Action (physics)
In physics, action is a scalar quantity that describes how the balance of kinetic versus potential energy of a physical system changes with trajectory. Action is significant because it is an input to the principle of stationary action, an approach to classical mechanics that is simpler for multiple objects. Action and the variational principle are used in Feynman's formulation of quantum mechanics and in general relativity. For systems with small values of action close to the Planck constant, quantum effects are significant.
In the simple case of a single particle moving with a constant velocity, the action is the momentum of the particle times the distance it moves, added up along its path; equivalently, action is the difference between the particle's kinetic energy and its potential energy, times the duration for which it has that amount of energy.
More formally, action is a mathematical functional which takes the trajectory of the system as its argument and has a real number as its result. Generally, the action takes different values for different paths. Action has dimensions of energy × time or momentum × length, and its SI unit is joule-second.
Introduction
Introductory physics often begins with Newton's laws of motion, relating force and motion; action is part of a completely equivalent alternative approach with practical and educational advantages. However, the concept took many decades to supplant Newtonian approaches and remains a challenge to introduce to students.Simple example
For a trajectory of a ball moving in the air on Earth the action is defined between two points in time, and as the kinetic energy minus the potential energy, integrated over time.The action balances kinetic against potential energy.
The kinetic energy of a ball of mass is where is the velocity of the ball; the potential energy is where is the acceleration due to gravity and its height. Then the action between and is
The action value depends upon the trajectory taken by the ball through and. This makes the action an input to the powerful stationary-action principle for classical and for quantum mechanics. Newton's equations of motion for the ball can be derived from the action using the stationary-action principle, but the advantages of action-based mechanics only begin to appear in cases where Newton's laws are difficult to apply. Replace the ball with an electron: classical mechanics fails but stationary action continues to work. The energy difference in the simple action definition, kinetic minus potential energy, is generalized and called the Lagrangian for more complex cases.
Planck's quantum of action
The Planck constant, written as, is the quantum of action. It is related to the quantum of angular momentum,, by the relation. These constants have units of energy times time. They appear in all significant quantum equations, such as the uncertainty principle and the de Broglie wavelength. Whenever the value of the action approaches the Planck constant, quantum effects are significant.History
and Leonhard Euler working in the 1740s developed early versions of the action principle. Joseph Louis Lagrange clarified the mathematics when he invented the calculus of variations. William Rowan Hamilton made the next big breakthrough, formulating Hamilton's principle in 1853. Hamilton's principle became the cornerstone for classical work with different forms of action until Richard Feynman and Julian Schwinger developed quantum action principles.Definitions
Expressed in mathematical language, using the calculus of variations, the evolution of a physical system corresponds to a stationary point of the action. Action has the dimensions of energy| × time|, and its SI unit is joule-second, which is identical to the unit of angular momentum.Several different definitions of "the action" are in common use in physics. The action is usually an integral over time. However, when the action pertains to fields, it may be integrated over spatial variables as well. In some cases, the action is integrated along the path followed by the physical system.
The action is typically represented as an integral over time, taken along the path of the system between the initial time and the final time of the development of the system:
where the integrand is called the Lagrangian. For the action integral to be well-defined, the trajectory has to be bounded in time and space.
Action (functional)
Most commonly, the term is used for a functional which takes a function of time and space as input and returns a scalar. In classical mechanics, the input function is the evolution of the system between times and, where represents the generalized coordinates. The action is defined as the integral of the Lagrangian for an input evolution between the two times:where the endpoints of the evolution are fixed and defined as and. According to Hamilton's principle, the true evolution is an evolution for which the action is stationary. This principle results in the equations of motion in Lagrangian mechanics.
Abbreviated action (functional)
In addition to the action functional, there is another functional called the [|abbreviated action]. In the abbreviated action, the input function is the path followed by the physical system without regard to its parameterization by time. For example, the path of a planetary orbit is an ellipse, and the path of a particle in a uniform gravitational field is a parabola; in both cases, the path does not depend on how fast the particle traverses the path.The abbreviated action is defined as the integral of the generalized momenta,
for a system Lagrangian along a path in the generalized coordinates :
where and are the starting and ending coordinates.
According to Maupertuis's principle, the true path of the system is a path for which the abbreviated action is stationary.
Hamilton's characteristic function
When the total energy rp|225math|Wrp|434Action of a generalized coordinate
A variable in the action-angle coordinates, called the "action" of the generalized coordinate, is defined by integrating a single generalized momentum around a closed path in phase space, corresponding to rotating or oscillating motion:The corresponding canonical variable conjugate to is its "angle", for reasons described more fully under action-angle coordinates. The integration is only over a single variable and, therefore, unlike the integrated dot product in the abbreviated action integral above. The variable equals the change in as is varied around the closed path. For several physical systems of interest, is either a constant or varies very slowly; hence, the variable is often used in perturbation calculations and in determining adiabatic invariants. For example, they are used in the calculation of planetary and satellite orbits.
Single relativistic particle
When relativistic effects are significant, the action of a point particle of mass travelling a world line parametrized by the proper time isIf instead, the particle is parametrized by the coordinate time of the particle and the coordinate time ranges from to, then the action becomes
where the Lagrangian is
where is the metric tensor.
Action principles and related ideas
Physical laws are frequently expressed as differential equations, which describe how physical quantities such as position and momentum change continuously with time, space or a generalization thereof. Given the initial and boundary conditions for the situation, the "solution" to these empirical equations is one or more functions that describe the behavior of the system and are called equations of motion.Action is a part of an alternative approach to finding such 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 generally, is stationary. In other words, the action satisfies a variational principle: the principle of stationary action. The action is defined by an integral, and the classical equations of motion of a system can be derived by minimizing the value of that integral.
The action principle provides deep insights into physics, and is an important concept in modern theoretical physics. Various action principles and related concepts are summarized below.
Maupertuis's principle
In classical mechanics, Maupertuis's principle states that the path followed by a physical system is the one of least length. Maupertuis's principle uses the abbreviated action between two generalized points on a path.Hamilton's principal function
states that the differential equations of motion for any physical system can be re-formulated as an equivalent integral equation. Thus, there are two distinct approaches for formulating dynamical models.Hamilton's principle 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 the path integral formulation of quantum mechanics makes use of the concept—where a physical system explores all possible paths, with the phase of the probability amplitude for each path being determined by the action for the path; the final probability amplitude adds all paths using their complex amplitude and phase.