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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.

The canonical context in which these are used is Classical Mechanics on cotangent bundles... Hamilton equations! In other words, the Hamiltonian is the energy of your system, and your symplectomorphisms describe the physical motion in phase space.

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. $\phi^*_tH_t=H_t$.

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