# Action (physics)

# Action (physics)

In physics, **action** is an attribute of the dynamics of a physical system from which the equations of motion of the system can be derived. 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.^{[1]} Action has the dimensions of [energy]⋅[time] or [momentum]⋅[length], and its SI unit is joule-second.

Introduction

Hamilton's principle 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.

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 the path integral formulation of quantum mechanics makes use of the concept—where a physical system randomly follows one of the possible paths, with the phase of the probability amplitude for each path being determined by the action for the path.^{[2]}

Solution of differential equation

Empirical 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*.

Minimization of action integral

*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 (see also below). 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.

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

History

*Action* was defined in several now obsolete ways during the development of the concept.^{[3]}

Gottfried Leibniz, Johann Bernoulli and Pierre Louis Maupertuis defined the action for light as the integral of its speed or inverse speed along its path length.

Leonhard Euler (and, possibly, Leibniz) defined action for a material particle as the integral of the particle's speed along its path through space.

Pierre Louis Maupertuis introduced several

*ad hoc*and contradictory definitions of action within a single article, defining action as potential energy, as virtual kinetic energy, and as a hybrid that ensured conservation of momentum in collisions.^{[4]}

Mathematical definition

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.

Several different definitions of "the action" are in common use in physics.^{[3]}^{[5]} 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:^{[3]}

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.

Action has the dimensions of [energy]⋅[time], and its SI unit is joule-second, which is identical to the unit of angular momentum.

Action in classical physics

In classical physics, the term "action" has a number of meanings.

Action (functional)

`Most commonly, the term is used for afunctionalwhich takes afunctionof time and (forfields) space as input and returns ascalar.`

^{[6]}^{[7]}Inclassical mechanics, the input function is the evolution**q**(*t*) of the system between two times*t*1and*t*2, where**q**represents thegeneralized coordinates. The actionis defined as theintegralof theLagrangian*L*for an input evolution between the two times:`where the endpoints of the evolution are fixed and defined asand. According toHamilton's principle, the true evolution`

**q**true(*t*) is an evolution for which the actionisstationary(a minimum, maximum, or asaddle point). This principle results in the equations of motion inLagrangian mechanics.Abbreviated action (functional)

`Usually denoted as, this is also afunctional. 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 actionis defined as the integral of the generalized momenta along a path in thegeneralized coordinates:`According toMaupertuis' principle, the true path is a path for which the abbreviated actionisstationary.`

Hamilton's principal function

`Hamilton's principal function is defined by theHamilton–Jacobi equations(HJE), another alternative formulation ofclassical mechanics. This function`

*S*is related to the functionalby fixing the initial time*t*1and the initial endpoint**q**1and allowing the upper limits*t*2and the second endpoint**q**2to vary; these variables are theargumentsof the function*S*. In other words, the action function*S*is theindefinite integralof the Lagrangian with respect to time.Hamilton's characteristic function

When the total energy *E* is conserved, the Hamilton–Jacobi equation can be solved with the additive separation of variables:

where the time-independent function *W*(*q*1, *q*2, … *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.

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".^{[3]}

Action of a generalized coordinate

This is a single variable *Jk* 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 *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

See tautological one-form.

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.^{[1]}^{[7]}

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 as

`where the initial and final times (`

*t*1and*t*2) and the final and initial positions are specified in advance asand. Let*x*true(*t*) represent the true evolution that we seek, and letbe 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 partsof the last term, together with the boundary conditions, yields the equation`

`The requirement thatbestationaryimplies 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 quantityis 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.- if, thenis constant in time.

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 angular 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.

The action principle

Classical fields

The **action principle** can be extended to obtain the equations of motion for fields, such as the electromagnetic field or gravitational field.

The Einstein equation utilizes the *Einstein–Hilbert action* as constrained by a variational principle.

The trajectory (path in spacetime) of a body in a gravitational field can be found using the action principle. For a free falling body, this trajectory is a geodesic.

Conservation laws

Implications of symmetries in a physical situation can be found 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.^{[2]}

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 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. It is best understood within quantum mechanics, particularly in Richard Feynman's path integral formulation, where it arises out of destructive interference of quantum amplitudes.

Maxwell's equations can also be derived as conditions of stationary action.

Single relativistic particle

`When relativistic effects are significant, the action of a point particle of mass`

*m*travelling aworld line*C*parametrized by theproper timeisIf instead, the particle is parametrized by the coordinate time *t* of the particle and the coordinate time ranges from *t*1 to *t*2, then the action becomes

where the Lagrangian is^{[8]}

Modern extensions

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.^{[6]}

See also

Calculus of variations

Functional derivative

Functional integral

Lagrangian

Lagrangian mechanics

Measure (physics)

Path integral formulation

Principle of least action

Principle of maximum entropy

Some actions: Nambu–Goto action Polyakov action Bagger–Lambert–Gustavsson action Einstein–Hilbert action

## References

*Œuvres de Mr de Maupertuis*(pre-1801 Imprint Collection at the Library of Congress).

*The Classical Theory of Fields*Addison-Wesley 1971 Sec. 8. p. 24–25.