Open main menu

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-French 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.

Contents

HistoryEdit

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]

StatementEdit

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 that   is differentiable,  , and  ;
  •  ; is the derivative of  :
     
  denotes the tangent space to   at the point  .
  •   is a real-valued function with continuous first partial derivatives:
     
  being the tangent bundle of   defined by
  ;

The Euler–Lagrange equation, then, is given by

 

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

If the dimension of the space   is greater than 1, this is a system of differential equations, one for each component:

 

ExamplesEdit

A standard example is finding the real-valued function f on the interval [a, b], such that f(a) = c and f(b) = d, for which the path length along the curve traced by f 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 derivativesEdit

Single function of single variable with higher derivativesEdit

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 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[5]

 

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[4]

 

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[4]

 

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

NotesEdit

  1. ^ Fox, Charles (1987). An introduction to the calculus of variations. Courier Dover Publications. ISBN 978-0-486-65499-7.
  2. ^ A short biography of Lagrange Archived 2007-07-14 at the Wayback Machine.
  3. ^ Courant & Hilbert 1953, p. 184
  4. ^ 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.
  5. ^ Weinstock, R. (1952). Calculus of Variations with Applications to Physics and Engineering. New York: McGraw-Hill.

ReferencesEdit