Canonical equations of Hamilton with beautiful symmetry

TL;DR

This paper introduces a symmetric form of canonical Hamilton equations applicable to both first- and second-order systems, enhancing the theoretical framework of Hamiltonian mechanics and demonstrating its application to the nonlinear Schrödinger equation.

Contribution

A new symmetric form of canonical Hamilton equations is derived, valid for both first- and second-order differential systems, clarifying the relationship between their canonical equations.

Findings

New symmetric canonical equations valid for first- and second-order systems

The number of canonical equations for first-order systems is half of that for second-order systems

Application of the new equations to the nonlinear Schrödinger equation

Abstract

The Hamiltonian formulation plays the essential role in constructing the framework of modern physics. In this paper, a new form of canonical equations of Hamilton with the complete symmetry is obtained, which are valid not only for the first-order differential system, but also for the second-order differential system. The conventional form of the canonical equations without the symmetry [Goldstein et al., Classical Mechanics, 3rd ed, Addison-Wesley, 2001] are only for the second-order differential system. It is pointed out for the first time that the number of the canonical equations for the first-order differential system is half of that for the second-order differential system. The nonlinear Schr\"{o}dinger equation, a universal first-order differential system, can be expressed with the new canonical equations in a consistent way.