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| {{About|the Lagrangian function in [[Lagrangian mechanics]]}}
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| The '''Lagrangian''', ''L'', of a [[dynamical system]] is a function that summarizes the dynamics of the system. It is named after [[Giuseppe Lodovico Lagrangia]], later known as [[Joseph Louis Lagrange]]. The concept of a Lagrangian was introduced in a reformulation of [[classical mechanics]] introduced by [[Giuseppe Lodovico Lagrangia | Lagrangia]], known as [[Lagrangian mechanics]].
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| ==Definition==
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| In classical mechanics, the natural form of the Lagrangian is defined as the [[kinetic energy]], ''T'', of the system minus its [[potential energy]], ''V''.<ref name="Torby1984">{{cite book |last=Torby |first=Bruce |title=Advanced Dynamics for Engineers |series=HRW Series in Mechanical Engineering |year=1984 |publisher=CBS College Publishing |location=United States of America |isbn=0-03-063366-4 |chapter=Energy Methods}}</ref> In symbols,
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| :<math>L = T - V.</math>
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| If the Lagrangian of a system is known, then the [[equation of motion|equations of motion]] of the system may be obtained by a direct substitution of the expression for the Lagrangian into the [[Euler–Lagrange equation]]. The Lagrangian of a given system is not unique, and two Lagrangians describing the same system can differ by the total derivative with respect to time of some function <math>f(q,t)</math>, but solving any equivalent Lagrangians will give the same equations of motion.<ref name="Goldstein">{{cite book | title=Classical Mechanics | last1=Goldstein | first1=Herbert |authorlink1=Herbert Goldstein |last2=Poole | first2=Charles P. | last3=Safko | first3=John L. |edition=3rd |publisher=Addison-Wesley | year=2002 | isbn=9780201657029 | page=21}}</ref><ref>{{cite book|last=Bell|first=L.D. Landau and E.M. Lifshitz ; translated from the Russian by J.B. Sykes and J.S.|title=Mechanics|year=1999|publisher=Butterworth-Heinemann|location=Oxford|isbn=9780750628969|page=4|edition=3rd ed.}}</ref>
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| == The Lagrangian formulation ==
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| === Simple example ===
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| The trajectory of a thrown ball is characterized by the sum of the Lagrangian values at each time being a (local) minimum.
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| The Lagrangian ''L'' can be calculated at several instants of time ''t'', and a graph of ''L'' against ''t'' can be drawn. The area under the curve is the [[action (physics)|action]]. Any different path between the initial and final positions leads to a larger action than that chosen by nature. Nature chooses the smallest action – this is the [[Principle of Least Action]].
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| {{cquote|If Nature has defined the mechanics problem of the thrown ball in so elegant a fashion, might She<!--sic--> have defined other problems similarly. So it seems now. Indeed, at the present time it appears that we can describe all the fundamental forces in terms of a Lagrangian. The search for Nature's One Equation, which rules all of the universe, has been largely a search for an adequate Lagrangian.|20px|20px|<ref>The Great Design: Particles, Fields, and Creation (New York: Oxford University Press, 1989), ROBERT K. ADAIR, p.22–24</ref>}}
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| Using only the principle of least action and the Lagrangian we can deduce the correct trajectory, by trial and error or the [[calculus of variations]].
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| === Importance ===
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| The Lagrangian formulation of mechanics is important not just for its broad applications, but also for its role in advancing deep understanding of [[physics]]. Although Lagrange only sought to describe [[classical mechanics]], the ''[[action principle]]'' that is used to derive the Lagrange equation was later recognized to be applicable to [[quantum mechanics]] as well.
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| Physical [[action (physics)|action]] and quantum-mechanical [[phase (waves)|phase]] are related via [[Planck's constant]], and the [[principle of stationary action]] can be understood in terms of [[constructive interference]] of [[wave function]]s.
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| The same principle, and the Lagrangian formalism, are tied closely to [[Noether's theorem]], which connects physical [[conserved quantity|conserved quantities]] to continuous [[symmetry|symmetries]] of a physical system.
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| Lagrangian mechanics and [[Noether's theorem]] together yield a natural formalism for [[first quantization]] by including [[commutators]] between certain terms of the Lagrangian equations of motion for a physical system.
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| === Advantages over other methods ===
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| *The formulation is not tied to any one coordinate system – rather, any convenient variables may be used to describe the system; these variables are called "[[generalized coordinates]]" ''q<sub>i</sub>'' and may be any quantitative attributes of the system (for example, strength of the [[magnetic field]] at a particular location; [[angle]] of a pulley; position of a particle in space; or degree of excitation of a particular [[eigenmode]] in a complex system) which are functions of the [[independent variable]](s). This trait makes it easy to incorporate constraints into a theory by defining coordinates that only describe states of the system that satisfy the constraints.
