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# 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;
• ; 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

whereanddenote the partial derivatives ofwith respect to the second and third arguments, respectively.
If the dimension of the spaceis greater than 1, this is a system of differential equations, one for each component:
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
We assume thatis twice continuously differentiable.[3] A weaker assumption can be used, but the proof becomes more difficult.Ifextremizes the functional subject to the boundary conditions, then any slight perturbation ofthat preserves the boundary values must either increase(ifis a minimizer) or decrease(ifis a maximizer). Letbe the result of such a perturbationof, whereis small andis a differentiable function satisfying. Then define
where. We now wish to calculate thetotal derivativeofwith respect to ε.
It follows from the total derivative that
So
When ε = 0 we have gε= f, *Fε= F(x, f(x), f'(x))* and *Jε*  has anextremumvalue, so that
The next step is to useintegration by partson the second term of the integrand, yielding
Using the boundary conditions,
Applying thefundamental lemma of calculus of variationsnow yields the Euler–Lagrange equation
Alternate derivation of one-dimensional Euler–Lagrange equation
Given a functional
onwith the boundary conditionsand, we proceed by approximating the extremal curve by a polygonal line withsegments and passing to the limit as the number of segments grows arbitrarily large. Divide the intervalintoequal segments with endpointsand let. Rather than a smooth functionwe consider the polygonal line with vertices, whereand. Accordingly, our functional becomes a real function ofvariables given by
Extremals of this new functional defined on the discrete pointscorrespond to points where
Evaluating this partial derivative gives
Dividing the above equation bygives
and taking the limit asof the right-hand side of this expression yields
The left hand side of the previous equation is thefunctional derivativeof the functional. 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.

## 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

can be obtained from the Euler–Lagrange equation[4]

under fixed boundary conditions for the function itself as well as for the firstderivatives (i.e. for all). The endpoint values of the highest derivativeremain flexible.

### Several functions of single variable with single derivative

If the problem involves finding several functions () of a single independent variable () that define an extremum of the functional

then the corresponding Euler–Lagrange equations are[5]

### Single function of several variables with single derivative

A multi-dimensional generalization comes from considering a function on n variables. Ifis some surface, then

is extremized only if f satisfies the partial differential equation

When n = 2 and functionalis theenergy functional, this leads to the soap-filmminimal surfaceproblem.

### Several functions of several variables with single derivative

If there are several unknown functions to be determined and several variables such that

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

then the Euler–Lagrange equation is[4]

which can be represented shortly as:

whereinare indices that span the number of variables, that is, here they go from 1 to 2. Here summation over theindices is only overin order to avoid counting the same partial derivative multiple times, for exampleappears only once in the previous equation.

### 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

whereare indices that span the number of variables, that is they go from 1 to m. Then the Euler–Lagrange equation is
where the summation over theis avoiding counting the same derivativeseveral times, just as in the previous subsection. This can be expressed more compactly as

## Generalization to manifolds

Letbe asmooth manifold, and letdenote the space ofsmooth functions. Then, for functionalsof the form
whereis the Lagrangian, the statementis equivalent to the statement that, for all, each coordinate frametrivializationof a neighborhood ofyields the followingequations:

## References

[1]
Citation Linkopenlibrary.orgFox, Charles (1987). An introduction to the calculus of variations. Courier Dover Publications. ISBN 978-0-486-65499-7.
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[2]
Citation Linkweb.archive.orgA short biography of Lagrange Archived 2007-07-14 at the Wayback Machine
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[3]
Citation Linkopenlibrary.orgCourant, R; Hilbert, D (1953). Methods of Mathematical Physics. Vol. I (First English ed.). New York: Interscience Publishers, Inc. ISBN 978-0471504474., p. 184
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[5]
Citation Linkopenlibrary.orgWeinstock, R. (1952). Calculus of Variations with Applications to Physics and Engineering. New York: McGraw-Hill.
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Citation Linknumericalmethods.eng.usf.eduA short biography of Lagrange
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