Lagmental Vicfred

Morse Critical Points Have a Quadratic Normal Form by Vicfred

tags: mathematics, symplectic geometry, Hamiltonian systems, Morse theory

Near a nondegenerate critical point, a smooth function is exactly a signed sum of squares in suitable coordinates. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

Statement

A symplectic manifold \((M^{2n},\omega)\) has a closed nondegenerate two-form. A Hamiltonian \(H:M\to\mathbf R\) determines a vector field through contraction with \(\omega\).

$$ df_p=0,\qquad\det\operatorname{Hess}_p(f)\ne0 $$

I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.

$$ f=f(p)-x_1^2-\cdots-x_\lambda^2+x_{\lambda+1}^2+\cdots+x_n^2 $$

Worked algebra

This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.

$$ \operatorname{Hess}_p(f)\sim\begin{pmatrix}-I_\lambda&0\\0&I_{n-\lambda}\end{pmatrix},\qquad\lambda=\operatorname{index}_p(f) $$

Conceptual compression

The abstraction earns its keep by explaining why the same computation reappears. The notation compresses repeated reasoning without erasing the hypothesis that licenses it.

$$ \begin{aligned} \mathsf{D}\;&:\quad df_p=0,\qquad\det\operatorname{Hess}_p(f)\ne0,\\[5pt] \mathsf{C}\;&:\quad f=f(p)-x_1^2-\cdots-x_\lambda^2+x_{\lambda+1}^2+\cdots+x_n^2. \end{aligned} $$

Caveat

Symplectic geometry has no preferred notion of distance. Nondegeneracy of a two-form is not positive definiteness of a metric.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] f=f(p)-x_1^2-\cdots-x_\lambda^2+x_{\lambda+1}^2+\cdots+x_n^2 \end{gathered}} $$

This is enough machinery for one note: an exact object, a worked case, a structural law, and a clearly marked boundary. Each layer can now be tested independently.

This article was posted on Tue 22 April 2025. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.