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| *If the Lagrangian is invariant under a symmetry, then the resulting equations of motion are also invariant under that symmetry. This characteristic is very helpful in showing that theories are consistent with either [[special relativity]] or [[general relativity]].
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| ===Cyclic coordinates and conservation laws===
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| <!-- [[Cyclic coordinate]] and [[Cyclic coordinates]] redirect here. Correct them accordingly if you modify the title of this section. -->
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| An important property of the Lagrangian is that conservation laws can easily be read off from it. For example, if the Lagrangian <math>\scriptstyle L</math> does ''not'' depend on <math>\scriptstyle q_i</math> itself, then the ''generalized momentum'' (<math>\scriptstyle p_i</math>), given by:
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| :<math>p_i =\frac{\partial L}{\partial\dot q_i},</math>
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| is a ''conserved'' quantity, because of [[Lagrange's equations]]:
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| :<math>\dot{p}_i =\frac{\partial L}{\partial q_i}=0.</math>
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| It doesn't matter if <math>\scriptstyle L</math> depends on the ''[[time derivative]]'' <math>\scriptstyle \dot q_i</math> of that generalized coordinate, since the Lagrangian independence of the coordinate always makes the above partial derivative zero. This is a special case of [[Noether's theorem]]. Such coordinates are called "cyclic" or "ignorable".
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| For example, the conservation of the generalized momentum,
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| : <math>p_2 =\frac{\partial L}{\partial\dot q_2},</math>
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| say, can be directly seen if the Lagrangian of the system is of the form
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| :<math>L(q_1,q_3,q_4, \dots; \dot q_1,\dot q_2,\dot q_3,\dot q_4, \dots;t)\,.</math>
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| Also, if the time ''t'', does not appear in ''L'', then the [[Hamiltonian mechanics|Hamiltonian]] is conserved. This is the energy conservation unless the potential energy depends on velocity, as in [[electrodynamics]].<ref>Classical Mechanics, T.W.B. Kibble, European Physics Series, McGraw-Hill (UK), 1973, ISBN 07-084018-0</ref><ref>Analytical Mechanics, L.N. Hand, J.D. Finch, Cambridge University Press, 2008, ISBN 978-0-521-57572-0</ref>
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| == Explanation ==
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| The Lagrangian in many classical systems is a function of generalized coordinates ''q''<sub>''i''</sub> and their velocities d''q''<sub>''i''</sub>/d''t''. These coordinates (and velocities) are, in their turn, parametric functions of time. In the classical view, time is an independent variable and ''q''<sub>''i''</sub> (and d''q''<sub>''i''</sub>/d''t'') are dependent variables as is often seen in [[phase space]] explanations of systems. This formalism was generalized further to handle [[classical field theory|field theory]]. In field theory, the independent variable is replaced by an event in [[spacetime]] (''x'', ''y'', ''z'', ''t''), or more generally still by a point ''s'' on a manifold. The dependent variables (''q'') are replaced by the value of a field at that point in spacetime ''φ(x,y,z,t)'' so that the [[equation of motion|equations of motion]] are obtained by means of an [[action (physics)|action]] principle, written as:
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| :<math>\frac{\delta \mathcal{S}}{\delta \varphi_i} = 0,\,</math>
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| where the ''action'', <math>\scriptstyle\mathcal{S}</math>, is a [[functional (mathematics)|functional]] of the dependent variables φ<sub>''i''</sub>(''s'') with their derivatives and ''s'' itself
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| :<math>\mathcal{S}\left[\varphi_i, \frac{\partial \varphi_i} {\partial s}\right] = \int{ \mathcal{L} \left[\varphi_i [s], \frac{\partial \varphi_i [s]}{\partial s^\alpha}, s^\alpha\right] \, \mathrm{d}^n s }</math>
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| and where ''s'' = { ''s<sup>α</sup>''} denotes the [[Set (mathematics)|set]] of ''n'' [[independent variable]]s of the system, indexed by ''α'' = 1, 2, 3,..., ''n''. Notice ''L'' is used in the case of one independent variable (''t'') and <math>\scriptstyle\mathcal{L} \,</math> is used in the case of multiple independent variables (usually four: ''x, y, z, t'').
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| The equations of motion obtained from this [[functional derivative]] are the [[Euler–Lagrange equations]] of this action. For example, in the [[classical mechanics]] of particles, the only independent variable is time, ''t''. So the Euler–Lagrange equations are
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| :<math>\frac{\mathrm{d}}{\mathrm{d} t} \frac{\partial L}{\partial\dot \varphi_i} = \frac{\partial L}{\partial\varphi_i} \,.</math>
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| Dynamical systems whose equations of motion are obtainable by means of an action principle on a suitably chosen Lagrangian are known as ''Lagrangian dynamical systems''. Examples of Lagrangian dynamical systems range from the classical version of the [[Standard Model]], to [[Newton's laws|Newton's equations]], to purely mathematical problems such as [[geodesic]] equations and [[Plateau's problem]].
