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{{Distinguish|Action at a distance}}

In [[physics]], '''action''' is an attribute of the [[dynamics (physics)|dynamics]] of a [[physical system]] from which the [[equations of motion]] of the system can be derived through the [[principle of stationary action]]. Action is a [[functional (mathematics)|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.<ref name="mcgraw1">McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, {{ISBN|0-07-051400-3}}</ref> Action has the [[dimensional analysis|dimensions]] of [[energy|[energy]]]⋅[[time|[time]]] or [[momentum|[momentum]]]⋅[[length|[length]]], and its [[SI unit]] is [[joule]]-second.

== Introduction ==
{{main article|Hamilton's principle}}

[[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 field]]s such as the [[electromagnetism|electromagnetic]] and [[gravity|gravitational]] [[field (physics)|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.<ref name="abers1">Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, {{ISBN|978-0-13-146100-0}}</ref>

===Solution of differential equation===
Empirical laws are frequently expressed as [[differential equation]]s, which describe how physical quantities such as [[position vector|position]] and [[momentum]] change [[continuous function|continuously]] with [[Time in physics|time]], [[space]] or a generalization thereof. Given the [[Initial value problem|initial]] and [[Boundary value problem|boundary]] conditions for the situation, the "solution" to these empirical equations is one or more [[function (mathematics)|function]]s 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 point|stationary]]. In other words, the action satisfies a [[Calculus of variations|variational]] principle: the [[principle of stationary action]] (see also below). The action is defined by an [[integral (calculus)|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.<ref name="handfinch">Analytical Mechanics, L.N. Hand, J.D. Finch, Cambridge University Press, 2008, {{ISBN|978-0-521-57572-0}}</ref>
* [[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 [[s:Translation:Derivation of the laws of motion and equilibrium from a metaphysical principle|article]], defining action as potential energy, as virtual kinetic energy, and as a hybrid that ensured conservation of momentum in collisions.<ref>''Œuvres de Mr de Maupertuis'' (pre-1801 Imprint Collection at the [[Library of Congress]]).</ref>

== Mathematical definition ==

Expressed in mathematical language, using the [[calculus of variations]], the [[time evolution|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.<ref name="handfinch" /><ref>Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, {{ISBN|3-527-26954-1}} (Verlagsgesellschaft), {{ISBN|0-89573-752-3}} (VHC Inc.)</ref> The action is usually an [[integral]] over time. However, when the action pertains to [[field (physics)|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:<ref name="handfinch" />
:<math>\mathcal{S} = \int_{t_1}^{t_2} L \, dt,</math>
where the integrand ''L'' is called the [[Lagrangian mechanics|Lagrangian]]. For the action integral to be well-defined, the trajectory has to be bounded in time and space.

Action has the [[dimensional analysis|dimensions]] of [[energy|[energy]]]⋅[[time|[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 a [[functional (mathematics)|functional]] <math>\mathcal{S}</math> which takes a [[function (mathematics)|function]] of time and (for [[field (physics)|fields]]) space as input and returns a [[scalar (physics)|scalar]].<ref name="penrose">The Road to Reality, Roger Penrose, Vintage books, 2007, {{ISBN|0-679-77631-1}}</ref><ref name="kibble">Classical Mechanics, T.W.B. Kibble, European Physics Series, McGraw-Hill (UK), 1973, {{ISBN|0-07-084018-0}}</ref> In [[classical mechanics]], the input function is the evolution '''q'''(''t'') of the system between two times ''t''<sub>1</sub> and ''t''<sub>2</sub>, where '''q''' represents the [[generalized coordinate]]s. The action <math>\mathcal{S}[\mathbf{q}(t)]</math> is defined as the [[integral]] of the [[Lagrangian mechanics|Lagrangian]] ''L'' for an input evolution between the two times:

:<math>
\mathcal{S}[\mathbf{q}(t)] = \int_{t_1}^{t_2} L[\mathbf{q}(t),\dot{\mathbf{q}}(t),t]\, dt,
</math>

where the endpoints of the evolution are fixed and defined as <math>\mathbf{q}_{1} = \mathbf{q}(t_{1})</math> and <math>\mathbf{q}_{2} = \mathbf{q}(t_{2})</math>. According to [[Hamilton's principle]], the true evolution '''q'''<sub>true</sub>(''t'') is an evolution for which the action <math>\mathcal{S}[\mathbf{q}(t)]</math> is [[stationary point|stationary]] (a minimum, maximum, or a [[saddle point]]). This principle results in the equations of motion in [[Lagrangian mechanics]].

