time-$t$ flow map of vector field

Question

I met the following expression in some article "... where $\Phi_t$ is the time-1 flow map of the Hamiltonian vector field produced by the Hamiltonian function $H$ = ..." I haven't met any explicit definition of such thing yet. Would you please give a clear definition of " time-$t$ flow map of a Hamiltonian vector field " ? For example, what is the time-$t$ flow map of the vector field produced by $$H: \mathbb{R}^d \to \mathbb{R}^d: q:=(q_1, ...,q_d)\mapsto H(q)= q_d.$$ Thank you in advanced

Answer 1

I believe you are reading symplectic geometry. Let$(M,\omega)$ a symplectic manifold $H:M\rightarrow R$ an Hamiltonian (a differentiable function), write $i_X\omega=-dH$. You obtain a (Hamiltonian) vector field on $M$ whose flow is the Hamiltonian flow.

Example: $M=R^{2n}, \omega= \sum dx_i\wedge dx_{n+1}$.