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| ==An example from classical mechanics==
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| ===In Cartesian coordinates===
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| Suppose we have a [[three-dimensional space]] in which a particle of mass ''m'' moves under the influence of a [[conservative force]] <math>\vec{F}</math>. Since the force is conservative, it corresponds to a potential energy function <math>V(\vec{x})</math> given by <math>\vec{F} = -\nabla V(\vec{x})</math>. The Lagrangian of the particle can be written
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| :<math>L(\vec{x}, \dot{\vec{x}}) = \frac{1}{2} m \dot{\vec{x}}^2 - V(\vec{x}).</math>
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| The equations of motion for the particle are found by applying the [[Euler–Lagrange equation]]
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| :<math>\frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{\partial L}{\partial \dot{x}_i} \right) - \frac{\partial L}{\partial x_i} = 0,</math>
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| where ''i'' = 1, 2, 3.
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| Then
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| :<math>\frac{\partial L}{\partial x_i} = - \frac{\partial V}{\partial x_i},</math>
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| :<math>\frac{\partial L}{\partial \dot{x}_i} = \frac{\partial ~}{\partial \dot{x}_i} \left( \frac{1}{2} m \dot{\vec{x}}^2 \right) = \frac{1}{2} m \frac{\partial ~}{\partial \dot{x}_i} \left( \dot{x}_i\dot{x}_i \right) = m \dot{x}_i,
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| </math>
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| and
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| :<math>\frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{\partial L}{\partial \dot{x}_i} \right) = m \ddot{x}_i.</math>
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| Thus
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| :<math>m\ddot{\vec{x}}+\nabla V=0,</math>
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| which is [[Newton's second law of motion]] for a particle subject to a conservative force. Here the time derivative is written conventionally as a dot above the quantity being differentiated, and ∇ is the [[del| del operator]].
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| ===In spherical coordinates===
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| Suppose we have a three-dimensional space using [[spherical coordinates]] (''r, θ, φ'') with the Lagrangian
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| :<math>L = \frac{m}{2}(\dot{r}^2+r^2\dot{\theta}^2 +r^2\sin^2\theta \, \dot{\varphi}^2)-V(r).</math>
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| Then the Euler–Lagrange equations are:
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| :<math>m\ddot{r}-mr(\dot{\theta}^2+\sin^2\theta \, \dot{\varphi}^2)+V' =0,</math>
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| :<math>\frac{\mathrm{d}}{\mathrm{d}t}(mr^2\dot{\theta}) -mr^2\sin\theta\cos\theta \, \dot{\varphi}^2=0,</math>
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| :<math>\frac{\mathrm{d}}{\mathrm{d}t}(mr^2\sin^2\theta \, \dot{\varphi})=0.</math>
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| Here the set of parameters ''s<sub>i</sub>'' is just the time ''t'', and the dynamical variables ''ϕ<sub>i</sub>''(''s'') are the trajectories <math>\scriptstyle\vec x(t)</math> of the particle.
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| Despite the use of standard variables such as ''x'', the Lagrangian allows the use of any coordinates, which do not need to be [[Orthogonal coordinates|orthogonal]]. These are "[[generalized coordinates]]".
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| ==Lagrangian of a test particle==
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| A test particle is a particle whose [[mass]] and [[electric charge|charge]] are assumed to be so small that its effect on external system is insignificant. It is often a hypothetical simplified point particle with no properties other than mass and charge. Real particles like [[electron]]s and [[up quark]]s are more complex and have additional terms in their Lagrangians.
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| ===Classical test particle with Newtonian gravity===
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| Suppose we are given a particle with mass ''m'' kilograms, and position <math>\scriptstyle\vec{x}</math> meters in a Newtonian gravitation field with potential ζ in J·kg<sup>−1</sup>. The particle's world line is parameterized by time ''t'' seconds. The particle's kinetic energy is:
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| :<math> T[t] = {1 \over 2} m \dot{\vec{x}}[t] \cdot \dot{\vec{x}}[t] </math>
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| and the particle's gravitational potential energy is:
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| :<math> V[t] = m \zeta [\vec{x} [t],t] .</math>
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| Then its Lagrangian is ''L'' joules where
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| :<math> L[t] = T[t] - V[t] = {1 \over 2} m \dot{\vec{x}}[t] \cdot \dot{\vec{x}}[t] - m \zeta [\vec{x} [t],t] .</math>
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| Varying <math>\scriptstyle\vec{x}\!</math> in the integral (equivalent to the Euler–Lagrange differential equation), we get
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| :<math>0 = \delta\int{L[t] \, \mathrm{d}t} = \int{\delta L[t] \, \mathrm{d}t} </math>
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| :<math>= \int{(m \dot{\vec{x}}[t] \cdot \dot{\delta \vec{x}}[t] - m \nabla \zeta [\vec{x} [t],t] \cdot \delta \vec{x}[t]) \, \mathrm{d}t}.</math>
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| Integrate the first term by parts and discard the total integral. Then divide out the variation to get
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| :<math>0 = - m \ddot{\vec{x}}[t] - m \nabla \zeta [\vec{x} [t],t] </math>
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| and thus
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| {{NumBlk|:|
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| <math>m \ddot{\vec{x}}[t] = - m \nabla \zeta [\vec{x} [t],t] </math>
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| |{{EquationRef|1}}}}
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| is the equation of motion – two different expressions for the force.
