Is a hamiltonian isotopy real-valued of manifold-valued?

Question

I have looked everywhere and I can't find a clear definition of hamiltonian isotopy. I have these definitions in my lecture notes but they are rather confusing

Acording to the last definition a Hamiltonian isotopy $\phi_t$ (the one they given in the definition of Hamiltonian symplectomorphism) is a map $M \to M$, otherwise the equality $\Phi=\phi_1$ would not make sense right? Since a symplectomorphism is a diffeomorphism $\Phi : M → M$ . Moreover since the isotopy $\phi_t: M \to M $ is a flow , it is by definition the identity on M at t=0 so more reason for it to be a map $M → M$

On the other hand it says that a Hamiltonian isotopy is the isotopy generated by a Hamiltonian vector field $X_t$ so $\phi_t$ plays the role of $H_t$ in the definition above in the next-to-last paragraph and it is a map $M \to \Bbb R$

I ran into trouble when I was trying to prove that Ham(M, ω) is a subgroup of Symp(M, ω) I started by taking $\Phi,\Psi\in Ham(M,\omega)$ so by definition there exist Hamiltonian isotopies $\phi_t$ and $\psi_t$ s.t $\phi_1=\Phi$ and $\psi_t=\Psi$ and to start with I wanted to argue that the composition $\phi_t\circ\psi_t$ is smooth, but composing them makes no sense if they are maps $M \to \Bbb R$. So I thought the definition must be wrong but I have checked in several places and in some places they do start proofs by taking Hamiltonian isotopies as real-valued , while in others as $M \to M$. But if both definitions are correct how do I make sense of the composition?

Can someone clarify these points? What are the correct definitions and how to I fix the start of the proof (make sense of composition)?

Edit------------unrelated to this question:

Answer 1

I think your confusion stems from abuse of terminology. There are several different things in symplectic geometry which all have the adjective "Hamiltonian":

1. Time-dependent Hamiltonian functions on a manifold $M$, which are smooth functions
   $$ H: M\times [0,1]\to {\mathbb R}. $$ Such $H$ can be regarded as a homotopy between the functions $H(\cdot, 0)$ and $H(\cdot, 1)$.

2. Time-dependent Hamiltonian vector fields, denoted $X_t$, which are maps $$ M\times [0,1]\to TM, (p,t)\mapsto X_t(p)\in T_pM $$ (satisfying further properties).

3. Time-dependent Hamiltonian maps (which is suboptimal) or Hamiltonian isotopies (which is better), which are maps $$ G: M\times [0,1]\to M $$ (satisfying further properties). Such a map is an isotopy between the identity map $G(\cdot, 0)$ and the diffeomorphism $G(\cdot, 1)$ (the time 1 map of the isotopy $G$).

4. Hamiltonian maps or Hamiltonian symplectomorphisms, which are maps $$ \Phi: M\to M $$ for which there exists a Hamiltonian isotopy $G$ such that $\Phi=G(\cdot, 1)$ (i.e. $\Phi$ is the time 1 map of a Hamiltonian isotopy).

While you are still learning basics of the subject, it is best to avoid conflating these notions and make sure you use different words for different concepts. For instance, maps in (4) should never be referred to as isotopies and functions in (1) should never be referred to as Hamiltonian maps. In physics literature one routinely calls functions in (1) "Hamiltonians." I suggest, you avoid doing so at this stage.