Lagmental Vicfred

Hamilton's Equations Come from Contracting dH with omega by Vicfred

Last updated: Wed 23 August 2023

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

The Hamiltonian vector field is symplectically dual to the differential of the energy. I will separate the object being defined from the consequence being claimed.

Objects and notation

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 \(\omega\).

$$ \iota_{X_H}\omega=dH $$

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.

$$ \dot q_i=\frac{\partial H}{\partial p_i},\qquad\dot p_i=-\frac{\partial H}{\partial q_i} $$

Push the symbols

The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.

$$ X_H=\sum_{i=1}^{n}\left(\frac{\partial H}{\partial p_i}\frac{\partial}{\partial q_i}-\frac{\partial H}{\partial q_i}\frac{\partial}{\partial p_i}\right) $$

Structural reading

The two-row display is also a debugging tool: if the conclusion changes when only notation changes, some hidden choice has entered the argument.

$$ \begin{aligned} \mathsf{D}\;&:\quad \iota_{X_H}\omega=dH,\\[5pt] \mathsf{C}\;&:\quad \dot q_i=\frac{\partial H}{\partial p_i},\qquad\dot p_i=-\frac{\partial H}{\partial q_i}. \end{aligned} $$

A hypothesis worth keeping

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \dot q_i=\frac{\partial H}{\partial p_i},\qquad\dot p_i=-\frac{\partial H}{\partial q_i} \end{gathered}} $$

The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.

This article was posted on Mon 21 September 2015. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.