In physics, the action is a particular quantity in a physical system that can be used to describe its operation. Action is an alternative to differential equations. The action is not necessarily the same for different types of systems.
The action yields the same results as using differential equations. Action only requires the states of the physical variable to be specified at two points, called the initial and final states. The values of the physical variable at all intermediate points may then be determined by "minimizing" the 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.
Hamilton's principle has also been extended to quantum mechanics and quantum field theory.
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:
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
and
, where
represent 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 (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
According to Maupertuis' principle, the true path is a path for which the abbreviated action is stationary.
Hamilton's principal function is defined by the Hamilton–Jacobi equations (HJE), another alternative formulation of classical mechanics. This function is related to the functional
by fixing the initial time
and endpoint
and allowing the upper limits
and the second endpoint
to vary; these variables are the arguments of the function
. In other words, the action function
is the indefinite integral of the Lagrangian with respect to time.
When the total energy is conserved, the HJE can be solved with the additive separation of variables
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.
The Hamilton–Jacobi equations are often solved by additive separability; in some cases, the individual terms of the solution, e.g., , are also called an "action".
This is a single variable in the action-angle coordinates, defined by integrating a single generalized momentum around a closed path in phase space, corresponding to rotating or oscillating motion
The variable is called the "action" of the generalized coordinate
; 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.
See tautological one-form.
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 ( and
) and the final and initial positions are specified in advance as
and
. Let
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
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 eqations 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.
When relativistic effects are significant, the action of a point particle of mass m traveling a world line C parameterized by the proper time is
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
The action principle can be extended to obtain the equations of motion for fields, such as the electromagnetic field or gravity.
The Einstein equation utilizes the Einstein-Hilbert action as constrained by a variational principle.
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.
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.
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.
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 [1] who lists, among other things, the following books