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| In [[calculus of variations]], the '''Euler–Lagrange equation''', '''Euler's equation''',<ref>{{cite book|first=Charles|last=Fox|title=An introduction to the calculus of variations|publisher=Courier Dover Publications|year=1987|isbn=978-0-486-65499-7}}</ref> or '''Lagrange's equation''' although the latter name is ambiguous (see [[Lagrange's formula (disambiguation)|disambiguation page]]), is a [[differential equation]] whose solutions are the [[function (mathematics)|function]]s for which a given [[functional (mathematics)|functional]] is [[stationary point|stationary]]. It was developed by Swiss mathematician [[Leonhard Euler]] and Italian mathematician [[Joseph Louis Lagrange]] in the 1750s.
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| Because a differentiable functional is stationary at its local [[maxima and minima]], the Euler–Lagrange equation is useful for solving [[optimization (mathematics)|optimization]] problems in which, given some functional, one seeks the function minimizing (or maximizing) it. This is analogous to [[Fermat's theorem (stationary points)|Fermat's theorem]] in [[calculus]], stating that at any point where a differentiable function attains a local extremum, its [[derivative (mathematics)|derivative]] is zero.
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| In [[Lagrangian mechanics]], because of [[Hamilton's principle]] of stationary action, the evolution of a physical system is described by the solutions to the Euler–Lagrange equation for the [[action (physics)#Action (functional)|action]] of the system. In [[classical mechanics]], it is equivalent to [[Newton's laws of motion]], but it has the advantage that it takes the same form in any system of [[generalized coordinate]]s, and it is better suited to generalizations. In [[classical field theory]] there is an [[classical field theory#Lagrangian dynamics|analogous equation]] to calculate the dynamics of a [[field (physics)|field]].
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| ==History==
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| The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the [[tautochrone]] problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point.
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| Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it to [[mechanics]], which led to the formulation of [[Lagrangian mechanics]]. Their correspondence ultimately led to the calculus of variations, a term coined by Euler himself in 1766.<ref>[http://numericalmethods.eng.usf.edu/anecdotes/lagrange.pdf A short biography of Lagrange]</ref>
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| ==Statement==
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| The Euler–Lagrange equation is an equation satisfied by a function, ''q'',
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| of a [[real number|real]] argument, ''t'', which is a stationary point of the [[functional (mathematics)|functional]] | |
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| :<math>\displaystyle S(q) = \int_a^b L(t,q(t),q'(t))\, \mathrm{d}t</math>
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| where:
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| *''q'' is the function to be found:
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| *:<math>\begin{align}
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| q \colon [a, b] \subset \mathbb{R} & \to X \\
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| t & \mapsto x = q(t)
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| \end{align}</math>
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| :such that ''q'' is differentiable, ''q''(''a'') = ''x''<sub>''a''</sub>, and ''q''(''b'') = ''x''<sub>''b''</sub>;
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| *''q''′ is the derivative of ''q'':
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| *:<math>\begin{align}
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| q' \colon [a, b] & \to T_{q(t)}X \\
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| t & \mapsto v = q'(t)
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| \end{align}</math>
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| :''TX'' being the [[tangent bundle]] of ''X'' defined by
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| : <math> TX = \bigcup_{x \in X} \{ x \} \times T_{x}X </math> ;
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| * ''L'' is a real-valued function with [[continuous function|continuous]] first [[partial derivatives]]:
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| *:<math>\begin{align}
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| L \colon [a, b] \times TX & \to \mathbb{R} \\
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| (t, x, v) & \mapsto L(t, x, v).
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| \end{align}</math>
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| The Euler–Lagrange equation, then, is given by
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| {{Equation box 1
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| |indent =:
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| |equation = <math>L_x(t,q(t),q'(t))-\frac{\mathrm{d}}{\mathrm{d}t}L_v(t,q(t),q'(t)) = 0.</math>
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| |border colour = #50C878
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| |background colour = #ECFCF4}}
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| where ''L''<sub>''x''</sub> and ''L''<sub>''v''</sub> denote the partial derivatives of ''L'' with respect to the second and third arguments, respectively.