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| ===Special relativistic test particle with electromagnetism===
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| In special relativity, the energy (rest energy plus kinetic energy) of a free test particle is
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| :<math>m c^2 \frac{dt}{d \tau [t]} = \frac{m c^2}{\sqrt {1 - \frac{v^2 [t]}{c^2}}} = +m c^2 + {1 \over 2} m v^2 [t] + {3 \over 8} m \frac{v^4 [t]}{c^2} + \dots \,.</math>
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| However, the term in the Lagrangian that gives rise to the derivative of the momentum is no longer the kinetic energy. It must be changed to
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| :<math>- m c^2 \frac{d \tau[t]}{d t} = - m c^2 \sqrt {1 - \frac{v^2 [t]}{c^2}} = -m c^2 + {1 \over 2} m v^2 [t] + {1 \over 8} m \frac{v^4 [t]}{c^2} + \dots </math>
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| where ''c'' is the vacuum [[speed of light]] in m·s<sup>−1</sup>, τ is the [[proper time]] in seconds (i.e. time measured by a clock moving with the particle) and <math>\scriptstyle v^2 [t] = \dot{\vec{x}}[t] \cdot \dot{\vec{x}}[t].</math> The second term in the series is just the classical kinetic energy. Suppose the particle has electrical charge ''q'' coulombs and is in an electromagnetic field with [[scalar potential]] ϕ volts (a volt is a joule per coulomb) and [[vector potential]] <math>\scriptstyle\vec{A}</math> V·s·m<sup>−1</sup>. The Lagrangian of a special relativistic test particle in an electromagnetic field is:
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| :<math> L[t] = - m c^2 \sqrt {1 - \frac{v^2 [t]}{c^2}} - q \phi [\vec{x}[t],t] + q \dot{\vec{x}}[t] \cdot \vec{A} [\vec{x}[t],t].</math>
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| Varying this with respect to <math>\scriptstyle\vec{x}</math>, we get
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| :<math>0 = - \frac{d}{d t}\left(\frac{m \dot{\vec{x}}[t]} {\sqrt {1 - \frac{v^2 [t]}{c^2}}}\right) - q \nabla\phi [\vec{x}[t],t] - q \partial_t{\vec{A}} [\vec{x}[t],t]
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| - q \dot{\vec{x}}[t] \cdot \nabla\vec{A} [\vec{x}[t],t]
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| + q \nabla{\vec{A}} [\vec{x}[t],t] \cdot \dot{\vec{x}}[t]
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| </math>
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| which is
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| :<math>\frac{d}{d t}\left(\frac{m \dot{\vec{x}}[t]} {\sqrt {1 - \frac{v^2 [t]}{c^2}}}\right) = q \vec{E}[\vec{x}[t],t] + q \dot{\vec{x}}[t] \times \vec{B} [\vec{x}[t],t] </math>
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| which is the equation for the [[Lorentz force]], where:
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| :<math>\vec{E}[\vec{x},t] = - \nabla\phi [\vec{x},t] - \partial_t{\vec{A}} [\vec{x},t] </math>
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| :<math>\vec{B}[\vec{x},t] = \nabla \times \vec{A} [\vec{x},t] </math>
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| are the fields and potentials.
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| ===General relativistic test particle===
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| In [[general relativity]], the first term generalizes (includes) both the classical kinetic energy and the interaction with the gravitational field. It becomes:
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| :<math>- m c^2 \frac{d \tau[t]}{d t} = - m c \sqrt {- g_{\alpha\beta}[x[t]] \frac{d x^{\alpha}[t]}{d t} \frac{d x^{\beta}[t]}{d t}}.</math>
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| The Lagrangian of a general relativistic test particle in an electromagnetic field is:
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| :<math> L[t] = - m c \sqrt {- g_{\alpha\beta}[x[t]] \frac{d x^{\alpha}[t]}{d t} \frac{d x^{\beta}[t]}{d t}} + q \frac{d x^{\gamma}[t]}{d t} A_{\gamma}[x[t]].</math>
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| If the four spacetime coordinates ''x''<sup>α</sup> are given in arbitrary units (i.e. unitless), then ''g''<sub>αβ</sub> in m<sup>2</sup> is the rank 2 symmetric [[metric tensor]] which is also the gravitational potential. Also, ''A''<sub>γ</sub> in V·s is the electromagnetic 4-vector potential. Notice that a factor of ''c'' has been absorbed into the square root because it is the equivalent of
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| :<math>c\, \sqrt {1 - \frac{v^2 [t]}{c^2}} = \sqrt {- ( - c^2 + v^2 [t])} .</math>
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| This notion has been directly generalized from special relativity.
