Numerical Evolution from the Hamiltonian

TL;DR

This paper introduces a numerical method for approximating the time evolution of point particle systems using only the Hamiltonian and initial conditions, demonstrated with a Python implementation.

Contribution

It presents a new numerical approach that generates and solves equations of motion directly from the Hamiltonian, with a practical implementation for educational purposes.

Findings

Method successfully computes time evolution for simple systems

Implementation demonstrates pedagogical value

Numerical effort limits practical use for complex systems

Abstract

We propose a numerical method for approximate calculations of the time evolution of point particle systems given only the system's Hamiltonian function and initial conditions. The method both generates and solves the equations of motion numerically. For demonstration purposes, a working implementation written in Python is described and applied to standard problems. The method may have some pedagogical merits but the numerical effort of generating the equations of motion makes it unsuitable for actual numerical solution of "real" problems with any but just a few degrees of freedom.