In the calculus of variations, the Euler–Lagrange equation, Euler's equation, 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.
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.
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
- 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:
|Derivation of 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. A weaker assumption can be used, but the proof becomes more difficult.
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
We now wish to calculate the total derivative of with respect to ε.
It follows from the total derivative that
When ε = 0 we have gε = f, Fε = F(x, f(x), f'(x)) and Jε 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 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 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 derivatives
Single function of single variable with higher derivatives
The stationary values of the functional
can be obtained from the Euler–Lagrange equation
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 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
Single function of several variables with single derivative
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 derivative
If there are several unknown functions to be determined and several variables such that
the system of Euler–Lagrange equations is
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
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 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
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 manifolds
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:
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