Editing Action (physics) (section)

== 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>