Emmy Noether's Theorem on Integral Invariants in the Context of the Calculus of Variations

Noether’s theorem, named for early twentieth century German mathematician Emmy Noether, is an important theorem in physics. Colloquially, Noether’s theorem states that a physical system with a symmetry property will have an associated conservation law and provides the means for determining it. Although an impressive derivation in its own right, the fact that the theorem’s author was a Jewish woman during a time in which both of these facets of her identity, particularly her womanhood, posed significant barriers to her advancement and acceptance in the held of mathematics, makes it all the more remarkable. A biographical sketch of Noether highlighting the inequities she faced is given in the first chapter of this paper.To fully appreciate Noether’s theorem, it is essential to view it in the context in which it was developed. Noether’s derivation of her theorem relied heavily on the calculus of variations and can be viewed as an extension of the subject. An account of the history of the calculus of variations until the beginning of the twentieth century, with an emphasis on the history of the Euler-Lagrange equations, is given in Chapter 2. During the early twentieth century, Einstein’s theory of general relativity was of great interest to many mathematicians and physicists and it served as inspiration for Noether’s work. Noether also used ideas from Lie theory in her derivation. Brief historical accounts of general relativity and of Lie theory are given Section 3.1.The final chapters of the paper focus on Noether's derivation of her theorem and the implications of her theorem. In her paper “Invariante Variationsprobleme” written on her theorem, Noether leaves out many of the mathematical steps necessary to obtain her results. An expanded account of her derivation is given in Chapter 3. In the following chapter, a demonstration of obtaining conservation laws from Noether’s theorem is given for several classic introductory physics problems.