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Morse–Palais lemma

From Wikipedia, the free encyclopedia

In mathematics, the Morse–Palais lemma is a result in the calculus of variations and theory of Hilbert spaces. Roughly speaking, it states that a smooth enough function near a critical point can be expressed as a quadratic form after a suitable change of coordinates.

The Morse–Palais lemma was originally proved in the finite-dimensional case by the American mathematician Marston Morse, using the Gram–Schmidt orthogonalization process. This result plays a crucial role in Morse theory. The generalization to Hilbert spaces is due to Richard Palais and Stephen Smale.

Statement of the lemma

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Let be a real Hilbert space, and let be an open neighbourhood of the origin in Let be a -times continuously differentiable function with that is, Assume that and that is a non-degenerate critical point of that is, the second derivative defines an isomorphism of with its continuous dual space by

Then there exists a subneighbourhood of in a diffeomorphism that is with inverse, and an invertible symmetric operator such that

Corollary

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Let be such that is a non-degenerate critical point. Then there exists a -with--inverse diffeomorphism and an orthogonal decomposition such that, if one writes then

See also

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References

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  • Lang, Serge (1972). Differential manifolds. Reading, Mass.–London–Don Mills, Ont.: Addison–Wesley Publishing Co., Inc.