Classical Hamiltonian system

It is a symplectic manifold $(M, ω)$ in which we have a prescribed a smooth function $H$ called the Hamiltonian of the system.
In other words, a Hamiltonian system is a classical mechanical system in which the state of the system is described by a point in a symplectic manifold $(M, ω)$ , and the evolution of the system is determined by the Hamiltonian function $H : M \to R$ . The Hamiltonian function generates one Hamiltonian vector field $X_{H}$ , whose flow represents the time evolution of the system. The dynamics of the system is then given by Hamilton's equations, which can be written in the form $\dot{q} = \frac{\partial H}{\partial p}$ and $\dot{p} = - \frac{\partial H}{\partial q}$ , where $(q, p)$ are local coordinates on $M$ . The Hamiltonian function encapsulates all the information about the energy and forces of the system, and the symplectic form $ω$ describes the underlying geometric structure that governs the evolution of the system.

It is important the notion of Hamiltonian symmetry.

In Poisson manifolds

A slightly more general idea of Hamiltonian system is given in @olver86 page 395. A system of first order ODEs is a Hamiltonian systems if there is a function $H$ and a matrix of functions $J$ coming from a Poisson bracket#In local coordinates such that the system can be written as

\frac{d x}{d t} = J (x) \nabla H (x) .

They are then generalized in @olver86 page 408 to the case

\frac{d x}{d t} = J (x) \nabla H (x, t)

although not explicitly defined. I interpret that the vector field of the system is, in this case, $\partial_{t} + X_{H}$ .
A first integral of the Hamiltonian system is not a first integral of $X_{H}$ but a first integral of $\partial_{t} + X_{H}$ , that is, a function $P (x, t)$ such that $(\partial_{t} + X_{H}) (P) = 0$ , i.e.,

\frac{\partial P}{\partial t} + {P, H} = 0.

The function $P (x, t)$ generates itself a Hamiltonian vector field $X_{P}$ , which depends on $t$ as a parameter.