On the Hamiltonian structure of 2D ODE possessing an invariant

The authors discuss the Hamiltonian structure with respect to the canonical bracket for a 2D dynamical system and prove that this structure can be obtained simply with a rescaling of the independent variable. The rescaling will provide a Hamiltonian when a time-independent invariant is known, the Hamiltonian coinciding with the invariant. Applications to different systems of ODE in two dimensions are presented.