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| ==Lagrangians and Lagrangian densities in field theory==
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| The [[time integral]] of the Lagrangian is called the [[action (physics)|action]] denoted by ''S''. In [[field theory (physics)|field theory]], a distinction is occasionally made between the Lagrangian ''L'', of which the action is the time integral:
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| :<math>\mathcal{S} = \int{L \, \mathrm{d}t}</math>
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| and the ''Lagrangian density'' <math>\scriptstyle\mathcal{L}</math>, which one integrates over all [[spacetime]] to get the action:
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| :<math>\mathcal{S} [\varphi_i] = \int{\mathcal{L} [\varphi_i (x)]\, \mathrm{d}^4x}.</math>
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| * General form of Lagrangian density: <math>\mathcal{L}=\mathcal{L}(\varphi_i,\varphi_{i,\mu})</math><ref name=Mandl>Mandl F., Shaw G., ''Quantum Field Theory'', chapter 2</ref> where <math>\varphi_{i,\mu}\equiv\frac{\partial\varphi_i}{\partial x^\mu}\equiv\partial_\mu\varphi_i</math> (see [[4-gradient]])
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| * The relationship between <math>\mathcal{L}</math> and <math>L</math>: <math>L = \int{ \mathcal{L} \, d x d y d z}</math>,<ref name=Mandl /> similar to <math>q = \int \rho \, dV</math>.
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| * In field theory, the independent variable t was replaced by an event in spacetime (x, y, z, t) or still more generally by a point s on a manifold.
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| The Lagrangian is then the spatial integral of the Lagrangian density. However, <math>\scriptstyle \mathcal{L}</math> is also frequently simply called the Lagrangian, especially in modern use; it is far more useful in [[special relativity|relativistic]] theories since it is a [[principle of locality|locally]] defined, [[Lorentz covariance|Lorentz]] [[Lorentz scalar|scalar]] field. Both definitions of the Lagrangian can be seen as special cases of the general form, depending on whether the spatial variable <math>\scriptstyle\vec x</math> is incorporated into the index ''i'' or the parameters ''s'' in ''φ<sub>i</sub>''(''s''). [[quantum field theory|Quantum field theories]] in [[particle physics]], such as [[quantum electrodynamics]], are usually described in terms of <math>\scriptstyle\mathcal{L}</math>, and the terms in this form of the Lagrangian translate quickly to the rules used in evaluating [[Feynman diagram]]s.
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| ==Selected fields==
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| To go with the section on test particles above, here are the equations for the fields in which they move. The equations below pertain to the fields in which the test particles described above move and allow the calculation of those fields. The equations below will not give you the equations of motion of a test particle in the field but will instead give you the potential (field) induced by quantities such as mass or charge density at any point <math>\scriptstyle[\vec{x},t]</math>. For example, in the case of Newtonian gravity, the Lagrangian density integrated over spacetime gives you an equation which, if solved, would yield <math>\scriptstyle\zeta [\vec{x},t]</math>. This <math>\scriptstyle\zeta [\vec{x},t]</math>, when substituted back in equation ({{EquationNote|1}}), the Lagrangian equation for the test particle in a Newtonian gravitational field, provides the information needed to calculate the acceleration of the particle.
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| ===Newtonian gravity===
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| The Lagrangian (density) is <math>\scriptstyle\mathcal{L}</math> in J·m<sup>−3</sup>. The interaction term ''mζ'' is replaced by a term involving a continuous mass density ''μ'' in kg·m<sup>−3</sup>. This is necessary because using a point source for a field would result in mathematical difficulties. The resulting Lagrangian for the classical gravitational field is:
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| :<math>\mathcal{L}[\vec{x},t] = - \mu [\vec{x},t] \zeta [\vec{x},t] - {1 \over 8 \pi G} (\nabla \zeta [\vec{x},t])^2 </math>
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| where ''G'' in m<sup>3</sup>·kg<sup>−1</sup>·s<sup>−2</sup> is the [[gravitational constant]]. Variation of the integral with respect to ''ζ'' gives:
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| :<math>0 = - \mu [\vec{x},t] \delta\zeta [\vec{x},t] - {2 \over 8 \pi G} (\nabla \zeta [\vec{x},t]) \cdot (\nabla \delta\zeta [\vec{x},t]) .</math>
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| Integrate by parts and discard the total integral. Then divide out by ''δζ'' to get:
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| :<math>0 = - \mu [\vec{x},t] + {1 \over 4 \pi G} \nabla \cdot \nabla \zeta [\vec{x},t] </math>
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| and thus
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| :<math>4 \pi G \mu [\vec{x},t] = \nabla^2 \zeta [\vec{x},t] </math>
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| which yields [[Gauss's law for gravity]].