===Abbreviated action (functional)=== <!-- [[Symplectic action]] redirects here -->
Usually denoted as <math>\mathcal{S}_{0}</math>, this is also a [[functional (mathematics)|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 <math>\mathcal{S}_{0}</math> is defined as the integral of the generalized momenta along a path in the [[generalized coordinates]]:

:<math>
\mathcal{S}_0 = \int \mathbf{p} \cdot d\mathbf{q} = \int p_i \,dq_i.
</math>

According to [[Maupertuis' principle]], the true path is a path for which the abbreviated action <math>\mathcal{S}_{0}</math> is [[stationary point|stationary]].

=== Hamilton's principal function ===
{{Main article|Hamilton–Jacobi equation}}
Hamilton's principal function is defined by the [[Hamilton–Jacobi equation]]s (HJE), another alternative formulation of [[classical mechanics]].  This function ''S'' is related to the functional <math>\mathcal{S}</math> by fixing the initial time ''t''<sub>1</sub> and the initial endpoint '''q'''<sub>1</sub> and allowing the upper limits ''t''<sub>2</sub> and the second endpoint '''q'''<sub>2</sub> to vary; these variables are the [[independent variable|arguments]] of the function ''S''.  In other words, the action function ''S'' is the [[antiderivative|indefinite integral]] of 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 [[Hamilton–Jacobi equation#Separation of variables|additive separation of variables]]:

:<math>S(q_1, \dots, q_N, t) = W(q_1, \dots, q_N) - E \cdot t,</math>

where the time-independent function ''W''(''q''<sub>1</sub>, ''q''<sub>2</sub>, … ''q<sub>N</sub>'') is called ''Hamilton's characteristic function''. The physical significance of this function is understood by taking its total time derivative

:<math>\frac{d W}{d t} = \frac{\partial W}{\partial q_i} \dot q_i = p_i \dot q_i.</math>

This can be integrated to give

:<math>W(q_1, \dots, q_N) = \int p_i\dot q_i \,dt = \int p_i \,dq_i,</math>

which is just the [[#Abbreviated action (functional)|abbreviated action]].

=== Other solutions of Hamilton–Jacobi equations ===
The [[Hamilton–Jacobi equation]]s are often solved by additive separability; in some cases, the individual terms of the solution, e.g., ''S<sub>k</sub>''(''q<sub>k</sub>''), are also called an "action".<ref name="handfinch" />

=== Action of a generalized coordinate ===
This is a single variable ''J<sub>k</sub>'' 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:

:<math>
J_k = \oint p_k \,dq_k
</math>

The variable ''J<sub>k</sub>'' is called the "action" of the generalized coordinate ''q<sub>k</sub>''; the corresponding canonical variable conjugate to ''J<sub>k</sub>'' is its "angle" ''w<sub>k</sub>'', for reasons described more fully under [[action-angle coordinates]].  The integration is only over a single variable ''q<sub>k</sub>'' and, therefore, unlike the integrated dot product in the abbreviated action integral above.  The ''J<sub>k</sub>'' variable equals the change in ''S<sub>k</sub>''(''q<sub>k</sub>'') as ''q<sub>k</sub>'' is varied around the closed path.  For several physical systems of interest, J<sub>k</sub> is either a constant or varies very slowly; hence, the variable ''J<sub>k</sub>'' is often used in perturbation calculations and in determining [[adiabatic invariant]]s.

=== 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 point|stationary]] under small perturbations of the evolution is equivalent to a set of [[differential equation]]s (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.<ref name="mcgraw1" /><ref name="kibble" />

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

:<math>
\mathcal{S} = \int_{t_1}^{t_2} L(x, \dot{x}, t) \,dt,
</math>

where the initial and final times (''t''<sub>1</sub> and ''t''<sub>2</sub>) and the final and initial positions are specified in advance as <math>x_1 = x(t_1)</math> and <math>x_2 = x(t_2)</math>.  Let ''x''<sub>true</sub>(''t'') represent the true evolution that we seek, and let <math>x_\text{per}(t)</math> be a slightly perturbed version of it, albeit with the same endpoints, <math>x_\text{per}(t_1) = x_1</math> and <math>x_\text{per}(t_2) = x_2</math>.  The difference between these two evolutions, which we will call <math>\varepsilon(t)</math>, is infinitesimally small at all times:

:<math>
\varepsilon(t) = x_\text{per}(t) - x_\text{true}(t).
</math>

At the endpoints, the difference vanishes, i.e., <math>\varepsilon(t_1) = \varepsilon(t_2) = 0</math>.

