Hamiltonian flow?

I was wondering what the Hamiltonian flow actually is?

Here is my idea, I just wanted to know if I am correct about this.

So let $(x(t),p(t))' = X_{H}(x(t),p(t))$ are the Hamilton's equations and $X_H$ the Hamiltonian vector field.

Then the Hamiltonian flow is the map $\phi^{t}(x(0),p(0)) = (x(t),p(t))$ and in particular $\phi^{0}= \operatorname{id}.$

Moreover we have that $d_t \phi^{t} = X_H(x(t),p(t)).$ Is this correct?

• "Is this correct?" questions are not good for this SE, since the potential answer "Yes" is too short to even submit as an answer. – ACuriousMind May 4 '15 at 21:52
• WP. – Cosmas Zachos Aug 16 '18 at 1:22

1 Answer

The evolution of systems in the Hamiltonian formalism is called a flow, not merely because it can be described by a mapping, but because it is described by a particular mapping: one whose evolution in (q,p)-space resembles fluid flow.

This resemblance gives rise to Liouville's theorem, where the Hamiltonian flow, like certain fluid flows, is shown to be incompressible (constant density).