Morse–Palais lemma

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, 〈 , 〉) be a real Hilbert space, and let U be an open neighbourhood of 0 in H. Let f : U → R be a (k + 2)-times continuously differentiable function with k ≥ 1, i.e. f ∈ C^k+2(U; R). Assume that f(0) = 0 and that 0 is a non-degenerate critical point of f, i.e. the second derivative D²f(0) defines an isomorphism of H with its continuous dual space H^∗ by

$H \ni x \mapsto \mathrm{D}^{2} f(0) ( x, - ) \in H^{*}. \,$

Then there exists a subneighbourhood V of 0 in U, a diffeomorphism φ : V → V that is C^k with C^k inverse, and an invertible symmetric operator A : H → H, such that

$f(x) = \langle A \varphi(x), \varphi(x) \rangle$

for all x ∈ V.

Corollary

Let f : U → R be C^k+2 such that 0 is a non-degenerate critical point. Then there exists a C^k-with-C^k-inverse diffeomorphism ψ : V → V and an orthogonal decomposition

$H = G \oplus G^{\perp},$

such that, if one writes

$\psi (x) = y + z \mbox{ with } y \in G, z \in G^{\perp},$

then

$f (\psi(x)) = \langle y, y \rangle - \langle z, z \rangle$

for all x ∈ V.