Noether’s Theorem

Nature is thrifty in all its actions.

    — Pierre Louis Maupertuis

From Wikipedia:

Noether’s (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918. The action of a physical system is the integral over time of a Lagrangian function (which may or may not be an integral over space of a Lagrangian density function), from which the system’s behavior can be determined by the principle of least action.

Noether’s theorem can be stated informally:

If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time.

Note:

Symmetries are transformations or exchanges in space or time that leave systems structurally or functionally equivalent to what they were before. The equivalence may or may not be an identity, but only the same in appearance or behavior.

Conservation laws are equivalences for quantitative properties of systems. A given property of matter or energy is quantitatively the same before and after, or continuously through space or time. The functional measure of this property remains constant.

So consider an analogy between Noether’s Theorem and the concept of Equivalent Exchange: for (symmetrical, differentiable) exchanges, there are properties that are equivalent (conserved)!

http://en.wikipedia.org/wiki/Noether’s_theorem

http://en.wikipedia.org/wiki/Action_%28physics%29

http://en.wikipedia.org/wiki/Lagrangian

http://en.wikipedia.org/wiki/Principle_of_least_action

http://math.ucr.edu/home/baez/noether.html

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3 Responses to “Noether’s Theorem”

  1. Wu Wei or Natural Action « Equivalent eXchange Says:

    […] Equivalent eXchange Philosophy, semiotics, and logic « Noether’s Theorem […]

  2. Matter, Energy, Space, and Time | Equivalent eXchange Says:

    […] of Matter. I have also used the pairs space-time and matter-energy as two parts of the fourfold Noether’s Theorem, the other two parts being symmetry and conservation. There is also the fourfold Spacetime that has […]

  3. What can we do? (Part Two) | BoyDownTheLane Says:

    […] principle of least action (or stationary action) seen in the previous entry Noether’s Theorem immediately makes me think of the Taoist concept of wu wei – literally no action or effortless […]

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