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