# A basic question related to 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 $$\frac{\mathrm{d}}{\mathrm{d}t}\phi_t = X_t \circ \phi_t.$$

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?

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Why not look at it in specific examples? Say the height function $H$ on the sphere $S^2\subset\mathbb{R}^3$. Or how it doesn't work for the obvious rotations on the torus.
In terms of visualization: Note that the vector field $X_t$ preserves $H_t$ (using Lie derivative, noting that $\omega$ is skew-commutative), and so the integral curves $\lbrace\phi_t(x)\rbrace_{t\in\mathbb{R}}$ are contained in level sets of $H$, i.e. $\phi^*_tH_t=H_t$.