Hamiltonian form for general autonomous ODE systems: Low dimensional examples

TL;DR

This paper develops a Hamiltonian canonical form for even-dimensional autonomous ODE systems, introducing effectively conserved quantities and illustrating applications across physics, biology, and chemistry with potential for geometric numerical methods.

Contribution

It presents a new Hamiltonian framework for autonomous ODEs, including a novel class of effectively conserved quantities applicable beyond physics.

Findings

Constructed Hamiltonian form for autonomous systems

Introduced effectively conserved quantities with unique properties

Provided examples from multiple scientific disciplines

Abstract

Paper is devoted to maintaining the simple objective: We want to provide Hamiltonian canonical form for autonomous dynamical system reducible to even-dimensional one. Along the road we construct new class of conserved quantities, called effectively conserved, that have dissimilar properties to traditional first integrals (e.g. differential of effectively conserved quantity being a Pfaffian form). We do not confine the discussion to physics; we consider examples from biology and chemistry, giving direct recipe for how to engage the framework in occurring problems. Perspective for future application in geometric numerical methods is given.