Time-dependent Hamiltonian isotopy and Hamiltonian symplectomorphism

Question

I am having some difficulty understanding the definitions of Hamiltonian isotopy and Hamiltonian symplectomorphism. I know that if we have a Hamiltonian function $H: M \to \mathbb{R}$, we have a Hamiltonian vector field $X_H$ associated with it using the symplectic form. Now, every vector field generates a flow $\phi^t_{H}:M \to M$ and, if we consider $t=1$, we have a diffeomorphism $\phi^1_{H}: M \to M$. Some people call this a Hamiltonian symplectomophism.

The problem is, I have seen another definition which considers a time-dependent Hamiltonian function $H:[0,1] \times M \to \mathbb{R}$. Denoting $H_s(x) := H(s,x)$, each $H_s: M \to \mathbb{R}$ is a Hamiltonian function in the former sense, so we have for each $s$ the associated Hamiltonian vector field $X_{H_s}$, which generates a flow $\phi^{t}_{H_s}$.

Now here is where I get confused. Some people just denote this $\phi^t_{H}$ and call it the Hamiltonian isotropy, but I think the time-dependence gets "forgotten" as if this were a flow, not a time-dependent family of flows. After that, they define a Hamiltonian symplectomorphism as $\phi^1_{H}$, but I have no idea which $s \in [0,1]$ and $H_s$ is being considered, so I don't know how this $\phi^1_{H}$, which for me should be $\phi^1_{H_s}$, is a symplectomophism from $M$ to $M$.

Answer 1

There is a standard way to deal with time dependent vector fields, in fact, there is a one to one correspondence between time dependent vector fields and isotopies, see, e.g. Exercise 58.6 of [C. Golé, Symplectic Twist map, 2001, pg.252].

Let me write here the case of Hamiltonian vector fields. Let $H (z, t) = H_t(z)$ be a smooth real valued function defined on $ M \times \mathbb{R}$. If H is constant on $t$, $H$ is called time time-independent or shortly: autonomous Hamiltonian. The Hamiltonian vector field associated to $H$ is the (possibly time dependent) vector field $X_H$ defined by: \begin{equation} \label{hamiltonian vector field} \omega\left(X_{H_t}, \cdot \right) = dH_t\left( \cdot \right). \end{equation} If $H$ is time independent, then Hamilton's equation: $\dot{z} = X_H(z,t)$ generates a (local) flow on $M$. Now, if $H$ is time dependent, then $X_H$ generates a (local) flow in the space $M \times \mathbb{R}$. More precisely, one solves the following time independent system on $M \times \mathbb{R}$ \begin{align} \dot{z} &= X_H\left(z,s \right) \\ \dot{s} &= 1 \end{align} which generates a flow $\phi^t$ in $M \times \mathbb{R}$ satisfying: $$\phi^t(z,s) = \left(h^{t+s}_s(z), s+t\right)$$ where $h^{t}_s$ is a 2-parameter (smooth in both parameter) family of smooth diffeomorphisms of $M$. Each diffeomorphism $h^{t}_{s}$ is called Hamiltonian map and, for each fixed $s$ the curve $t \mapsto h_s^t$ is a Hamiltonian isotopy. What happen to omit the parameter $s$ is the following: Another way of describing $h^{t}_s(z)$ is noting that it is the unique solution $z(t)$ of Hamilton's equation with initial condition $z(s) = z$. One can often fixes $s = 0$ and denotes $h^t_0$ by $h^t$. WARNING: Note that in the time dependent case $h^t$ is not necessarily a flow on $M$, the group property may not be true.