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| ===Electromagnetism in special relativity===
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| The interaction terms
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| :<math>- q \phi [\vec{x}[t],t] + q \dot{\vec{x}}[t] \cdot \vec{A} [\vec{x}[t],t]</math>
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| are replaced by terms involving a continuous charge density ρ in A·s·m<sup>−3</sup> and current density <math>\scriptstyle\vec{j}</math> in A·m<sup>−2</sup>. The resulting Lagrangian for the electromagnetic field is:
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| :<math>\mathcal{L}[\vec{x},t] = - \rho [\vec{x},t] \phi [\vec{x},t] + \vec{j} [\vec{x},t] \cdot \vec{A} [\vec{x},t] + {\epsilon_0 \over 2} {E}^2 [\vec{x},t] - {1 \over {2 \mu_0}} {B}^2 [\vec{x},t] .</math>
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| Varying this with respect to ϕ, we get
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| :<math>0 = - \rho [\vec{x},t] + \epsilon_0 \nabla \cdot \vec{E} [\vec{x},t] </math>
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| which yields [[Gauss' law]].
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| Varying instead with respect to <math>\scriptstyle\vec{A}</math>, we get
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| :<math>0 = \vec{j} [\vec{x},t] + \epsilon_0 \partial_t \vec{E} [\vec{x},t] - {1 \over \mu_0} \nabla \times \vec{B} [\vec{x},t] </math>
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| which yields [[Ampère's law]].
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| ===Electromagnetism in general relativity===
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| For the Lagrangian of gravity in general relativity, see [[Einstein–Hilbert action]]. The Lagrangian of the electromagnetic field is:
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| :<math>\mathcal{L}[x] = + J^{\gamma}[x] A_{\gamma}[x]
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| - {1 \over 4\mu_0} F_{\mu \nu}[x] F_{\alpha \beta}[x] g^{\mu\alpha}[x] g^{\nu\beta}[x] \sqrt{\frac{-1}{c^2} \mathrm{det} [g[x]]}. </math>
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| If the four spacetime coordinates ''x<sup>α</sup>'' are given in arbitrary units, then: <math>\scriptstyle\mathcal{L}</math> in J·s is the Lagrangian, a scalar density; <math>\scriptstyle J^{\gamma} </math> in coulombs is the current, a vector density; and <math>\scriptstyle F_{\mu \nu}\!</math> in V·s is the [[electromagnetic tensor]], a covariant antisymmetric tensor of rank two. Notice that the determinant under the square root sign is applied to the matrix of components of the covariant metric tensor ''g<sub>αβ</sub>'', and ''g<sup>αβ</sup>'' is its inverse. Notice that the units of the Lagrangian changed because we are integrating over (''x''<sup>0</sup>, ''x''<sup>1</sup>, ''x''<sup>2</sup>, ''x''<sup>3</sup>) which are unitless rather than over (''t, x, y, z'') which have units of s·m<sup>3</sup>. The electromagnetic field tensor is formed by anti-symmetrizing the partial derivative of the electromagnetic vector potential; so it is not an independent variable. The square root is needed to convert that term into a scalar density instead of just a scalar, and also to compensate for the change in the units of the variables of integration. The factor of (−c<sup>−2</sup>) inside the square root is needed to normalize it so that the square root will reduce to one in special relativity (since the determinant is (−c<sup>2</sup>) in special relativity).
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| ===Electromagnetism using differential forms===
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| Using [[differential forms]], the electromagnetic action ''S'' in vacuum on a (pseudo-) Riemannian manifold <math>\scriptstyle\mathcal M</math> can be written (using [[natural units]], {{nowrap|1=''c'' = ''ε''<sub>0</sub> = 1}}) as
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| :<math>\mathcal S[\mathbf{A}] = \int_{\mathcal{M}} \left(-\frac{1}{2}\mathrm{d}\mathbf{A} \wedge *\mathrm{d}\mathbf{A} + \mathbf{A} \wedge \mathbf{J}\right) .</math>
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| Here, '''A''' stands for the electromagnetic potential 1-form, and '''J''' is the current 3-form. This is exactly the same Lagrangian as in the section above, except that the treatment here is coordinate-free; expanding the integrand into a basis yields the identical, lengthy expression. Variation of the action leads to
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| :<math>\mathrm{d}\, {*\mathrm{d}\mathbf{A}} = \mathbf{J} .</math>
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| These are Maxwell's equations for the electromagnetic potential. Substituting {{nowrap|1='''F''' = d'''A'''}} immediately yields the equations for the fields,
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| :<math>\mathrm{d}\mathbf{F} = 0</math>
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| :<math>\mathrm{d}\, {*\mathbf{F}} = \mathbf{J} .</math>
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| ===Dirac Lagrangian===
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| The Lagrangian density for a [[Dirac field]] is:<ref>Itzykson-Zuber, eq. 3-152</ref>
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| :<math>\mathcal{L} = i \hbar c \bar \psi {\partial}\!\!\!/\ \psi - mc^2 \bar\psi \psi</math>
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| where ''ψ'' is a [[Dirac spinor]] ([[annihilation operator]]), <math>\scriptstyle\bar \psi = \psi^\dagger \gamma^0</math> is its [[Dirac adjoint]] ([[creation operator]]) and <math>{\partial}\!\!\!/</math> is [[Feynman slash notation|Feynman notation]] for <math>\scriptstyle\gamma^\sigma \partial_\sigma\!</math>.
