Friday, September 8, 2023

Conservation Laws and Noether Theorem: UNIZOR.COM - Relativity 4 All - C...

Notes to a video lecture on UNIZOR.COM

Conservation Laws
and Noether Theorem

In this course we are trying to support every formula or a statement with a relatively rigorous proof, based on earlier proven properties and formulas.

This particular lecture is about an issue, which, on one hand, is extremely important for the theory, but, on the other hand, involves a significant mathematical effort that goes beyond the scope of this course.

So, the following is a theoretical item (Noether's Theorem) presented without a proof.

The Noether's Theorem was published in 1918 by Emmy Noether, a brilliant German mathematician, who was regarded by such scientists as Albert Einstein as the most important woman in the history of mathematics.
Her main contribution to Physics was to validate the only experimentally confirmed laws of conservation by a theoretical proof based on much more fundamental properties of the Universe.

We wish every statement to be based on some earlier proven statement or a theorem. Analyzing these earlier statements, we find them to be based on even earlier ones etc.

Inevitably, we would come to a statement that we have to take as an axiom without a proof.
Obviously, we would like to accept as axioms such statements that correspond to our general feelings about the Universe, seem to us as natural and not contradicting our intuition.

The laws of conservation of energy, linear momentum or angular momentum were widely accepted by physicists for centuries based on experimental data.
However, there was always a doubt whether these laws were indeed the universal laws of nature or just our not always perfect results of experiments.
The Nouther's Theorem actually stated that these laws of conservation are intimately related to properties of our space and time.
In particular, they are consequences of uniformity of our space and time.

Emmy Nouther took as an axiom some quite fundamental and well accepted statements that our time is uniform and totally symmetrical, our space is uniform and symmetrical, one direction in space is no different than another.
Based just on this uniformity and symmetry, she has proven the validity of our laws of conservation.

More precisely,
(a) if we accept the uniformity of time, the law of conservation of energy follows as a mathematical consequence;
(b) if we accept the linear uniformity of space, the law of conservation of linear momentum follows as a mathematical consequence;
(c) if we accept the directional uniformity of space, the law of conservation of angular momentum follows as a mathematical consequence.

There is no doubts that it's significantly easier and much more natural to accept the uniformity of time as an axiom than the law of conservation of energy, and, similarly, the rest of the prepositions of the Nouther's Theorem.
That's why this theoretical result is as fundamentally important for Physics as axioms of Euclid for Geometry.

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