Morse–Palais lemma explained

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

(H, \langle ⋅ , ⋅ \rangle) be a real Hilbert space, and let U be an open neighbourhood of the origin in H. Let f : U \to \R be a (k +2 ) -times continuously differentiable function with k \geq 1 ; that is, f \in C ^{k+2} (U; \R). Assume that f( 0 ) = 0 and that 0 is a non-degenerate critical point of f; that is, the second derivative D ^{2} f( 0 ) defines an isomorphism of H with its continuous dual space H ^{*} by$$H\; \backslash ni\; x\; \backslash mapsto\; \backslash mathrm^2\; f(0)\; (x,\; -)\; \backslash in\; H^*.$$

Then there exists a subneighbourhood

V of 0 in U, a diffeomorphism \varphi : V \to V that is C ^{k} with C ^{k} inverse, and an invertible symmetric operator A : H \to H, such that$$f(x)\; =\; \backslash langle\; A\; \backslash varphi(x),\; \backslash varphi(x)\; \backslash rangle\; \backslash quad\; \backslash text\; x\; \backslash in\; V.$$

Corollary

Let

f : U \to \R be f \in C ^{k+2} such that 0 is a non-degenerate critical point. Then there exists a C ^{k} -with-C ^{k} -inverse diffeomorphism \psi : V \to V and an orthogonal decomposition$$H\; =\; G\; \backslash oplus\; G^,$$such that, if one writes$$\backslash psi\; (x)\; =\; y\; +\; z\; \backslash quad\; \backslash mbox\; y\; \backslash in\; G,\; z\; \backslash in\; G^,$$then$$f\; (\backslash psi(x))\; =\; \backslash langle\; y,\; y\; \backslash rangle\; -\; \backslash langle\; z,\; z\; \backslash rangle\; \backslash quad\; \backslash text\; x\; \backslash in\; V.$$

References

• Book: Lang, Serge. Serge Lang

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