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![{\displaystyle H\ni x\mapsto \mathrm {D} ^{2}f(0)(x,-)\in H^{*}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Then there exists a subneighbourhood
of
in
a diffeomorphism
that is
with
inverse, and an invertible symmetric operator
such that![{\displaystyle f(x)=\langle A\varphi (x),\varphi (x)\rangle \quad {\text{ for all }}x\in V.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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![{\displaystyle f(\psi (x))=\langle y,y\rangle -\langle z,z\rangle \quad {\text{ for all }}x\in V.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
See also
References
- Lang, Serge (1972). Differential manifolds. Reading, Mass.–London–Don Mills, Ont.: Addison–Wesley Publishing Co., Inc.