Expanded to first order, the difference between the actions integrals for the two evolutions is

:<math>\begin{align}
\delta \mathcal{S} &= \int_{t_1}^{t_2}
 \left[ L(x_\text{true} + \varepsilon, \dot x_\text{true} + \dot\varepsilon, t) -
 L(x_\text{true}, \dot x_\text{true}, t) \right] \,dt \\
&= \int_{t_1}^{t_2}
 \left(\varepsilon\frac{\partial L}{\partial x} +
 \dot\varepsilon\frac{\partial L}{\partial \dot x} \right)\,dt.
\end{align}</math>

[[Integration by parts]] of the last term, together with the boundary conditions <math>\varepsilon(t_1) = \varepsilon(t_2) = 0</math>, yields the equation

:<math>
\delta \mathcal{S} = \int_{t_1}^{t_2}
 \left(
  \varepsilon\frac{\partial L}{\partial x} -
  \varepsilon\frac{d}{dt}\frac{\partial L}{\partial \dot x}
 \right)\,dt.
</math>

The requirement that <math>\mathcal{S}</math> be [[stationary point|stationary]] implies that the first-order change must be zero for ''any'' possible perturbation ε(''t'') about the true evolution:

{{Equation box 1
|indent =:
|title= '''[[Principle of stationary action]]'''
|equation = <math>\delta\mathcal{S}=0</math>
|cellpadding
|border=1
|border colour = black
|background colour = white}}

This can be true only if

{{Equation box 1
|indent =:
|title= '''[[Euler–Lagrange equation]]'''
|equation = <math> {\partial L\over\partial x} - {d\over dt }{\partial L\over\partial \dot{x}} = 0 </math>
|cellpadding
|border=1
|border colour = black
|background colour = white}}

The Euler–Lagrange equation is obeyed provided the [[functional derivative]] of the action integral is identically zero:

:<math>\frac{\delta \mathcal{S}}{\delta x(t)} = 0.</math>

The quantity <math>\frac{\partial L}{\partial\dot x}</math> 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.

:if <math>\frac{\partial L}{\partial x} = 0</math>, then <math>\frac{\partial L}{\partial\dot x}</math> is 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
:<math>L = \frac{1}{2} mv^2 = \frac{1}{2}m \left( \dot{x}^2 + \dot{y}^2 \right)</math>
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

:<math>
L = \frac{1}{2}m \left( \dot{r}^2 + r^2 \dot\varphi^2 \right).
</math>

The radial ''r'' and angular φ components of the Euler–Lagrangian equations become respectively

:<math>\begin{align}
\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{r}} \right) - \frac{\partial L}{\partial r} &= 0
 & &\Rightarrow &
 \ddot{r} - r\dot{\varphi}^2 &= 0, \\
\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\varphi}} \right) - \frac{\partial L}{\partial \varphi} &= 0
 & &\Rightarrow &
\ddot{\varphi} + \frac{2}{r}\dot{r}\dot{\varphi} &= 0.
\end{align}</math>

The solution of these two equations is given by

:<math>\begin{align}
r\cos\varphi &= a t + b, \\
r\sin\varphi &= c t + d
\end{align}</math>

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 ==

{{Main article|principle of stationary action}}

=== Classical fields ===

{{See also|Einstein–Hilbert action}}

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 ===

{{Main article|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 (physics)|conservation law]] (and conversely). This deep connection requires that the action principle be assumed.<ref name="abers1" />

=== Quantum mechanics and quantum field theory ===

{{Main article|Quantum mechanics|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 formulation|path integral]], that gives the [[probability amplitude]]s 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 [[Electromagnetic tensor#Lagrangian formulation of classical electromagnetism|be derived as conditions of stationary action]].

