In modern physics, action is an attribute of the development of a physical system over a period of time, namely amount by which the phase of the wave function has changed. It has units of energy × time (joule-seconds in SI units). Planck's constant is the quantum of action. The Lagrangian function, , is the rate at which action occurs, that is, the rate (frequency) at which the phase changes. So the classical action, , associated with a period of time from to is calculated as the integral:
where the integral is taken over the path (that is, the sequence of intermediate states) taken by the system between the initial state at time and final state at time . Typically, the action will take different values for different paths. Classical mechanics shows that the path actually followed by a real physical system is that for which the action is minimized (or, more strictly, is stationary). The classical (differential) equations of motion of a system can be derived from this principle of least action.
The stationary action formulation of classical mechanics extends readily to quantum mechanics, and is at the heart of the Feynman path integral. It also provides a basis for the development of string theory.
History of term action
The term action was defined in several (now obsolete) ways during its development.
Physical laws are most often expressed as differential equations, which specify how a physical variable changes from its present value with infinitesimally small changes in time, position, or some other variable. By adding up these small changes, a differential equation provides a recipe for determining the value of the physical variable at any point, given only its starting value at one point and possibly some initial derivatives.
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.
Expressed in mathematical language, using the calculus of variations, the evolution of a physical system (i.e., how the system actually progresses from one state to another) corresponds to an extremum (usually, a minimum) of the action.
Several different definitions of 'the action' are in common use in physics:
Disambiguation of "action" in classical physics
In classical physics, the term "action" has at least eight distinct meanings.
Most commonly, the term is used for a functional which takes a function of time and (for fields) space as input and returns a scalar. In classical mechanics, the input function is the evolution of the system between two times t1 and t2, where represent the generalized coordinates. The action is defined as the integral of the Lagrangian L 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 (a minimum, maximum, or a saddle point). This principle results in the equations of motion in Lagrangian mechanics.
Abbreviated action (functional)
Usually denoted as , this is also a functional. Here 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 along a path in the generalized coordinates
Hamilton's principal function
Hamilton's principal function is defined by the Hamilton–Jacobi equations (HJE), another alternative formulation of classical mechanics. This function S is related to the functional by fixing the initial time t1 and endpoint and allowing the upper limits t2 and the second endpoint to vary; these variables are the arguments of the function S. In other words, the action function S is the indefinite integral of the Lagrangian with respect to time.
Hamilton's characteristic function
where the time independent function is called Hamilton's characteristic function. The physical significance of this function is understood by taking its total time derivative
This can be integrated to give
which is just the abbreviated action.
Other solutions of Hamilton–Jacobi equations
The Hamilton–Jacobi equations are often solved by additive separability; in some cases, the individual terms of the solution, e.g., Sk(qk), are also called an "action".
Action of a generalized coordinate
The variable Jk is called the "action" of the generalized coordinate qk; the corresponding canonical variable conjugate to Jk is its "angle" wk, for reasons described more fully under action-angle coordinates. The integration is only over a single variable qk and, therefore, unlike the integrated dot product in the abbreviated action integral above. The Jk variable equals the change in Sk(qk) as qk is varied around the closed path. For several physical systems of interest, Jk is either a constant or varies very slowly; hence, the variable Jk is often used in perturbation calculations and in determining adiabatic invariants.
Action for a Hamiltonian flow
Euler–Lagrange equations for the action integral
As noted above, the requirement that the action integral be stationary under small perturbations of the evolution is equivalent to a set of differential equations (called the Euler–Lagrange equations) that may be determined using the calculus of variations. We illustrate this derivation here using only one coordinate, x; the extension to multiple coordinates is straightforward.
Adopting Hamilton's principle, we assume that the Lagrangian L (the integrand of the action integral) depends only on the coordinate x(t) and its time derivative dx(t)/dt, and does not depend on time explicitly. In that case, the action integral can be written
where the initial and final times (t1 and t2) and the final and initial positions are specified in advance as x1 = x(t1) and x2 = x(t2). Let xtrue(t) represent the true evolution that we seek, and let xper(t) be a slightly perturbed version of it, albeit with the same endpoints, xper(t1) = x1 and xper(t2) = x2. The difference between these two evolutions, which we will call , is infinitesimally small at all times
At the endpoints, the difference vanishes, i.e., .
Expanded to first order, the difference between the actions integrals for the two evolutions is
Integration by parts of the last term, together with the boundary conditions , yields the equation
The requirement that be stationary implies that the first-order change must be zero for any possible perturbation about the true evolution. This can be true only if
Those familiar with functional analysis will note that the Euler–Lagrange equations simplify to
The quantity is called the conjugate momentum for the coordinate x. An important consequence of the Euler–Lagrange equations is that if L does not explicitly contain coordinate x, i.e.
In such cases, the coordinate x is called a cyclic coordinate, and its conjugate momentum is conserved.
Example: Free particle in polar coordinates
Simple examples help to appreciate the use of the action principle via the Euler–Lagrangian equations. A free particle (mass m and velocity v) in Euclidean space moves in a straight line. Using the Euler–Lagrange equations, this can be shown in polar coordinates as follows. In the absence of a potential, the Lagrangian is simply equal to the kinetic energy
in orthonormal (x,y) coordinates, where the dot represents differentiation with respect to the curve parameter (usually the time, t). In polar coordinates (r, φ) the kinetic energy and hence the Lagrangian becomes
The radial r and φ components of the Euler–Lagrangian equations become, respectively
The solution of these two equations is given by
for a set of constants a, b, c, d determined by initial conditions. Thus, indeed, the solution is a straight line given in polar coordinates.
Action principle for single relativistic particle
If instead, the particle is parameterized by the coordinate time t of the particle and the coordinate time ranges from t1 to t2, then the action becomes
where the Lagrangian is
Action principle for classical fields
The path of a body in a gravitational field (i.e. free fall in space time, a so called geodesic) can be found using the action principle.
Action principle in quantum mechanics and quantum field theory
In quantum mechanics, the system does not follow a single path whose action is stationary, but the behavior of the system depends on all imaginable paths and the value of their action. The action corresponding to the various paths is used to calculate the path integral, that gives the probability amplitudes of the various outcomes.
Although equivalent in classical mechanics with Newton's laws, the action principle is better suited for generalizations and plays an important role in modern physics. Indeed, this principle is one of the great generalizations in physical science. In particular, it is fully appreciated and best understood within quantum mechanics. Richard Feynman's path integral formulation of quantum mechanics is based on a stationary-action principle, using path integrals. Maxwell's equations can be derived as conditions of stationary action.
Action principle and conservation laws
Symmetries in a physical situation can better be treated with the action principle, together with the Euler–Lagrange equations, which are derived from the action principle. An example is Noether's theorem, which states that to every continuous symmetry in a physical situation there corresponds a conservation law (and conversely). This deep connection requires that the action principle be assumed.
Modern extensions of the action principle
The action principle can be generalized still further. For example, the action need not be an integral because nonlocal actions are possible. The configuration space need not even be a functional space given certain features such as noncommutative geometry. However, a physical basis for these mathematical extensions remains to be established experimentally.
For an annotated bibliography, see Edwin F. Taylor  who lists, among other things, the following books
Published - July 2009
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