Para-Hamiltonian form for General Autonomous ODE Systems: Introductory Results

We propose a new tool to deal with autonomous ODE systems for which the solution to the Hamiltonian inverse problem is not available in the usual, classical sense. Our approach allows a class of formally conserved quantities to be constructed for dynamical systems showing dissipative behavior and other, more general, phenomena. The only ingredients of this new framework are Hamiltonian geometric mechanics (to sustain certain desirable properties) and the direct reformulation of the notion of the derivative along the phase curve. This seemingly odd and inconsistent marriage of apparently remote ideas leads to the existence of the generator of motion for every autonomous ODE system. Having constructed the generator, we obtained the Lie invariance of the symplectic form ω for free. Various examples are presented, ranging from mathematics, classical mechanics, and thermodynamics, to chemical kinetics and population dynamics in biology. Applications of these ideas to geometric integration techniques of numerical analysis are suggested.