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| If the dimension of the space ''X'' is greater than 1, this is a system of differential equations, one for each component:
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| :<math>\frac{\partial L(t,q(t),q'(t))}{\partial x_i}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L(t,q(t),q'(t))}{\partial v_i} = 0
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| \quad \text{for } i = 1, \dots, n.</math>
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| :{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"
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| !Derivation of one-dimensional Euler–Lagrange equation
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| |-
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| The derivation of the one-dimensional Euler–Lagrange equation is one of the classic proofs in [[mathematics]]. It relies on the [[fundamental lemma of calculus of variations]]. | |
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| We wish to find a function <math>f</math> which satisfies the boundary conditions <math>f(a) = A</math>, <math>f(b) = B</math>, and which extremizes the functional
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| : <math> J = \int_a^b F(x,f(x),f'(x))\, dx\ . \,\!</math> | |
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| We assume that <math>F</math> has continuous first partial derivatives. A weaker assumption can be used, but the proof becomes more difficult.{{Cn|date=September 2013}}
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| If <math>f</math> extremizes the functional subject to the boundary conditions, then any slight perturbation of <math>f</math> that preserves the boundary values must either increase <math>J</math> (if <math>f</math> is a minimizer) or decrease <math>J</math> (if <math>f</math> is a maximizer).
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| Let <math>g_{\varepsilon} (x) = f (x) + \varepsilon \eta (x)</math> be the result of such a perturbation <math>\varepsilon \eta (x)</math> of <math>f</math>, where <math>\varepsilon</math> is small and <math>\eta (x)</math> is a differentiable function satisfying <math>\eta (a) = \eta (b) = 0</math>. Then define
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| : <math> J_\varepsilon = \int_a^b F(x,g_\varepsilon(x), g_\varepsilon'(x) ) \, \mathrm{d}x = \int_a^b F_\varepsilon\, dx \,\! </math>
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| where <math> F_\varepsilon = F(x, \, g_\varepsilon (x), \, g_\varepsilon' (x) ) </math> .
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| We now wish to calculate the [[total derivative]] of <math> J_\varepsilon</math> with respect to ''ε''.
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| : <math> \frac{\mathrm{d} J_\varepsilon}{\mathrm{d} \varepsilon} = \frac{\mathrm d}{\mathrm d\varepsilon}\int_a^b F_\varepsilon\, \mathrm{d}x = \int_a^b \frac{\mathrm{d} F_\varepsilon}{\mathrm{d}\varepsilon} \, \mathrm{d}x </math>
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| It follows from the total derivative that
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| :<math>
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| \begin{align}
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| \frac{\mathrm d F_\varepsilon}{\mathrm d\varepsilon} & =\frac{\mathrm d x}{\mathrm d\varepsilon}\frac{\partial F_\varepsilon}{\partial x} + \frac{\mathrm d g_\varepsilon}{\mathrm d\varepsilon}\frac{\partial F_\varepsilon}{\partial g_\varepsilon} + \frac{\mathrm d g_\varepsilon'}{\mathrm d\varepsilon}\frac{\partial F_\varepsilon}{\partial g_\varepsilon'} \\
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| & = \frac{\mathrm d g_\varepsilon}{\mathrm d\varepsilon}\frac{\partial F_\varepsilon}{\partial g_\varepsilon}+\frac{\mathrm d g'_\varepsilon}{\mathrm d\varepsilon}\frac{\partial F_\varepsilon}{\partial g'_\varepsilon} \\ | |
| & = \eta(x) \frac{\partial F_\varepsilon}{\partial g_\varepsilon} + \eta'(x) \frac{\partial F_\varepsilon}{\partial g_\varepsilon'} \ . \\ | |
| \end{align}
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| </math>
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| So
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| : <math> \frac{\mathrm{d} J_\varepsilon}{\mathrm{d} \varepsilon} = \int_a^b \left[\eta(x) \frac{\partial F_\varepsilon}{\partial g_\varepsilon} + \eta'(x) \frac{\partial F_\varepsilon}{\partial g_\varepsilon'} \, \right]\,\mathrm{d}x \ . </math>
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| When ''ε'' = 0 we have ''g''<sub>''ε''</sub> = ''f'', ''F<sub>ε</sub> = F(x, f(x), f'(x))'' and ''J<sub>ε</sub>'' has an [[extremum]] value, so that
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| : <math> \frac{\mathrm d J_\varepsilon}{\mathrm d\varepsilon}\bigg|_{\varepsilon=0} = \int_a^b \left[ \eta(x) \frac{\partial F}{\partial f} + \eta'(x) \frac{\partial F}{\partial f'} \,\right]\,\mathrm{d}x = 0 \ .</math>
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| The next step is to use [[integration by parts]] on the second term of the integrand, yielding
