Euler–Lagrange equation

The Euler–Lagrange equation or Lagrange's equation, developed by Leonhard Euler and Joseph-Louis Lagrange in the 1750s, is the major formula of the calculus of variations. It provides a way to solve for functions which extremize a given cost functional. It is widely used to solve optimization problems, and in conjunction with the action principle to calculate trajectories. It is analogous to the result from calculus that when a smooth function attains its extreme values its derivative goes to zero.

Statement

The Euler–Lagrange equation is an equation satisfied by a function f of a real parameter t which extremises the functional

$S=\int _{a}^{b}L(t,q(t),q'(t))\,\mathrm {d} t\,\!$

where L is a given function

$L:R\times X\times Y\,\to R\,\!$

with continuous first partial derivatives. Here R denotes the set of real numbers and q is an X-valued function on the reals

$q:R\to X$

whereas the derivative of f is defined as

$q':R\to Y$

so Y is the space of values of the derivative of q, i.e., Y = TX (the space of tangents to X).

The Euler–Lagrange equation then is the ordinary differential equation

$L_{x}(t,q(t),q'(t))-{\frac {\mathrm {d} }{\mathrm {d} t}}L_{y}(t,q(t),q'(t))=0.$

where L_x and L_y denote the partial derivatives of L with respect to the second and third arguments, respectively.

Field theory

Field theories, both classical field theory and quantum field theory, deal with continuous coordinates, and like classical mechanics, has its own Euler-Lagrange equation of motion for a field,

$\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}\right)-{\frac {\partial {\mathcal {L}}}{\partial \psi }}=0.\,$

where

$\psi \,$ is the field, and

$\partial \,$ is a vector differential operator:

$\partial _{\mu }=\left({\frac {1}{c}}{\frac {\partial }{\partial t}},{\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right).\,$

Note: Not all classical fields are assumed commuting/bosonic variables, some of them (like the Dirac field, the Weyl field, the Rarita-Schwinger field) are fermionic and so, when trying to get the field equations from the Lagrangian density, one must choose whether to use the right or the left derivative of the Lagrangian density (which is a boson) with respect to the fields and their first space-time derivatives which are fermionic/anticommuting objects.

There are several examples of applying the Euler–Lagrange equation to various Lagrangians.

• Dirac equation
• Electromagnetic tensor
• Korteweg–de Vries equation
• Quantum electrodynamics

Examples

A standard example is finding the shortest path between two points in the plane. Assume that the points to be connected are (a,c) and (b,d). The length of a path $y=f(x)$ between these two points is

$L=\int _{a}^{b}{\sqrt {1+\left({\frac {\mathrm {d} y}{\mathrm {d} x}}\right)^{2}}}\ \mathrm {d} x.\,\!$

The Euler–Lagrange equation yields the differential equation

${\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {\partial }{\partial y'}}{\sqrt {1+\left({\frac {\mathrm {d} y}{\mathrm {d} x}}\right)^{2}}}=0\Rightarrow {\frac {\mathrm {d} y}{\mathrm {d} x}}=C.\,\!$

In other words, a straight line.

Multidimensional variations

There are also various multi-dimensional versions of the Euler–Lagrange equations. If q is a path in n-dimensional space, then it extremizes the cost functional

$S=\int _{t_{1}}^{t_{2}}L(t,q(t),q'(t))\,\mathrm {d} t\,\!$

only if it satisfies

${\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial q'_{k}}}-{\frac {\partial L}{\partial q_{k}}}=0{\text{ for }}k=1,2,\dots ,n.$

This formulation is particularly useful in physics when L is taken to be the Lagrangian.

Another multi-dimensional generalization comes from considering a function on n variables. If Ω is some surface, then

$S=\int _{\Omega }L(f,x_{1},\dots ,x_{n},f_{x_{1}},\dots ,f_{x_{n}})\,\mathrm {d} \Omega \,\!$

is extremized only if f satisfies the partial differential equation

${\frac {\partial L}{\partial f}}-\sum _{i=1}^{n}{\frac {d}{dx_{i}}}{\frac {\partial L}{\partial f_{x_{i}}}}=0.\,\!$

When n = 2 and L is the energy functional, this leads to the soap-film minimal surface problem.

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. The two 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.

Proof

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.

We wish to find a function $f$ which satisfies the boundary conditions f(a) = c, f(b) = d, and which extremizes the cost functional

$J=\int _{a}^{b}F(x,f(x),f'(x))\,dx.\,\!$

We assume that F has continuous first partial derivatives. A weaker assumption can be used, but the proof becomes more difficult.

If f extremizes the cost functional subject to the boundary conditions, then any slight perturbation of f that preserves the boundary values must either increase J (if f is a minimizer) or decrease J (if f is a maximizer).

