Action (physics) - Introduction

Introduction

Physical laws are frequently expressed as differential equations, which describe how physical quantities such as position and momentum change continuously with time. Given the initial and boundary conditions for the situation, the solution to the equation is a function describing the behavior of the system (positions and momenta of the particles) at all times and all positions within the set boundaries.

There is an alternative approach to finding equations of motion. Classical mechanics postulates that the path actually followed by a physical system is that for which the action is minimized, or, more strictly, is stationary. That is to say, the action satisfies a variational principle: the principle of stationary action (see also below). The action is defined by an integral, and the classical equations of motion of a system can be derived from minimizing the value of the action integral, rather than solving differential equations.

This simple principle provides deep insights into physics, and is an important concept in modern theoretical physics.

The equivalence of these two approaches is contained in Hamilton's principle, which states that the differential equations of motion for any physical system can be re-formulated as an equivalent integral equation. It applies not only to the classical mechanics of a single particle, but also to classical fields such as the electromagnetic and gravitational fields. Hamilton's principle has also been extended to quantum mechanics and quantum field theory—in particular path integral formulation makes use of the concept—where a physical system follows simultaneously all possible paths with probability amplitudes for each path being determined by the action for the path.