Editing Action (physics)

{{Short description|Physical quantity of dimension energy × time}}
{{About|a property of a trajectory|the central force concept|action at a distance}}
{{Infobox physical quantity
| name = Action
| image = 
| caption = 
| unit = [[joule-second]]
| otherunits = J&sdot;Hz{{superscript|−1}}
| symbols = ''S''
| baseunits = kg&sdot;m{{superscript|2}}&sdot;s{{superscript|−1}}
| dimension = <math>\mathsf{M} \cdot \mathsf{L}^{2} \cdot \mathsf{T}^{-1}</math>
| extensive = 
| conserved = 
| derivations =
}}

In [[physics]], '''action''' is a [[Scalar (physics)|scalar quantity]] that describes how the balance of kinetic versus potential energy of a [[physical system]] changes with trajectory. Action is significant because it is an input to the [[principle of stationary action]], an approach to classical mechanics that is simpler for multiple objects.<ref name="pubs.aip.org">{{Cite journal |last1=Neuenschwander |first1=Dwight E. |last2=Taylor |first2=Edwin F. |last3=Tuleja |first3=Slavomir |date=2006-03-01 |title=Action: Forcing Energy to Predict Motion |url=https://pubs.aip.org/pte/article/44/3/146/274422/Action-Forcing-Energy-to-Predict-Motion |journal=The Physics Teacher |language=en |volume=44 |issue=3 |pages=146–152 |doi=10.1119/1.2173320 |bibcode=2006PhTea..44..146N |issn=0031-921X}}</ref> Action and the variational principle are used in [[Path integral formulation|Feynman's formulation of quantum mechanics]]<ref>{{Cite journal |last1=Ogborn |first1=Jon |last2=Taylor |first2=Edwin F |date=2005-01-01 |title=Quantum physics explains Newtons laws of motion |url=https://www.eftaylor.com/pub/QMtoNewtonsLaws.pdf |journal=Physics Education |volume=40 |issue=1 |pages=26–34 |doi=10.1088/0031-9120/40/1/001 |bibcode=2005PhyEd..40...26O |s2cid=250809103 |issn=0031-9120}}</ref> and in general relativity.<ref>{{Cite journal |last=Taylor |first=Edwin F. |date=2003-05-01 |title=A call to action |url=https://pubs.aip.org/ajp/article/71/5/423/1044678/A-call-to-action |journal=American Journal of Physics |language=en |volume=71 |issue=5 |pages=423–425 |doi=10.1119/1.1555874 |bibcode=2003AmJPh..71..423T |issn=0002-9505}}</ref> For systems with small values of action close to the [[Planck constant]], quantum effects are significant.<ref name=FeynmanII/>

In the simple case of a single particle moving with a constant velocity (thereby undergoing [[uniform linear motion]]), the action is the [[momentum]] of the particle times the distance it moves, [[integral (mathematics)|added up]] along its path; equivalently, action is the difference between the particle's [[kinetic energy]] and its [[potential energy]], times the duration for which it has that amount of energy. 

More formally, 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">{{cite encyclopedia |last1=Goodman |first1=Bernard |title=Action |date=1993|encyclopedia=McGraw-Hill Encyclopaedia of Physics |publisher=McGraw-Hill |location=New York |editor=Parker, S. P.|isbn=0-07-051400-3|page=22 |edition=2nd |url=https://archive.org/details/mcgrawhillencycl1993park/page/22/mode/2up}}</ref> Action has [[dimensional analysis|dimensions]] of [[energy]]&nbsp;×&nbsp;[[time]] or [[momentum]]&nbsp;×&nbsp;[[length]], and its [[SI unit]] is [[joule]]-second (like the [[Planck constant]] ''h'').<ref>{{cite encyclopedia |last1=Stehle |first1=Philip M. |title=Least-action principle |date=1993|encyclopedia=McGraw-Hill Encyclopaedia of Physics |publisher=McGraw-Hill |location=New York |editor=Parker, S. P.|isbn=0-07-051400-3|page=670 |edition=2nd |url=https://archive.org/details/mcgrawhillencycl1993park/page/670/mode/2up}}</ref>

== Introduction ==
Introductory physics often begins with [[Newton's laws of motion]], relating force and motion; action is part of a completely equivalent alternative approach with practical and educational advantages.<ref name="pubs.aip.org"/> However, the concept took many decades to supplant Newtonian approaches and remains a challenge to introduce to students.<ref name=Fee1942>{{Cite journal |last=Fee |first=Jerome |date=1942 |title=Maupertuis and the Principle of Least Action |url=https://www.jstor.org/stable/27825934 |journal=American Scientist |volume=30 |issue=2 |pages=149–158 |jstor=27825934 |issn=0003-0996}}</ref>

