Flow of Hamiltonian vector fields, time dependent flow

Question

I have trouble understanding the notion of time dependent flows of Hamiltonian vector fields:

Let $(M, \omega)$ be a symplectic manifold, $H:M \rightarrow M$ a Hamiltonian function.

1. question: In my lecture, we haven't properly defined what Hamiltonian function means. Is it correct, that it's just a smooth function?

Now let $X_H$ be the Hamiltonian vector field which means $\omega(X_H,.)=-dH$). Then the flow of $X_H$, $\varphi_t$, is the Hamiltonian flow.

Now I know the definition of the flow from differential geometry. Here it means that $X_H(\varphi_t)= \dfrac{d}{dt} \varphi_t$.

Now we have the notion of a time-$t$ map, which I don't understand. For example, I don't understand the following exercise:

Let $\varphi_t:(M, \omega) \rightarrow (M, \omega)$ be the family of diffeomorphisms determined by the time-dependent Hamiltonian function $H : [0, 1] \times M \rightarrow \mathbb{R}$ via $\overset{.}{\varphi} = X_{H_{t}} \circ \varphi_t$.

For each $t \in (0,1)$, write $\varphi_t$ as time one map of a family of diffeomorphisms determined by a new Hamiltonian function built from $H$.

I don't understand this exercise at all. As I see it, you can get a family of flows $\{\varphi^{s}_t\}_{s \in [0,1]}$ such that for each fixed $s$, $\varphi^s_t$ is the Hamiltonian flow of the function $H_s: M \rightarrow M$. But I don't see why the parameter $t$ of the flow here is identified with the parameter $t \in [0,1]$ of the function $H$.

Answer 1

1. Usually the Hamiltonian function $H: M\rightarrow \mathbb{R}$ goes from $M$ to $\mathbb{R}$. It is the function corresponding to the $H$ in the equation $\iota(X_H)\omega=-dH$ you have given. So in in general it is just a smooth function from $M\rightarrow \mathbb{R}$. But note that some author require some extra condition for example that H has compact support or that H is normalized. Furthermore there will be some time dependency. i.e. the Hamiltonian function becomes $H: I\times M \rightarrow \mathbb{R}$, where $I$ is some interval in $\mathbb{R}$. If it helps for your intuition: In Physics Hamiltonians are the so called total energy functions. It describes the total energy of a particle. And by conservation of Energy a particle has to move on the level sets of the Hamiltonian function.

2. A time $t$ map is just the individual diffeomorphism $\phi_t$. Of particular importance are time one maps which is just $\phi_1$. The goal of the exercise is to find a new flow $\psi_t$ corresponding to $H'$ such that $\psi_1=\phi_t$ and you have to show that you can do this for every $t\in (0,1)$. One reason for this exercise might be, that many authors define the Hamiltonian group of diffeomorphisms as (heuristically speaking) time one maps of so called Hamiltonian isotopies. But with your exercise you show that one can reparametrize and so it follows that actually every diffeomorphism in your flow $\phi_t$ is a Hamiltonian diffeomorphism, not just the last one.