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| : <math> \int_a^b \left[ \frac{\partial F}{\partial f} - \frac{\mathrm{d}}{\mathrm{d}x} \frac{\partial F}{\partial f'} \right] \eta(x)\,dx + \left[ \eta(x) \frac{\partial F}{\partial f'} \right]_a^b = 0 \ . </math>
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| Using the boundary conditions <math>\eta (a) = \eta (b) = 0</math>,
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| : <math> \int_a^b \left[ \frac{\partial F}{\partial f} - \frac{\mathrm{d}}{\mathrm{d}x} \frac{\partial F}{\partial f'} \right] \eta(x)\,dx = 0 \ . \,\!</math>
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| Applying the [[fundamental lemma of calculus of variations]] now yields the Euler–Lagrange equation
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| : <math> \frac{\partial F}{\partial f} - \frac{\mathrm{d}}{\mathrm{d}x} \frac{\partial F}{\partial f'} = 0 \ . </math>
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| |}
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| :{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"
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| !Alternate derivation of one-dimensional Euler–Lagrange equation
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| |-
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| Given a functional
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| :<math>J = \int^b_aF(t, y(t), y'(t))\,\mathrm{d}t</math>
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| on <math>C^1([a, b])</math> with the boundary conditions <math>y(a) = A</math> and <math>y(b) = B</math>, we proceed by approximating the extremal curve by a polygonal line with <math>n</math> segments and passing to the limit as the number of segments grows arbitrarily large.
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| Divide the interval <math>[a, b]</math> into <math>n</math> equal segments with endpoints <math>t_0 = a, t_1, t_2, \ldots, t_n, t_{n + 1} = b</math> and let <math>\Delta t = t_k - t_{k - 1}</math>. Rather than a smooth function <math>y(t)</math> we consider the polygonal line with vertices <math>(t_0, y_0),\ldots,(t_{n + 1}, y_{n + 1})</math>, where <math>y_0 = A</math> and <math>y_{n + 1} = B</math>. Accordingly, our functional becomes a real function of <math>n</math> variables given by
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| :<math>J(y_1, \ldots, y_n) \approx \sum^n_{k = 0}F\left(t_k, y_k, \frac{y_{k + 1} - y_k}{\Delta t}\right)\Delta t.</math>
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| Extremals of this new functional defined on the discrete points <math>t_0,\ldots,t_{n + 1}</math> correspond to points where
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| :<math>\frac{\partial J(y_1,\ldots,y_n)}{\partial y_m} = 0.</math>
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| Evaluating this partial derivative gives
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| :<math>\frac{\partial J}{\partial y_m} = F_y\left(t_m, y_m, \frac{y_{m + 1} - y_m}{\Delta t}\right)\Delta t + F_{y'}\left(t_{m - 1}, y_{m - 1}, \frac{y_m - y_{m - 1}}{\Delta t}\right) - F_{y'}\left(t_m, y_m, \frac{y_{m + 1} - y_m}{\Delta t}\right).</math>
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| Dividing the above equation by <math>\Delta t</math> gives
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| :<math>\frac{\partial J}{\partial y_m \Delta t} = F_y\left(t_m, y_m, \frac{y_{m + 1} - y_m}{\Delta t}\right) - \frac{1}{\Delta t}\left[F_{y'}\left(t_m, y_m, \frac{y_{m + 1} - y_m}{\Delta t}\right) - F_{y'}\left(t_{m - 1}, y_{m - 1}, \frac{y_m - y_{m - 1}}{\Delta t}\right)\right],</math>
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| and taking the limit as <math>\Delta t \to 0</math> of the right-hand side of this expression yields
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| :<math>F_y - \frac{\mathrm{d}}{\mathrm{d}t}F_{y'} = 0.</math>
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| The left hand side of the previous equation is the [[functional derivative]] <math>\delta J/\delta y</math> of the functional <math>J</math>. A necessary condition for a differentiable functional to have an extremum on some function is that its functional derivative at that function vanishes, which is granted by the last equation. | |
| |}
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| ==Examples==
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| A standard example is finding the real-valued function on the interval [''a'', ''b''], such that ''f''(''a'') = ''c'' and ''f''(''b'') = ''d'', the [[arc length|length]] of whose [[graph of a function|graph]] is as short as possible. The length of the graph of ''f'' is:
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| :<math> \ell (f) = \int_{a}^{b} \sqrt{1+(f'(x))^2}\,\mathrm{d}x,</math>
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| the integrand function being {{nowrap|1=''L''(''x'', ''y'', ''y''′) = {{radic|1 + ''y''′ ²}}}} evaluated at {{nowrap|1=(''x'', ''y'', ''y''′) = (''x'', ''f''(''x''), ''f''′(''x''))}}.
