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Ricardo Andrade
Ricardo Andrade
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A basic questoinquestion related to Hamiltonian Isotopyisotopy in Symplecticsymplectic 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.$$\phi_t : M \rightarrow M$. It is generated by a unique family of vector fields $X_t: M \rightarrow TM$ such that $$\frac{\mathrm{d}}{\mathrm{dt}}\phi_t = X_t \circ \phi_t.$$$$\frac{\mathrm{d}}{\mathrm{d}t}\phi_t = X_t \circ \phi_t.$$

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?

A basic questoin 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 $$\frac{\mathrm{d}}{\mathrm{dt}}\phi_t = X_t \circ \phi_t.$$

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?

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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Qiao
Qiao
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A basic questoin 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 $$\frac{\mathrm{d}}{\mathrm{dt}}\phi_t = X_t \circ \phi_t.$$

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?