# Hamiltonian Isotopy in Symplectic geometry

In any standard symplectic geometry/topology textbook, the concept of Hamiltonian isotopy was introduced:

$(M, \omega)$ is a sympplectic manifold. Given a symplectic isotopy $\phi_t : M \rightarrow M.$ It is generated by a unique family of vector fields $X_t: M \rightarrow TM$ such that $$\frac{\mathrm{d}}{\mathrm{dt}}\phi_t = X_t \circ \phi_t.$$

The vector fields$X_t$ are symplectic vector fields.

When $$\iota(X_t)\omega = dH_t$$ for a smooth family of Hamiltonian functions $H_t: M \rightarrow \mathbb{R},$ the isotopies above is called Hamiltonian.

My question is: How do we 'visualize' the relation between the Hamiltonian functions $H_t$ and the symplectomorphisms $\phi_t$ in the isotopy? What are some of the good contexts in which Hamiltonian isotopies are used?

-
Can you give a bit more detail about what you are looking for? What do you mean by "visualize"? what makes a context "good"? Have you studied any classical physics where the Hamiltonian is giving the total energy of a conservative system? if so, this is the right visualization (at least on cotangent bundles). –  Sam Lisi Apr 27 '13 at 21:53