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

Artur Kobus,Jan L Cieśliński

doi:10.3390/e24030338

Abstract

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 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.

Highlights

Faculty of Physics, University of Bialystok, ul
Hamiltonian systems, which are model examples of how conserved quantities fit into physics, are very convenient since their nice formal features and associated structure provided by symplectic geometry reveal several hidden links between objects engaged in the theory [1]
We found the main motivation of the undertaken research in the geometric numerical treatment of ordinary differential equations (ODEs) known as Geometric Numerical Integration (GNI), Academic Editors: Giuseppe

Summary

Introduction

Physics clearly distinguishes systems with conservation laws obeyed, as this simplifies the treatment of physical input given by initial or boundary conditions and a formal description in terms of admissible mathematical tools. The last reference here is an excellent example of accessing the second invariant quantity through discrete gradients Such a numerical treatment was applied to systems with first integrals and Lyapunov functions, but for arbitrary systems, it generally ceased to function properly because of a lack of a structural property guiding the evolution of the system. We show that Hamiltonian systems are ubiquitous when we consider some consequences of the theorem on the existence and uniqueness of solutions to ordinary differential equations (ODEs) [19–21] and seek for the structure described above even if it is apparently absent. This approach leads to an abundance of so-called effective integrals of motion.

Hamiltonian Mechanics

M-Systems of Ecology and Chemical Kinetics

Para-Hamiltonian Description of Non-Conservative Systems

Particular Non-Potential Systems

Conclusions