MorseTheory

Morse–Palais lemma

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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 ( H , ⟨ ⋅ , ⋅ ⟩ ) {\displaystyle (H,\langle \cdot ,\cdot \rangle )} be a real Hilbert space, and let U U be an open neighbourhood of the origin in H . 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 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 H with its continuous dual space H ∗ 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 V of 0 {\displaystyle 0} in U , {\displaystyle U,} a diffeomorphism φ : V → V {\displaystyle \varphi :V\to V} that is C k C^{k} with C k 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 C^{k}-with- C k C^{k}-inverse diffeomorphism ψ : V → V {\displaystyle \psi :V\to V} and an orthogonal decomposition H = G ⊕ G ⊥ ,Morse–Palais lemma

[Article](https://en.wikipedia.org/wiki/Morse%E2%80%93Palais_lemma)
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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 ( H , ⟨ ⋅ , ⋅ ⟩ ) {\displaystyle (H,\langle \cdot ,\cdot \rangle )} be a real Hilbert space, and let U U be an open neighbourhood of the origin in H . 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 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 H with its continuous dual space H ∗ 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 V of 0 {\displaystyle 0} in U , {\displaystyle U,} a diffeomorphism φ : V → V {\displaystyle \varphi :V\to V} that is C k C^{k} with C k 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 C^{k}-with- C k 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.} {\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.}