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.

tatement 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 D²"f"(0) defines an isomorphism of "H" with its continuous dual space "H"^∗ by

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

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

:$f(x)\; =\; langle\; A\; varphi(x),\; varphi(x)\; angle$

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

:$f\; (psi(x))\; =\; langle\; y,\; y\; angle\; -\; langle\; z,\; z\; angle$

for all "x" ∈ "V".

References

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