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

H ∋ x ↦ D 2 f ( 0 ) ( x , − ) ∈ H ∗ . {\displaystyle H\ni x\mapsto \mathrm {D} ^{2}f(0)(x,-)\in H^{*}.}

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

f ( x ) = ⟨ A φ ( x ) , φ ( x ) ⟩  for all  x ∈ V . {\displaystyle f(x)=\langle A\varphi (x),\varphi (x)\rangle \quad {\text{ for all }}x\in V.}

Corollary

Let f : U → R {\displaystyle f:U\to \mathbb {R} } be f ∈ C k + 2 {\displaystyle f\in C^{k+2}} such that 0 {\displaystyle 0} is a non-degenerate critical point. Then there exists a C k {\displaystyle C^{k}} -with- C k {\displaystyle C^{k}} -inverse diffeomorphism ψ : V → V {\displaystyle \psi :V\to V} and an orthogonal decomposition

H = G ⊕ G ⊥ , {\displaystyle H=G\oplus G^{\perp },}
such that, if one writes
ψ ( x ) = y + z  with  y ∈ G , z ∈ G ⊥ , {\displaystyle \psi (x)=y+z\quad {\mbox{ with }}y\in G,z\in G^{\perp },}
then
f ( ψ ( x ) ) = ⟨ y , y ⟩ − ⟨ z , z ⟩  for all  x ∈ V . {\displaystyle f(\psi (x))=\langle y,y\rangle -\langle z,z\rangle \quad {\text{ for all }}x\in V.}

See also

References

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