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Noether conservation theorems
Posted:
Oct 12, 2007 10:34 AM
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Many texts give a simple form of Emmy Noether's conservation theorem:
Each one-parameter group of symmetries of a Lagrangian system gives a
conserved quantity of that system. It is a beautiful result and
entirely clear -- once you are shown the proof.
But Noether proved two more involved versions of the theorem: Any
finite dimensional Lie gorup action that preserves a Lagrangian gives
some kind of family of conserved quantities that I do not yet
understand, and an infinite-dimensional Lie group gives something more
complicated that I do not yet understand.
Several sources claim that the difference between the finite- and
infinite-dimensional versions says something important about the
failure of momentum-energy conservation in General Relativity.
Can anyone explain what they mean, or point me to sources?
Actually, I don't see how either version can say anything definitive
about failure of conservation, since each seems to give sufficient but
NOT necessary conditions for conservation of a quantity.
I'd appreciate any advice I can get on this.
best, Colin
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