Especially Klein's second Note and the present developments have been mutually influenced by each other, in which regard I may refer to the concluding remarks of Klein's Note.