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| The partial derivatives of ''L'' are:
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| :<math>\frac{\partial L(x, y, y')}{\partial y'} = \frac{y'}{\sqrt{1 + y'^2}} \quad \text{and} \quad
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| \frac{\partial L(x, y, y')}{\partial y} = 0.</math>
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| By substituting these into the Euler–Lagrange equation, we obtain
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| :<math>
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| \begin{align}
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| \frac{\mathrm{d}}{\mathrm{d}x} \frac{f'(x)}{\sqrt{1 + (f'(x))^2}} &= 0 \\
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| \frac{f'(x)}{\sqrt{1 + (f'(x))^2}} &= C = \text{constant} \\
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| \Rightarrow f'(x)&= \frac{C}{\sqrt{1-C^2}} := A \\
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| \Rightarrow f(x) &= Ax + B
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| \end{align}
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| </math>
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| that is, the function must have constant first derivative, and thus its graph is a [[straight line]].
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| ===Classical mechanics===
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| ====Basic method====
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| To find the equations of motions for a given system (whose potential energy is time-independent), one only has to follow these steps:
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| * From the kinetic energy <math>T</math>, and the potential energy <math>V</math>, compute the [[Lagrangian]] <math>L = T - V</math>.
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| * Compute <math>\frac{\partial L}{\partial q}</math>.
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| * Compute <math>\frac{\partial L}{\partial \dot{q}}</math> and from it, <math>\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}}</math>. It is important that <math>\dot{q}</math> be treated as a complete variable in its own right, and not as a derivative.
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| * Equate <math>\frac{\partial L}{\partial q} = \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}}</math>. This is the Euler–Lagrange equation.
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| * Solve the differential equation obtained in the preceding step. At this point, <math>\dot{q}</math> is treated "normally". Note that the above might be a system of equations and not simply one equation.
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| ====Particle in a conservative force field====
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| The motion of a single particle in a [[conservative force]] field (for example, the gravitational force) can be determined by requiring the [[Action (physics)#Action (functional)|action]] to be stationary, by [[Hamilton's principle]]. The action for this system is
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| :<math>S = \int_{t_0}^{t_1} L(t, \mathbf{x}(t), \mathbf{\dot{x}}(t))\,\mathrm{d}t</math>
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| where '''x'''(''t'') is the position of the particle at time ''t''. The dot above is [[Newton's notation]] for the time derivative: thus '''ẋ'''(''t'') is the particle velocity, '''v'''(''t''). In the equation above, ''L'' is the [[Lagrangian]] (the [[kinetic energy]] minus the [[potential energy]]):
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| :<math>L(t, \mathbf{x}, \mathbf{v}) = \frac{1}{2}m \sum_{i=1} ^{3} v_i^2 - U(\mathbf{x}),</math>
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| where:
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| *''m'' is the [[mass (physics)|mass]] of the particle ([[conservation of mass|assumed to be constant]] in classical physics);
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| *''v''<sub>''i''</sub> is the ''i''-th component of the vector '''v''' in a Cartesian coordinate system (the same notation will be used for other vectors);
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| *''U'' is the potential of the conservative force.
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| In this case, the Lagrangian does not vary with its first argument ''t''. (By [[Noether's theorem]], such symmetries of the system correspond to [[conservation law]]s. In particular, the invariance of the Lagrangian with respect to time implies the [[conservation of energy]].)
