Euler–Lagrange equation

In the calculus of variations and classical mechanics, the Euler–Lagrange equations[1] are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered in the 1750s by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange.

Because a differentiable functional is stationary at its local extrema, 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, according to Hamilton's principle of stationary action, the evolution of a physical system is described by the solutions to the Euler equation for the action of the system. In this context Euler equations are usually called Lagrange equations. In classical mechanics,[2] it is equivalent to Newton's laws of motion; indeed, the Euler-Lagrange equations will produce the same equations as Newton's Laws. This is particularly useful when analyzing systems whose force vectors are particularly complicated. 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.


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.[3]


Let   be a mechanical system with   degrees of freedom. Here   is the configuration space and   the Lagrangian, i.e. a smooth real-valued function such that   and   is an  -dimensional "vector of speed". (For those familiar with differential geometry,   is a smooth manifold, and   where   is the tangent bundle of  

Let   be the set of smooth paths   for which   and   The action functional   is defined via


A path   is a stationary point of   if and only if


Here,   is the time derivative of  

Derivation of the one-dimensional Euler–Lagrange equation

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   which satisfies the boundary conditions  ,  , and which extremizes the functional


We assume that   is twice continuously differentiable.[4] A weaker assumption can be used, but the proof becomes more difficult.[citation needed]

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

Let   be the result of such a perturbation   of  , where   is small and   is a differentiable function satisfying  . Then define

where   .

We now wish to calculate the total derivative of   with respect to ε.


It follows from the total derivative that


The second line follows from the fact that   does not depend on  , i.e.  .


When   we have  ,   and   has an extremum value, so that

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


Using the boundary conditions  ,


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

Alternate derivation of the one-dimensional Euler–Lagrange equation

Given a functional

on   with the boundary conditions   and  , we proceed by approximating the extremal curve by a polygonal line with   segments and passing to the limit as the number of segments grows arbitrarily large.

Divide the interval   into   equal segments with endpoints   and let  . Rather than a smooth function   we consider the polygonal line with vertices  , where   and  . Accordingly, our functional becomes a real function of   variables given by


Extremals of this new functional defined on the discrete points   correspond to points where


Evaluating this partial derivative gives


Dividing the above equation by   gives

and taking the limit as   of the right-hand side of this expression yields

The left hand side of the previous equation is the functional derivative   of 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.


A standard example is finding the real-valued function y(x) 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 a constant first derivative, and thus its graph is a straight line.


Single function of single variable with higher derivativesEdit

The stationary values of the functional


can be obtained from the Euler–Lagrange equation[5]


under fixed boundary conditions for the function itself as well as for the first   derivatives (i.e. for all  ). The endpoint values of the highest derivative   remain flexible.

Several functions of single variable with single derivativeEdit

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[6]


Single function of several variables with single derivativeEdit

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


is extremized only if f satisfies the partial differential equation


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

Several functions of several variables with single derivativeEdit

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


the system of Euler–Lagrange equations is[5]


Single function of two variables with higher derivativesEdit

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[5]


which can be represented shortly as:


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

Several functions of several variables with higher derivativesEdit

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


where   are 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 the   is avoiding counting the same derivative   several times, just as in the previous subsection. This can be expressed more compactly as


Generalization to manifoldsEdit

Let   be a smooth manifold, and let   denote the space of smooth functions  . Then, for functionals   of the form


where   is the Lagrangian, the statement   is equivalent to the statement that, for all  , each coordinate frame trivialization   of a neighborhood of   yields the following   equations:


See alsoEdit


  1. ^ Fox, Charles (1987). An introduction to the calculus of variations. Courier Dover Publications. ISBN 978-0-486-65499-7.
  2. ^ Goldstein, H.; Poole, C.P.; Safko, J. (2014). Classical Mechanics (3rd ed.). Addison Wesley.
  3. ^ A short biography of Lagrange Archived 2007-07-14 at the Wayback Machine
  4. ^ Courant & Hilbert 1953, p. 184
  5. ^ a b c Courant, R; Hilbert, D (1953). Methods of Mathematical Physics. Vol. I (First English ed.). New York: Interscience Publishers, Inc. ISBN 978-0471504474.
  6. ^ Weinstock, R. (1952). Calculus of Variations with Applications to Physics and Engineering. New York: McGraw-Hill.