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| ===Quantum electrodynamic Lagrangian===
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| The Lagrangian density for [[Quantum electrodynamics|QED]] is:
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| :<math>\mathcal{L}_{\mathrm{QED}} = i\hbar c \bar \psi {D}\!\!\!\!/\ \psi - mc^2 \bar\psi \psi - {1 \over 4\mu_0} F_{\mu \nu} F^{\mu \nu}</math>
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| where <math>\scriptstyle F^{\mu \nu}\!</math> is the [[electromagnetic tensor]], ''D'' is the [[gauge covariant derivative]], and <math>{D}\!\!\!\!/</math> is [[Feynman slash notation|Feynman notation]] for <math>\scriptstyle\gamma^\sigma D_\sigma\!</math>.
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| ===Quantum chromodynamic Lagrangian===
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| The Lagrangian density for [[quantum chromodynamics]] is:<ref>http://www.fuw.edu.pl/~dobaczew/maub-42w/node9.html</ref><ref>http://smallsystems.isn-oldenburg.de/Docs/THEO3/publications/semiclassical.qcd.prep.pdf</ref><ref>http://www-zeus.physik.uni-bonn.de/~brock/teaching/jets_ws0405/seminar09/sluka_quark_gluon_jets.pdf</ref>
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| :<math>\mathcal{L}_{\mathrm{QCD}} = \sum_n \left ( i\hbar c\bar\psi_n{D}\!\!\!\!/\ \psi_n - m_n c^2 \bar\psi_n \psi_n \right) - {1\over 4} G^\alpha {}_{\mu\nu} G_\alpha {}^{\mu\nu}</math>
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| where ''D'' is the QCD [[gauge covariant derivative#Quantum chromodynamics|gauge covariant derivative]],
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| ''n'' = 1, 2, ...6 counts the quark types, and <math>\scriptstyle G^\alpha {}_{\mu\nu}\!</math> is the gluon [[field strength]] tensor.
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| ==Mathematical formalism==
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| Suppose we have an ''n''-dimensional [[manifold]], ''M'', and a target manifold, ''T''. Let <math>\scriptstyle\mathcal{C}</math> be the configuration space of [[smooth function]]s from ''M'' to ''T''.
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| === Examples ===
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| * In [[classical mechanics]], in the [[Hamiltonian mechanics|Hamiltonian]] formalism, ''M'' is the one-dimensional manifold <math>\scriptstyle\mathbb{R}</math>, representing time and the target space is the [[cotangent bundle]] of [[space]] of generalized positions.
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| * In field theory, ''M'' is the [[spacetime]] manifold and the target space is the set of values the fields can take at any given point. For example, if there are m [[real number|real]]-valued [[scalar field]]s, ''ϕ''<sub>1</sub>, ..., ''ϕ<sub>m</sub>'', then the target manifold is <math>\scriptstyle\mathbb{R}^m</math>. If the field is a real [[vector field]], then the target manifold is [[isomorphic]] to <math>\scriptstyle\mathbb{R}^n</math>. There is actually a much more elegant way using [[tangent bundle]]s over ''M'', but we will just stick to this version.
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| === Mathematical development ===
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| Consider a [[Functional analysis|functional]],
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| :<math>\mathcal{S}:\mathcal{C}\rightarrow \mathbb{R}</math>,
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| called the [[Action (physics)|action]]. Physical considerations require it be a [[Map (mathematics)|mapping]] to <math>\scriptstyle\mathbb{R}</math> (the set of all [[real numbers]]), not <math>\scriptstyle\mathbb{C}</math> (the set of all [[complex numbers]]).