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| By partial differentiation of the above Lagrangian, we find:
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| :<math>\frac{\partial L(t,\mathbf{x},\mathbf{v})}{\partial x_i} = -\frac{\partial U(\mathbf{x})}{\partial x_i} = F_i (\mathbf{x})\quad \text{and} \quad
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| \frac{\partial L(t,\mathbf{x},\mathbf{v})}{\partial v_i} = m v_i = p_i,</math>
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| where the force is '''F''' = −'''∇'''''U'' (the negative [[gradient]] of the potential, by definition of conservative force), and '''p''' is the [[momentum]].
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| By substituting these into the Euler–Lagrange equation, we obtain a system of second-order differential equations for the coordinates on the particle's trajectory,
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| :<math>F_i(\mathbf{x}(t)) = \frac{\mathrm d}{\mathrm{d}t} m \dot{x}_i(t) = m \ddot{x}_i(t),</math> | |
| which can be solved on the interval [''t''<sub>0</sub>, ''t''<sub>1</sub>], given the boundary values ''x''<sub>''i''</sub>(''t''<sub>0</sub>) and ''x''<sub>''i''</sub>(''t''<sub>1</sub>).
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| In vector notation, this system reads
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| :<math>\mathbf{F}(\mathbf{x}(t)) = m\mathbf{\ddot x}(t)</math>
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| or, using the momentum,
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| :<math> \mathbf{F} = \frac {\mathrm{d}\mathbf{p}} {\mathrm{d}t}</math>
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| which is [[Newton's second law]].
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| == Variations for several functions, several variables, and higher derivatives ==
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| === Single function of single variable with higher derivatives ===
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| The stationary values of the functional
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| :<math>
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| I[f] = \int_{x_0}^{x_1} \mathcal{L}(x, f, f', f'', \dots, f^{(n)})~\mathrm{d}x ~;~~
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| f' := \cfrac{\mathrm{d}f}{\mathrm{d}x}, ~f'' := \cfrac{\mathrm{d}^2f}{\mathrm{d}x^2}, ~
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| f^{(n)} := \cfrac{\mathrm{d}^nf}{\mathrm{d}x^n}
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| </math>
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| can be obtained from the Euler–Lagrange equation<ref name=Courant>Courant, R. and Hilbert, D., 1953, '''Methods of Mathematical Physics: Vol I''', Interscience Publishers, New York.</ref>
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| :<math>
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| \cfrac{\partial \mathcal{L}}{\partial f} - \cfrac{\mathrm{d}}{\mathrm{d} x}\left(\cfrac{\partial \mathcal{L}}{\partial f'}\right) + \cfrac{\mathrm{d}^2}{\mathrm{d} x^2}\left(\cfrac{\partial \mathcal{L}}{\partial f''}\right) - \dots +
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| (-1)^n \cfrac{\mathrm{d}^n}{\mathrm{d} x^n}\left(\cfrac{\partial \mathcal{L}}{\partial f^{(n)}}\right) = 0
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| </math>
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| under fixed boundary conditions for the function itself as well as for the first <math>n-1</math> derivatives (i.e. for all <math>f^{(i)}, i \in \{0, ..., n-1\}</math>). The endpoint values of the highest derivative <math>f^{(n)}</math> remain flexible.
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| === Several functions of one variable ===
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| If the problem involves finding several functions (<math>f_1, f_2, \dots, f_n</math>) of a single independent variable (<math>x</math>) that define an extremum of the functional
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| :<math>
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| I[f_1,f_2, \dots, f_n] = \int_{x_0}^{x_1} \mathcal{L}(x, f_1, f_2, \dots, f_n, f_1', f_2', \dots, f_n')~\mathrm{d}x
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| ~;~~ f_i' := \cfrac{\mathrm{d}f_i}{\mathrm{d}x}
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| </math>
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| then the corresponding Euler–Lagrange equations are<ref name=Weinstock>Weinstock, R., 1952, '''Calculus of Variations With Applications to Physics and Engineering''', McGraw-Hill Book Company, New York.</ref>
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| :<math>
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| \begin{align}
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| \cfrac{\partial \mathcal{L}}{\partial f_i} - \cfrac{d}{dx}\left(\cfrac{\partial \mathcal{L}}{\partial f_i'}\right) = 0
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| \end{align}
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| </math> | |
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| === Single function of several variables ===
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| A multi-dimensional generalization comes from considering a function on ''n'' variables. If Ω is some surface, then
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| : <math>
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| I[f] = \int_{\Omega} \mathcal{L}(x_1, \dots , x_n, f, f_{x_1}, \dots , f_{x_n})\, \mathrm{d}\mathbf{x}\,\! ~;~~
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| f_{x_i} := \cfrac{\partial f}{\partial x_i}
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| </math>
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| is extremized only if ''f'' satisfies the [[partial differential equation]]
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| : <math> \frac{\partial \mathcal{L}}{\partial f} - \sum_{i=1}^{n} \frac{\partial}{\partial x_i} \frac{\partial \mathcal{L}}{\partial f_{x_i}} = 0. \,\!</math>
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| When ''n'' = 2 and <math>\mathcal{L}</math> is the [[energy functional]], this leads to the soap-film [[minimal surface]] problem.
