Euler–Lagrange equation
Euler–Lagrange equation
In the calculus of variations, the Euler–Lagrange equation, Euler's equation,[1] or Lagrange's equation (although the latter name is ambiguous—see disambiguation page), is a second-order partial differential equation whose solutions are the functions for which a given functional is stationary. It was developed by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange in the 1750s.
Because a differentiable functional is stationary at its local maxima and minima, the Euler–Lagrange equation is useful for solving optimization problems in which, given some functional, one seeks the function minimizing or maximizing it. This is analogous to Fermat's theorem in calculus, stating that at any point where a differentiable function attains a local extremum its derivative is zero.
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 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 coordinates, and it is better suited to generalizations. In classical field theory there is an analogous equation to calculate the dynamics of a field.
History
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.
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.[2]
Statement
The Euler–Lagrange equation is an equation satisfied by a function q of a real argument t, which is a stationary point of the functional
where:
is the function to be found:
- such thatis differentiable,, and;
![{\begin{aligned}{\boldsymbol {q}}\colon [a,b]\subset \mathbb {R} &\to X\\t&\mapsto x={\boldsymbol {q}}(t)\end{aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0449510bda481dee50eb2fea73b3226e90ea65a)
; is the derivative of :
- denotes thetangent spacetoat the point.
is a real-valued function with continuous first partial derivatives:
- being thetangent bundleofdefined by ;
The Euler–Lagrange equation, then, is given by
| Derivation of one-dimensional Euler–Lagrange equation |
|---|
| The derivation of the one-dimensional Euler–Lagrange equation is one of the classic proofs inmathematics. It relies on thefundamental lemma of calculus of variations.
We wish to find a functionwhich satisfies the boundary conditions,, and which extremizes the functional |
![{\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\ .](https://wikimedia.org/api/rest_v1/media/math/render/svg/347727c7ed98505991b9194ffbb36e2f4be280bc)
![{\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\ .](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2dcfa38166ff4468cbd8f7512f1c8635edd3f0e)
![\int _{a}^{b}\left[{\frac {\partial F}{\partial f}}-{\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {\partial F}{\partial f'}}\right]\eta (x)\,\mathrm {d} x+\left[\eta (x){\frac {\partial F}{\partial f'}}\right]_{a}^{b}=0\ .](https://wikimedia.org/api/rest_v1/media/math/render/svg/c833f69b728f7321f240fcb5db83e166e892300f)
![{\displaystyle \int _{a}^{b}\left[{\frac {\partial F}{\partial f}}-{\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {\partial F}{\partial f'}}\right]\eta (x)\,\mathrm {d} x=0\ .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d293d21e0c5ba28a97f40647e1b06ad98038797a)
| Alternate derivation of one-dimensional Euler–Lagrange equation |
|---|
| Given a functional |
![C^{1}([a,b])](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4d44d2584dad0d92346326d4eff220968446993)
![[a,b]](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
![{\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],](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bd33ff65f68b6ebc750ce5a31a9b9e7716bf370)
Examples
A standard example is finding the real-valued function y on the interval [a, b], such that y(a) = c and y(b) = d, for which the path length along the curve traced by y is as short as possible.
the integrand function being L(x, y, y′) = √1 + y′ ² .
The partial derivatives of L are:
By substituting these into the Euler–Lagrange equation, we obtain
that is, the function must have constant first derivative, and thus its graph is a straight line.
Generalizations for several functions, several variables, and higher derivatives
Single function of single variable with higher derivatives
The stationary values of the functional
![{\displaystyle I[f]=\int _{x_{0}}^{x_{1}}{\mathcal {L}}(x,f,f',f'',\dots ,f^{(k)})~\mathrm {d} x~;~~f':={\cfrac {\mathrm {d} f}{\mathrm {d} x}},~f'':={\cfrac {\mathrm {d} ^{2}f}{\mathrm {d} x^{2}}},~f^{(k)}:={\cfrac {\mathrm {d} ^{k}f}{\mathrm {d} x^{k}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0225dddd92f86184bf165d4b5e4a58844f9049d)
can be obtained from the Euler–Lagrange equation[4]
Several functions of single variable with single derivative
![{\displaystyle I[f_{1},f_{2},\dots ,f_{m}]=\int _{x_{0}}^{x_{1}}{\mathcal {L}}(x,f_{1},f_{2},\dots ,f_{m},f_{1}',f_{2}',\dots ,f_{m}')~\mathrm {d} x~;~~f_{i}':={\cfrac {\mathrm {d} f_{i}}{\mathrm {d} x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f781ed0b37670be01ea7542d64eeef89d72a87c1)
then the corresponding Euler–Lagrange equations are[5]
Single function of several variables with single derivative
![{\displaystyle I[f]=\int _{\Omega }{\mathcal {L}}(x_{1},\dots ,x_{n},f,f_{1},\dots ,f_{n})\,\mathrm {d} \mathbf {x} \,\!~;~~f_{j}:={\cfrac {\partial f}{\partial x_{j}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e42836df9987f316a4141697b98757b87c6d08f1)
is extremized only if f satisfies the partial differential equation
Several functions of several variables with single derivative
If there are several unknown functions to be determined and several variables such that
![{\displaystyle 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} \,\!~;~~f_{i,j}:={\cfrac {\partial f_{i}}{\partial x_{j}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80dedbd5d62ca4f9353f7a954124655f21e6266a)
the system of Euler–Lagrange equations is[4]
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 x1 and x2 and if the functional depends on higher derivatives of f up to n-th order such that
![{\displaystyle {\begin{aligned}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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c96618e8ff2b1fe9a72186bc017b213b567bb9e9)
then the Euler–Lagrange equation is[4]
which can be represented shortly as:
Several functions of several variables with higher derivatives
If there are p unknown functions fi to be determined that are dependent on m variables x1 ... xm and if the functional depends on higher derivatives of the fi up to n-th order such that
![{\displaystyle {\begin{aligned}I[f_{1},\ldots ,f_{p}]&=\int _{\Omega }{\mathcal {L}}(x_{1},\ldots ,x_{n};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_{i}}{\partial x_{\mu }}}\;,\quad f_{i,\mu _{1}\mu _{2}}:={\cfrac {\partial ^{2}f_{i}}{\partial x_{\mu _{1}}\partial x_{\mu _{2}}}}\;,\;\;\dots \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a96339cfbcc828deb3db05fadf37ed7262979c0)
Generalization to manifolds
![C^{\infty }([a,b])](https://wikimedia.org/api/rest_v1/media/math/render/svg/659892a168b4498e587161258ca400db12561727)
![f:[a,b]\to M](https://wikimedia.org/api/rest_v1/media/math/render/svg/712c3e0984c28a1648af02d818a9a4fdec0141f0)
![S:C^{\infty }([a,b])\to \mathbb {R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b5308c04282486e61284e50d918731f5d065af6)
![S[f]=\int _{a}^{b}(L\circ {\dot {f}})(t)\,\mathrm {d} t](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa07716cf37832981db890147592174562074312)
![t\in [a,b]](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7f3050ace6dc0dd95250c418528da28eb477ffe)
See also
Lagrangian mechanics
Analytical mechanics
Beltrami identity
Functional derivative