=== Simple example ===
For a trajectory of a ball moving in the air on Earth the '''action''' is defined between two points in time, <math>t_1</math> and <math>t_2</math> as the kinetic energy (KE) minus the potential energy (PE), integrated over time.<ref name=FeynmanII>{{Cite web |title=The Feynman Lectures on Physics Vol. II Ch. 19: The Principle of Least Action |url=https://www.feynmanlectures.caltech.edu/II_19.html |access-date=2023-11-03 |website=www.feynmanlectures.caltech.edu}}</ref> 
:<math>S = \int_{t_1}^{t_2} \left( KE(t) - PE(t)\right) dt</math>
The action balances kinetic against potential energy.<ref name=FeynmanII/> 
The kinetic energy of a ball of mass <math>m</math> is <math>(1/2)mv^2</math> where <math>v</math> is the velocity of the ball; the potential energy is <math>mgx</math> where <math>g</math> is the acceleration due to gravity. Then the action between <math>t_1</math> and <math>t_2</math> is
:<math>S = \int_{t_1}^{t_2} \left(\frac{1}{2}m v^2(t) - mg x(t) \right) dt</math>
The action value depends upon the trajectory taken by the ball through <math>x(t)</math> and <math>v(t)</math>. This makes the action an input to the powerful [[stationary-action principle]] for [[classical mechanics|classical]] and for [[quantum mechanics]]. Newton's equations of motion  for the ball can be derived from the action using the stationary-action principle, but the advantages of action-based mechanics only begin to appear in cases where Newton's laws are difficult to apply. Replace the ball with an electron: classical mechanics fails but stationary action continues to work.<ref name=FeynmanII/> The energy difference in the simple action definition, kinetic minus potential energy, is generalized and called [[Lagrangian (physics)#The Lagrangian|the Lagrangian]] for more complex cases.

=== Planck's quantum of action ===
The [[Planck constant]], written as <math>h</math> is the quantum of action.<ref>{{Cite web |title=Max Planck Nobel Lecture |url=https://www.nobelprize.org/prizes/physics/1918/planck/lecture/ |url-status=live |archive-url=https://web.archive.org/web/20230714164215/https://www.nobelprize.org/prizes/physics/1918/planck/lecture/ |archive-date=2023-07-14 |access-date=2023-07-14}}</ref> The quantum of [[angular momentum]] is <math>\hbar = \frac{h}{2\pi}</math>. These constants have units of energy times time. They appear in all significant quantum equations, like the [[uncertainty principle]] and the [[de Broglie wavelength]]. Whenever the value of the action approaches the Planck constant, quantum effects are significant.<ref name=FeynmanII/>

== History ==
{{Main | History of variational principles in physics}}
[[Pierre Louis Maupertuis]] and [[Leonhard Euler]] working in the 1740s developed early versions of the action principle. [[Joseph Louis Lagrange]] clarified the mathematics when he invented the [[calculus of variations]]. [[William Rowan Hamilton]] made the next big breakthrough, formulating Hamilton's principle in 1853.<ref name=Kline1972>{{cite book|last=Kline|first=Morris|title=Mathematical Thought from Ancient to Modern Times|url=https://archive.org/details/mathematicalthou0000unse|url-access=registration|publisher=Oxford University Press|location=New York|year=1972|pages= [https://archive.org/details/mathematicalthou0000unse/page/167 167]–168|isbn=0-19-501496-0}}</ref>{{rp|740}}  Hamilton's principle became the cornerstone for classical work with different forms of action until [[Richard Feynman]] and [[Julian Schwinger]] developed quantum action principles.<ref>{{Cite book |last1=Yourgrau |first1=Wolfgang |title=Variational principles in dynamics and quantum theory |last2=Mandelstam |first2=Stanley |date=1979 |publisher=Dover Publ |isbn=978-0-486-63773-0 |edition=Republ. of the 3rd ed., publ. in 1968 |series=Dover books on physics and chemistry |location=New York, NY}}</ref>{{rp|127}}

== Definitions ==
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. 
Action has the [[dimensional analysis|dimensions]] of [[energy|[energy]]]&nbsp;×&nbsp;[[time|[time]]], and its [[SI unit]] is [[joule]]-second, which is identical to the unit of [[angular momentum]].

Several different definitions of "the action" are in common use in physics.<ref name="handfinch">Analytical Mechanics, L.N. Hand, J.D. Finch, Cambridge University Press, 2008, {{ISBN|978-0-521-57572-0}}</ref><ref>Encyclopaedia of Physics (2nd Edition), [[Rita G. Lerner|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 display="block">\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 (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">T. W. B. Kibble, ''Classical Mechanics'', 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 display="block">
\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 -->{{anchor| Symplectic action}}{{anchor| abbreviated action}}
In addition to the action functional, there is another functional called the ''abbreviated action''. In the abbreviated action, 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> (sometime written as <math>W</math>) is defined as the integral of the generalized momenta,
<math display="block">p_i = \frac{\partial L(q,t)}{\partial \dot{q}_i},</math> 
for a system Lagrangian <math>L</math> along a path in the [[generalized coordinates]] <math>q_i</math>:
<math display="block">
\mathcal{S}_0 = \int_{q_1}^{q_2} \mathbf{p} \cdot d\mathbf{q} = \int_{q_1}^{q_2} \Sigma_i p_i \,dq_i.
</math>
where <math>q_1</math> and <math>q_2</math> are the starting and ending coordinates.
According to [[Maupertuis's principle]], the true path of the system is a path for which the abbreviated action is [[stationary point|stationary]].