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| === Several functions of several variables ===
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| If there are several unknown functions to be determined and several variables such that | |
| : <math>
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| I[f_1,f_2,\dots,f_m] = \int_{\Omega} \mathcal{L}(x_1, \dots , x_n, f_1, \dots, f_m, f_{1,1}, \dots , f_{1,n}, \dots, f_{m,1}, \dots, f_{m,n}) \, \mathrm{d}\mathbf{x}\,\! ~;~~
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| f_{j,i} := \cfrac{\partial f_j}{\partial x_i}
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| </math>
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| the system of Euler–Lagrange equations is<ref name=Courant/> | |
| : <math>
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| \begin{align}
| |
| \frac{\partial \mathcal{L}}{\partial f_1} - \sum_{i=1}^{n} \frac{\mathrm{d}}{\mathrm{d}x_i} \frac{\partial \mathcal{L}}{\partial f_{1,i}} &= 0 \\
| |
| \frac{\partial \mathcal{L}}{\partial f_2} - \sum_{i=1}^{n} \frac{\mathrm{d}}{\mathrm{d}x_i} \frac{\partial \mathcal{L}}{\partial f_{2,i}} &= 0 \\
| |
| \vdots \qquad \vdots \qquad &\quad \vdots \\
| |
| \frac{\partial \mathcal{L}}{\partial f_j} - \sum_{i=1}^{n} \frac{\mathrm{d}}{\mathrm{d}x_i} \frac{\partial \mathcal{L}}{\partial f_{j,i}} &= 0.
| |
| \end{align}
| |
| </math>
| |
| | |
| === Single function of two variables with higher derivatives ===
| |
| If there is a single unknown function ''f'' to be determined that is dependent on two variables ''x''<sub>1</sub> and ''x''<sub>2</sub> and if the functional depends on higher derivatives of ''f'' up to ''n''-th order such that
| |
| : <math>
| |
| \begin{align}
| |
| I[f] & = \int_{\Omega} \mathcal{L}(x_1, x_2, f, f_{,1}, f_{,2}, f_{,11}, f_{,12}, f_{,22},
| |
| \dots, f_{,22\dots 2})\, \mathrm{d}\mathbf{x} \\
| |
| & \qquad \quad
| |
| f_{,i} := \cfrac{\partial f}{\partial x_i} \; , \quad
| |
| f_{,ij} := \cfrac{\partial^2 f}{\partial x_i\partial x_j} \; , \;\; \dots
| |
| \end{align}
| |
| </math>
| |
| then the Euler–Lagrange equation is<ref name=Courant/>
| |
| :<math>
| |
| \begin{align}
| |
| \frac{\partial \mathcal{L}}{\partial f}
| |
| & - \frac{\partial}{\partial x_1}\left(\frac{\partial \mathcal{L}}{\partial f_{,1}}\right)
| |
| - \frac{\partial}{\partial x_2}\left(\frac{\partial \mathcal{L}}{\partial f_{,2}}\right)
| |
| + \frac{\partial^2}{\partial x_1^2}\left(\frac{\partial \mathcal{L}}{\partial f_{,11}}\right)
| |
| + \frac{\partial^2}{\partial x_1\partial x_2}\left(\frac{\partial \mathcal{L}}{\partial f_{,12}}\right)
| |
| + \frac{\partial^2}{\partial x_2^2}\left(\frac{\partial \mathcal{L}}{\partial f_{,22}}\right) \\
| |
| & - \dots
| |
| + (-1)^n \frac{\partial^n}{\partial x_2^n}\left(\frac{\partial \mathcal{L}}{\partial f_{,22\dots 2}}\right) = 0
| |
| \end{align}
| |
| </math>
| |
| which can be represented shortly as:
| |
| :<math>
| |
| \frac{\partial \mathcal{L}}{\partial f} +\sum_{i=1}^n (-1)^i \frac{\partial^i}{\partial x_{\mu_{1}}\dots \partial x_{\mu_{i}}} \left( \frac{\partial \mathcal{L} }{\partial f_{,\mu_1\dots\mu_i}}\right)=0
| |
| </math>
| |
| where <math>\mu_1 \dots \mu_i</math> are indices that span the number of variables, that is they go from 1 to 2. Here summation over the <math>\mu_1 \dots \mu_i</math> indices is implied according to [[Einstein notation]].
