On the Principle of Least Action

Vu B Ho

doi:10.12691/ijp-6-2-4

Abstract

Investigations into the nature of the principle of least action have shown that there is an intrinsic relationship between geometrical and topological methods and the variational principle in classical mechanics. In this work, we follow and extend this kind of mathematical analysis into the domain of quantum mechanics. First, we show that the identification of the momentum of a quantum particle with the de Broglie wavelength in 2-dimensional space would lead to an interesting feature; namely the action principle S=0 would be satisfied not only by the stationary path, corresponding to the classical motion, but also by any path. Thereupon the Bohr quantum condition possesses a topological character in the sense that the principal quantum number is identified with the winding number, which is used to represent the fundamental group of paths. We extend our discussions into 3-dimensional space and show that the charge of a particle also possesses a topological character and is quantised and classified by the homotopy group of closed surfaces. We then discuss the possibility to extend our discussions into spaces with higher dimensions and show that there exist physical quantities that can be quantised by the higher homotopy groups. Finally we note that if Einsteinâs field equations of general relativity are derived from Hilbertâs action through the principle of least action then for the case of n=2 the field equations are satisfied by any metric if the energy-momentum tensor is identified with the metric tensor, similar to the case when the momentum of a particle is identified with the curvature of the particleâs path.

Highlights

In classical physics, the principle of least action is a variational principle that can be used to determine uniquely the equations of motion for various physical systems
It has been shown that different formulations of the principle of least action can be constructed for a physical system that is described by a particular system of differential equations [1,2,3,4,5]
In the old quantum theory, the Bohr quantum condition ∮ pds = nh, where p is the momentum of a particle, h is Planck constant and n is a positive integer, played a crucial role in the quantum description of a physical system, it had been introduced into the quantum theory in an ad hoc manner [7]

Summary

Introduction

The principle of least action is a variational principle that can be used to determine uniquely the equations of motion for various physical systems. New approach to the theories of gravitation using the principle of least action has been considered [6] To extend these investigations, in this work we will consider the case in which the principle of least action can be applied to both classical physics and quantum physics. Except for the quantum condition imposed on the orbital angular momentum, the Bohr model was based entirely on the classical dynamics of Newtonian physics. In this case, it seems natural to raise the question as to whether the Bohr quantum condition can be described in a classical way. In this work we show that this problem may be investigated in terms of geometry and topology, and it transpires that topology may play an important role in the determination of the nature of a quantum observable [10,11]

Principle of Least Action in Classical Mechanics

Principle of Least Action in Quantum Mechanics

Principle of Least Action in General Relativity

Conclusion