=== 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]]:<ref name="handfinch" />{{rp|225}}
<math display="block">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 display="block">\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 display="block">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]].<ref name=Goldestein3/>{{rp|434}}

=== Action of a generalized coordinate ===
A variable ''J<sub>k</sub>'' in the [[action-angle coordinates]], called the "action" of the generalized coordinate ''q<sub>k</sub>'', is defined by integrating a single generalized momentum around a closed path in [[phase space]], corresponding to rotating or oscillating motion:<ref name=Goldestein3>{{Cite book |last1=Goldstein |first1=Herbert |title=Classical mechanics |last2=Poole |first2=Charles P. |last3=Safko |first3=John L. |date=2008 |publisher=Addison Wesley |isbn=978-0-201-65702-9 |edition=3, [Nachdr.] |location=San Francisco Munich}}</ref>{{rp|454}}

<math display="block">
J_k = \oint p_k \,dq_k
</math>

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. For example, they are used in the calculation of planetary and satellite orbits.<ref name=Goldestein3/>{{rp|477}}

=== Single relativistic particle ===
{{Main|Relativistic Lagrangian mechanics|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 display="block">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 display="block">S = \int_{t1}^{t2} L \, dt,</math>
where the [[Lagrangian mechanics|Lagrangian]] is<ref>L. D. Landau and E. M. Lifshitz (1971). ''The Classical Theory of Fields''. Addison-Wesley. Sec.&nbsp;8. p.&nbsp;24–25.</ref>
<math display="block">L = -mc^2 \sqrt{1 - \frac{v^2}{c^2}}.</math>

== Action principles and related ideas ==
{{Main|Principle of stationary action}}

Physical 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)|functions]] that describe the behavior of the system and are called ''[[equations of motion]]''.

''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.

The action principle provides deep insights into physics, and is an important concept in modern [[theoretical physics]]. Various action principles and related concepts are summarized below.

=== Maupertuis's principle ===
{{Main | Maupertuis's principle}}
In classical mechanics, [[Maupertuis's principle]] (named after Pierre Louis Maupertuis) states that the path followed by a physical system is the one of least length (with a suitable interpretation of path and length). Maupertuis's principle uses the [[#abbreviated action|abbreviated action]] between two generalized points on a path.

=== Hamilton's principal function ===
{{Main|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.

Hamilton's principle 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 explores all possible paths, with the phase of the probability amplitude for each path being determined by the action for the path; the final probability amplitude adds all paths using their complex amplitude and phase.<ref name="abers1">Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, {{ISBN|978-0-13-146100-0}}</ref>

=== Hamilton–Jacobi equation ===
{{Main|Hamilton–Jacobi equation}}
Hamilton's principal function <math>S=S(q,t;q_0,t_0)</math> is obtained from the action functional <math>\mathcal{S}</math> by fixing the initial time <math>t_0</math> and the initial endpoint <math>q_0,</math> while allowing the upper time limit <math>t</math> and the second endpoint <math>q</math> to vary. The Hamilton's principal function satisfies the Hamilton–Jacobi equation, a formulation of [[classical mechanics]]. Due to a similarity with the [[Schrödinger equation]], the Hamilton–Jacobi equation provides, arguably, the most direct link with [[quantum mechanics]].

=== Euler–Lagrange equations ===
{{Main|Euler–Lagrange equations}}

In Lagrangian mechanics, the requirement that the action integral be [[stationary point|stationary]] under small perturbations is equivalent to a set of [[differential equation]]s (called the Euler–Lagrange equations) that may be obtained using the [[calculus of variations]].

=== 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]]. 
[[Maxwell's equations]] can [[Electromagnetic tensor#Lagrangian formulation of classical electromagnetism|be derived as conditions of stationary action]].

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|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" />

=== Path integral formulation of quantum field theory ===
{{Main|Path integral formulation}}

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]], which 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.

=== 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 integration]]
* [[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}}

== Further reading ==
* ''The Cambridge Handbook of Physics Formulas'', G. Woan, Cambridge University Press, 2010, {{ISBN|978-0-521-57507-2}}.
* 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.

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

{{Classical mechanics derived SI units}}
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[[Category:Action (physics)| ]]
[[Category:Lagrangian mechanics]]
[[Category:Hamiltonian mechanics]]
[[Category:Calculus of variations]]
[[Category:Dynamics (mechanics)]]