| |
| | |
| | |
| === Several functions of several variables with higher derivatives ===
| |
| If there is are ''p'' unknown functions ''f''<sub>i</sub> to be determined that are dependent on ''m'' variables ''x''<sub>1</sub> ... ''x''<sub>m</sub> and if the functional depends on higher derivatives of the ''f''<sub>i</sub> up to ''n''-th order such that
| |
| : <math>
| |
| \begin{align}
| |
| I[f_1,\ldots,f_m] & = \int_{\Omega} \mathcal{L}(x_1, \ldots, x_m; f_1,\ldots,f_p; f_{1,1},\ldots,
| |
| f_{p,m}; f_{1,11},\ldots, f_{p,mm};\ldots f_{p,m\ldots m})\, \mathrm{d}\mathbf{x} \\
| |
| & \qquad \quad
| |
| f_{i,\mu} := \cfrac{\partial f}{\partial x_\mu} \; , \quad
| |
| f_{i,\mu_1\mu_2} := \cfrac{\partial^2 f}{\partial x_{\mu_1}\partial x_{\mu_1}} \; , \;\; \dots
| |
| \end{align}
| |
| </math>
| |
| | |
| where <math>\mu_1 \dots \mu_j</math> are indices that span the number of variables, that is they go from 1 to m. Then the Euler–Lagrange equation is
| |
| | |
| :<math>
| |
| \frac{\partial \mathcal{L}}{\partial f_i} +\sum_{j=1}^n (-1)^j \frac{\partial^j}{\partial x_{\mu_{1}}\dots \partial x_{\mu_{j}}} \left( \frac{\partial \mathcal{L} }{\partial f_{i,\mu_1\dots\mu_j}}\right)=0
| |
| </math>
| |
| | |
| where summation over the <math>\mu_1 \dots \mu_j</math> is implied according to [[Einstein notation]]. This can be expressed more compactly as
| |
| | |
| :<math>
| |
| \sum_{j=0}^n (-1)^j \partial_{ \mu_{1}\ldots \mu_{j} }^j \left( \frac{\partial \mathcal{L} }{\partial f_{i,\mu_1\dots\mu_j}}\right)=0
| |
| </math>
| |
| | |
| ==See also==
| |
| {{Wiktionary|Euler–Lagrange equation}}
| |
| *[[Lagrangian mechanics]]
| |
| *[[Hamiltonian mechanics]]
| |
| *[[Analytical mechanics]]
| |
| *[[Beltrami identity]]
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| * {{springer|title=Lagrange equations (in mechanics)|id=p/l057150}}
| |
| * {{mathworld|urlname=Euler-LagrangeDifferentialEquation|title=Euler-Lagrange Differential Equation}}
| |
| * {{planetmathref |id=1995|title=Calculus of Variations}}
| |
| * {{cite book |last=Gelfand |first=Izrail Moiseevich |authorlink=Israel Gelfand |title=Calculus of Variations |publisher=Dover |year=1963 |isbn=0-486-41448-5}}
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| | |
| {{DEFAULTSORT:Euler-Lagrange Equation}}
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| [[Category:Ordinary differential equations]]
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| [[Category:Partial differential equations]]
| |
| [[Category:Calculus of variations]]
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| [[Category:Articles containing proofs]]
| |