Hamiltonian vector field

In the context of Hamiltonian mechanics, a Hamiltonian vector field is a vector field that is derived from a Hamiltonian function, which typically represents the total energy of a physical system. The Hamiltonian formulation of classical mechanics describes the evolution of a system in phase space using this vector field. Suppose we have a Hamiltonian function \( H(q, p) \), where \( q \) represents generalized coordinates (position variables) and \( p \) represents generalized momenta.