=== Single relativistic particle ===

{{Main article|Theory of relativity}}

When relativistic effects are significant, the action of a point particle of mass ''m'' travelling a [[world line]] ''C'' parametrized by the [[proper time]] <math>\tau</math> is
:<math>S = - m c^2 \int_{C} \, d \tau. </math>

If instead, the particle is parametrized by the coordinate time ''t'' of the particle and the coordinate time ranges from ''t''<sub>1</sub> to ''t''<sub>2</sub>, then the action becomes
:<math>\int_{t1}^{t2} L \, dt,</math>

where the [[Lagrangian mechanics|Lagrangian]] is<ref>L. D. Landau and E. M. Lifshitz ''The Classical Theory of Fields'' Addison-Wesley 1971 Sec.&nbsp;8. p.&nbsp;24–25.</ref>
:<math>L = -mc^2 \sqrt{1 - \frac{v^2}{c^2}}.</math>

=== Modern extensions ===

The action principle can be generalized still further.  For example, the action need not be an integral, because [[Nonlocal Lagrangian|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.<ref name="penrose" />

== See also ==
{{Div col}}
* [[Calculus of variations]]
* [[Functional derivative]]
* [[Functional integral]]
* [[Hamiltonian mechanics]]
* [[Lagrangian (field theory)|Lagrangian]]
* [[Lagrangian mechanics]]
* [[Measure (physics)]]
* [[Noether's theorem]]
* [[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]]
{{Div col end}}

== References ==

{{reflist}}

== Sources and further reading ==

For an annotated bibliography, see Edwin F. Taylor who [http://www.eftaylor.com/pub/BibliogLeastAction12.pdf lists], among other things, the following books
* ''The Cambridge Handbook of Physics Formulas'', G. Woan, Cambridge University Press, 2010, {{ISBN|978-0-521-57507-2}}.
* [[Cornelius Lanczos]], [https://books.google.com/books?id=cmPDAgAAQBAJ The Variational Principles of Mechanics] (Dover Publications, New York, 1986). {{ISBN|0-486-65067-7}}. ''The'' reference most quoted by all those who explore this field.
* [[L. D. Landau]] and [[E. M. Lifshitz]], Mechanics, [[Course of Theoretical Physics]] (Butterworth-Heinenann, 1976), 3rd ed., Vol. 1. {{ISBN|0-7506-2896-0}}. Begins with the principle of least action.
* Thomas A. Moore "Least-Action Principle" in Macmillan Encyclopedia of Physics (Simon & Schuster Macmillan, 1996), Volume 2, {{ISBN|0-02-897359-3}},  {{OCLC|35269891}}, pages 840 – 842.
* [[Gerald Jay Sussman]] and [[Jack Wisdom]], [http://groups.csail.mit.edu/mac/users/gjs/6946/sicm-html/ Structure and Interpretation of Classical Mechanics] (MIT Press, 2001). Begins with the principle of least action, uses modern mathematical notation, and checks the clarity and consistency of procedures by programming them in computer language.
* Dare A. Wells, Lagrangian Dynamics, Schaum's Outline Series (McGraw-Hill, 1967) {{ISBN|0-07-069258-0}}, A 350-page comprehensive "outline" of the subject.
* Robert Weinstock, Calculus of Variations, with Applications to Physics and Engineering (Dover Publications, 1974). {{ISBN|0-486-63069-2}}.  An oldie but goodie, with the formalism carefully defined before use in physics and engineering.
* [[Wolfgang Yourgrau]] and [[Stanley Mandelstam]], [https://books.google.com/books/about/Variational_Principles_in_Dynamics_and_Q.html?id=OwTyrJJXZbYC Variational Principles in Dynamics and Quantum Theory] (Dover Publications, 1979). A nice treatment that does not avoid the philosophical implications of the theory and lauds the Feynman treatment of quantum mechanics that reduces to the principle of least action in the limit of large mass.
* Edwin F. Taylor's [http://www.eftaylor.com/leastaction.html page]

== External links ==
* [http://www.eftaylor.com/software/ActionApplets/LeastAction.html Principle of least action interactive] Interactive explanation/webpage

{{Classical mechanics derived SI units}}

[[Category:Lagrangian mechanics]]
[[Category:Hamiltonian mechanics]]
[[Category:Calculus of variations]]
[[Category:Dynamics (mechanics)]]