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||It has been suggested that this article or section be merged with Hamilton's principle. (Discuss) Proposed since November 2011.|
In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Generally, the action takes different values for different paths.
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.
Action was defined in several, now obsolete, ways during the development of the concept.
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 a stationary point (usually, a minimum) of the action.
where the integrand L is called the Lagrangian. For the action integral to be well defined the trajectory has to be bounded in time and space.
In classical physics, the term "action" has a number of 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 q(t) of the system between two times t1 and t2, where q 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 qtrue(t) 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.
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 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 q1 and allowing the upper limits t2 and the second endpoint q2 to vary; these variables are the arguments of the function S. In other words, the action function is the indefinite integral of the Lagrangian with respect to time.
where the time independent function W(q1, q2 ... qN) 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.
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.
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 may also depend explicitly on time. 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 and . Let xtrue(t) represent the true evolution that we seek, and let be a slightly perturbed version of it, albeit with the same endpoints, and . 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 ε(t) about the true evolution,
This can be true only if
The Euler–Lagrange equation is obeyed provided the functional derivative of the action integral is identically zero:
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.
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.
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. 
In quantum mechanics, the system does not follow a single path whose action is stationary, but the behavior of the system depends on all permitted 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.
If instead, the particle is parametrized 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
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