For a smooth function , the Hamiltonian vector field is defined by , with the sign depending on convention.
A Hamiltonian function is a smooth function whose differential determines a Hamiltonian vector field.
A Hamiltonian isotopy is the flow of a time-dependent Hamiltonian vector field. Its time-one map is a Hamiltonian diffeomorphism.
Articles by others on the same topic
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.