Determination of the Hamiltonian from the Equations of Motion with Illustration from Examples

TL;DR

This paper presents a method to determine the Hamiltonian from equations of motion by reformulating the problem as matrix factorization, providing explicit Hamiltonians and symplectic structures through a constructive proof.

Contribution

It introduces a new approach to derive Hamiltonians from equations of motion using matrix factorization and offers explicit construction methods.

Findings

Criterion for the existence of a Hamiltonian based on the evolution matrix

Constructive proof enabling explicit Hamiltonian derivation

Application to various dynamical systems

Abstract

In this paper, we study the determination of Hamiltonian from a given equations of motion. It can be cast into a problem of matrix factorization after reinterpretation of the system as first-order evolutionary equations in the phase space coordinates. We state the criterion on the evolution matrix for a Hamiltonian to exist. In addition, the proof is constructive and an explicit Hamiltonian with accompanied symplectic structure can be obtained. As an application, we will study a few classes of dynamical systems for illustration.