Nöther’s Theorem

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    We have noticed that the variables that show up in Heisenberg’s indeterminacy principle come in pairs that are correlated via the principle of least action. The German mathematician Amalie Emmy Nöther (1882 Mar 23 - 1935 Apr 14) found another correlation among those same pairs, one that bears a closer look. That correlation, encoded in what we call Nöther’s Theorem, asserts that if the laws of physics display a symmetry with respect to one variable in a correlated pair, then the other variable in the pair comes under the rule of a conservation law: symmetry of the laws of physics with respect to translations in space correlate with conservation of linear momentum, for example. In devising that theorem she added a large and important piece to the Map of Physics and she did it without putting any physics into it: here we have an example of pure mathematics describing Reality in a fundamental way.

    I can think of no better way to present Fraulein Professor Nöther’s theorem than to offer my own paraphrase of Dr. Morton Tavel’s 1971 translation, from the original German to English, of Nöther’s 1918 paper, in which she laid out her derivation and interpretation of the theorem. I divide the paper as Nöther did and include comments of my own where necessary to clarify Nöther’s statements. The introduction to group theory and transformations is my own. Thus we have now

INVARIANT VARIATION PROBLEMS

Emmy Nöther (1918)

Abstract

    In this paper we will consider problems in the calculus of variations that pertain to continuous groups, the algebraic entities devised and studied by the Norwegian mathematician Sophus Lie and we will obtain results from the corresponding differential equations, which results lead to the most general expression in the theorems formulated in the section Preliminary Remarks and Formulation of Theorems and given proof and verification in the following sections. We can make much more precise statements about the differential equations that come from the calculus of variations than we can make about arbitrary differential equations pertaining to an algebraic group, the subject of Lie’s study. Therefore, in what follows we apply a combination of the methods of the formal calculus of variations with the methods of Lie’s group theory. This combination of methods simply extends work that others have done with regard to special groups and problems in the calculus of variations.

GO TO: A Brief Introduction to Group Theory and Transformations

GO TO: Summary of Nöther’s Paper

GO TO: Hamilton’s Principle and Einstein’s Equation

GO TO: "Preliminary Remarks and Formulation of Theorems"

GO TO: "Divergence Relationships and Dependencies"

GO TO: "Converse in the Case of a Finite Group"

GO TO: "Converse in the Case of an Infinite Group"

GO TO: "Invariance of Several Constituents of the Relations"

GO TO: "A Hilbertian Assertion"

GO TO: The Nöther Identities

GO TO: The Conservation Laws

    To distinguish between Nöther’s text and my own comments I have put my text into the PTBarnum BT font and put Professor Nöther’s text into the Arial font.

    In marking the equations I have retained the numbering that I found in Nöther’s paper. For those equations that Nöther did not number I use lower-case Greek letters. To mark the equations that I have added in my commentary I use upper-case Latin letters. Of course, these comments apply only to the six sections derived directly from Nöther’s paper.

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