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| In order for the action to be [[Functional_(mathematics)#Local_vs_non-local|local]], we need additional restrictions on the [[action (physics)|action]]. If <math>\scriptstyle\varphi\ \in\ \mathcal{C}</math>, we assume <math>\scriptstyle\mathcal{S}[\varphi]</math> is the [[integral]] over ''M'' of a function of <math>\scriptstyle\varphi</math>, its [[derivative]]s and the position called the '''Lagrangian''', <math>\mathcal{L}(\varphi,\partial\varphi,\partial\partial\varphi, ...,x)</math>. In other words,
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| :<math>\forall\varphi\in\mathcal{C}, \ \ \mathcal{S}[\varphi]\equiv\int_M \mathrm{d}^nx \mathcal{L} \big( \varphi(x),\partial\varphi(x),\partial\partial\varphi(x), ...,x \big).</math>
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| It is assumed below, in addition, that the Lagrangian depends on only the field value and its first derivative but not the higher derivatives.
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| Given [[boundary condition]]s, basically a specification of the value of <math>\scriptstyle\varphi</math> at the [[Boundary (topology)|boundary]] if ''M'' is [[Compact space|compact]] or some limit on <math>\scriptstyle\varphi</math> as ''x'' → ∞ (this will help in doing [[integration by parts]]), the [[subspace topology|subspace]] of <math>\scriptstyle \mathcal{C}</math> consisting of functions, <math>\scriptstyle\varphi</math>, such that all [[functional derivative]]s of ''S'' at <math>\scriptstyle\varphi</math> are zero and <math>\scriptstyle\varphi</math> satisfies the given boundary conditions is the subspace of [[on shell]] solutions.
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| The solution is given by the [[Euler–Lagrange equations]] (thanks to the [[boundary condition]]s),
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| :<math>\frac{\delta\mathcal{S}}{\delta\varphi}=-\partial_\mu
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| \left(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\varphi)}\right)+ \frac{\partial\mathcal{L}}{\partial\varphi}=0.</math>
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| The left hand side is the [[functional derivative]] of the [[action (physics)|action]] with respect to <math>\scriptstyle\varphi</math>.
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| ==Uses in Engineering==
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| 50 years ago Lagrangians were a general part of the Engineering curriculum, but quarter of a century later, even with the ascendency of [[Dynamical system]]s, they were dropped as requirements from the majority of Engineering programs, and considered to be the domain of Physics. A decade ago this changed dramatically, and Lagrangians are not only a required part of many ME and EE curricula, but are now seen as far more than the province of Physics. This is true of pure and applied Engineering, as well as the more Physics related aspects of Engineering, or Engineering optimization, which itself is more the province of [[Lagrange multipliers]].
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| Today, Lagrangians find their way into hundreds of direct Engineering solutions, including [[Robotics]], turbulent flow analysis ([[Lagrangian and Eulerian specification of the flow field]]), [[Signal processing]], microscopic component contact and [[Nanotechnology]] (superlinear convergent augmented Lagrangians), gyroscopic forcing and dissipation, [[Semi-infinite]] supercomputing (which also involve [[Lagrange multipliers]] in the sub field of [[Semi infinite programming]]), [[Chemical engineering]] (specific heat linear Lagrangian interpolation in reaction planning), civil engineering (dynamic analysis of traffic flows), Optics engineering and design (Lagrangian and [[Hamiltonian optics]]) aerospace (Lagrangian interpolation), force stepping integrators, and even [[airbag]] deployment (coupled Eulerian-Lagrangians as well as SELM—the [[Stochastic Eulerian Lagrangian method]]). <ref name="Engineering Lagrangians">{{cite book |isbn= 978-1461439295 |url= http://www.amazon.com/gp/product/1461439299/ref=olp_product_details?ie=UTF8&me=&seller=|title= Engineering Dynamics: From the Lagrangian to Simulation |author=Roger F Gans |location=New York|publisher=Springer|year=2013}}</ref>
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| ==See also==
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| {{multicol}}
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| *[[Calculus of variations]]
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| *[[Covariant classical field theory]]
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| *[[Einstein–Maxwell–Dirac equations]]
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| *[[Functional derivative]]
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| *[[Functional integral]]
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| *[[Generalized coordinates]]
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| *[[Hamiltonian mechanics]]
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| {{multicol-break}}
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| *[[Lagrangian and Eulerian coordinates]]
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| *[[Euler–Lagrange equation]]
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| *[[Lagrangian mechanics]]
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| *[[Lagrangian point]]
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| *[[Lagrangian system]]
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| *[[Noether's theorem]]
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| *[[Onsager–Machlup function]]
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| *[[Principle of least action]]
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| *[[Scalar field theory]]
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| {{multicol-end}}
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| ==Notes==
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| <references />
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| ==References==
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| * David Tong [http://www.damtp.cam.ac.uk/user/tong/dynamics.html Classical Dynamics] (Cambridge lecture notes)
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| [[Category:Concepts in physics]]
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| [[Category:Lagrangian mechanics| ]]
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| [[Category:Dynamical systems]]
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| [[Category:Mathematical and quantitative methods (economics)]]
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