Let g_ε(x) = f(x) + εη(x) be such a perturbation of f, where η(x) is a differentiable function satisfying η(a) = η(b) = 0. Then define

$J(\epsilon )=\int _{a}^{b}F(x,g_{\epsilon }(x),g_{\varepsilon }'(x))\,dx.\,\!$

We now wish to calculate the total derivative of J with repect to ε

${\frac {\mathrm {d} J}{\mathrm {d} \varepsilon }}=\int _{a}^{b}{\frac {\mathrm {d} F}{\mathrm {d} \epsilon }}(x,g_{\varepsilon }(x),g_{\varepsilon }'(x))\,dx.$

It follows from the definition of the total derivative that

${\frac {\mathrm {d} F}{\mathrm {d} \epsilon }}={\frac {\partial F}{\partial x}}{\frac {\partial x}{\partial \varepsilon }}+{\frac {\partial F}{\partial g_{\varepsilon }}}{\frac {\partial g_{\varepsilon }}{\partial \varepsilon }}+{\frac {\partial F}{\partial g'_{\varepsilon }}}{\frac {\partial g'_{\varepsilon }}{\partial \varepsilon }}=\eta (x){\frac {\partial F}{\partial g_{\varepsilon }}}+\eta '(x){\frac {\partial F}{\partial g_{\varepsilon }'}}.$

So

${\frac {\mathrm {d} J}{\mathrm {d} \epsilon }}=\int _{a}^{b}\left[\eta (x){\frac {\partial F}{\partial g_{\varepsilon }}}+\eta '(x){\frac {\partial F}{\partial g_{\varepsilon }'}}\,\right]\,dx.$

When ε = 0 we have g_ε = f and since f is an extreme value it follows that J'(0) = 0, i.e.

$J'(0)=\int _{a}^{b}\left[\eta (x){\frac {\partial F}{\partial f}}+\eta '(x){\frac {\partial F}{\partial f'}}\,\right]\,dx=0.$

The next crucial step is to use integration by parts on the second term, yielding

$0=\int _{a}^{b}\left[{\frac {\partial F}{\partial f}}-{\frac {d}{dx}}{\frac {\partial F}{\partial f'}}\right]\eta (x)\,dx+\left[\eta (x){\frac {\partial F}{\partial f'}}\right]_{a}^{b}.$

Using the boundary conditions on η, we get that

$0=\int _{a}^{b}\left[{\frac {\partial F}{\partial f}}-{\frac {d}{dx}}{\frac {\partial F}{\partial f'}}\right]\eta (x)\,dx.\,\!$

Applying the fundamental lemma of calculus of variations now yields the Euler–Lagrange equation

$0={\frac {\partial F}{\partial f}}-{\frac {d}{dx}}{\frac {\partial F}{\partial f'}}.$

Alternate proof

Given a functional

$J=\int _{a}^{b}F(t,y(t),y'(t))dt$

on $C^{1}([a,b])$ with the boundary conditions $y(a)=A$ and $y(b)=B$, we proceed by approximating the extremal curve by a polygonal line with $n$ segments and passing to the limit as the number of segments grows arbitrarily large.

Divide the interval $[a,b]$ into $n+1$ equal segments with endpoints $t_{0}=a,t_{1},t_{2},\ldots ,t_{n},t_{n+1}=b$ and let $\Delta t=t_{k}-t_{k-1}$. Rather than a smooth function $y(t)$ we consider the polygonal line with vertices $(t_{0},y_{0}),\ldots ,(t_{n+1},y_{n+1})$, where $y_{0}=A$ and $y_{n+1}=B$. Accordingly, our functional becomes a real function of $n$ variables given by

$J(y_{1},\ldots ,y_{n})=\sum _{k=0}^{n}F\left(t_{k},y_{k},{\frac {y_{k+1}-y_{k}}{\Delta t}}\right)\Delta t.$

Extremals of this new functional defined on the discrete points $t_{0},\ldots ,t_{n+1}$ correspond to points where

${\frac {\partial J(y_{1},\ldots ,y_{n})}{\partial y_{m}}}=0.$

Evaluating this partial derivative gives that

${\frac {\partial J}{\partial y_{m}}}=F_{y}\left(t_{m},y_{m},{\frac {y_{m+1}-y_{m}}{\Delta t}}\right)\Delta t+F_{y'}\left(t_{m-1},y_{m-1},{\frac {y_{m}-y_{m-1}}{\Delta t}}\right)-F_{y'}\left(t_{m},y_{m},{\frac {y_{m+1}-y_{m}}{\Delta t}}\right).$

Dividing the above equation by $\Delta t$ gives

${\frac {\partial }{\partial y_{m}\Delta t}}=F_{y}\left(t_{m},y_{m},{\frac {y_{m+1}-y_{m}}{\Delta t}}\right)-{\frac {1}{\Delta t}}\left[F_{y'}\left(t_{m},y_{m},{\frac {y_{m+1}-y_{m}}{\Delta t}}\right)-F_{y'}\left(t_{m-1},y_{m-1},{\frac {y_{m}-y_{m-1}}{\Delta t}}\right)\right],$

and taking the limit as $\Delta t\to 0$ of the right-hand side of this expression yields

${\frac {\delta J}{\delta y}}=F_{y}-{\frac {d}{dt}}F_{y'}.$

The term ${\frac {\delta J}{\delta y}}$ denotes the variational derivative of the functional $J$, and a necessary condition for a differentiable functional to have an extremum on some function is that its variational derivative at that function vanishes.

References

• Weisstein, Eric W. "Euler-Lagrange Differential Equation". MathWorld.
• "Calculus of Variations". PlanetMath.
• Izrail Moiseevish Gelfand (1963). Calculus of Variations. Dover. ISBN 0-486-41448-5.
• Calculus of Variations at Example Problems.com (Provides examples of problems from the calculus of variations that involve the Euler–Lagrange equations.)