Euler–Lagrange equation
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Euler–Lagrange equation(only for study)
In calculus of variations, the Euler–Lagrange equation, Euler's equation,^{[1]} or Lagrange's equation, is a 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 where a differentiable function attains its local extrema, itsderivative 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 theaction 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.
Contents[hide] 
[edit]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]}
[edit]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:
 q is the function to be found:
 such that q is differentiable, q(a) = x_{a}, and q(b) = x_{b};
 q′ is the derivative of q:
 TX being the tangent bundle of X (the space of possible values of derivatives of functions with values in X);
 L is a realvalued function with continuous first partial derivatives:
The Euler–Lagrange equation, then, is given by
where L_{x} and L_{v} denote the partial derivatives of L with respect to the second and third arguments, respectively.
If the dimension of the space X is greater than 1, this is a system of differential equations, one for each component:

[show]Derivation of onedimensional Euler–Lagrange equation

[show]Alternate derivation of onedimensional Euler–Lagrange equation
[edit]Examples
A standard example is finding the realvalued function on the interval [a, b], such that f(a) = c and f(b) = d, the length of whose graph is as short as possible. The length of the graph of f is:
the integrand function being L(x, y, y′) = √1 + y′ ² evaluated at (x, y, y′) = (x, f(x), f′(x)).
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.
[edit]Classical mechanics
[edit]Basic method
To find the equations of motions for a given system, one only has to follow these steps:
 From the kinetic energy , and the potential energy , compute the Lagrangian .
 Compute .
 Compute and from it, . It is important that be treated as a complete variable in its own right, and not as a derivative.
 Equate . This is the Euler–Lagrange equation.
 Solve the differential equation obtained in the preceding step. At this point, is treated "normally". Note that the above might be a system of equations and not simply one equation.
[edit]Particle in a conservative force field
The motion of a single particle in a conservative force field (for example, the gravitational force) can be determined by requiring the action to be stationary, by Hamilton's principle. The action for this system is
where x(t) is the position of the particle at time t. The dot above is Newton's notation for the time derivative: thus ẋ(t) is the particle velocity, v(t). In the equation above, L is theLagrangian (the kinetic energy minus the potential energy):
where:
 m is the mass of the particle (assumed to be constant in classical physics);
 v_{i} is the ith component of the vector v in a Cartesian coordinate system (the same notation will be used for other vectors);
 U is the potential of the conservative force.
In this case, the Lagrangian does not vary with its first argument t. (By Noether's theorem, such symmetries of the system correspond to conservation laws. In particular, the invariance of the Lagrangian with respect to time implies the conservation of energy.)
By partial differentiation of the above Lagrangian, we find:
where the force is F = −∇U (the negative gradient of the potential, by definition of conservative force), and p is the momentum. By substituting these into the Euler–Lagrange equation, we obtain a system of secondorder differential equations for the coordinates on the particle's trajectory,
which can be solved on the interval [t_{0}, t_{1}], given the boundary values x_{i}(t_{0}) and x_{i}(t_{1}). In vector notation, this system reads
or, using the momentum,
which is Newton's second law.
[edit]Variations for several functions, several variables, and higher derivatives
[edit]Single function of single variable with higher derivatives
The stationary values of the functional
can be obtained from the Euler–Lagrange equation^{[3]}
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.
[edit]Several functions of one variable
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^{[4]}
[edit]Single function of several variables
A multidimensional generalization comes from considering a function on n variables. If Ω is some surface, then
is extremized only if f satisfies the partial differential equation
When n = 2 and is the energy functional, this leads to the soapfilm minimal surface problem.
[edit]Several functions of several variables
If there are several unknown functions to be determined and several variables such that
the system of Euler–Lagrange equations is^{[3]}
[edit]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 x_{1} and x_{2} and if the functional depends on higher derivatives of f up to nth order such that
then the Euler–Lagrange equation is^{[3]}
[edit]Notes
 ^ Fox, Charles (1987). An introduction to the calculus of variations. Courier Dover Publications. ISBN 9780486654997.
 ^ A short biography of Lagrange
 ^ ^{a} ^{b} ^{c} Courant, R. and Hilbert, D., 1953, Methods of Mathematical Physics: Vol I, Interscience Publishers, New York.
 ^ Weinstock, R., 1952, Calculus of Variations With Applications to Physics and Engineering, McGrawHill Book Company, New York.
[edit]References
 Hazewinkel, Michiel, ed. (2001), "Lagrange equations (in mechanics)", Encyclopedia of Mathematics, Springer, ISBN 9781556080104
 Weisstein, Eric W., "EulerLagrange Differential Equation" from MathWorld.
 Calculus of Variations, PlanetMath.org.
 Izrail Moiseevich Gelfand (1963). Calculus of Variations. Dover. ISBN 0486414485.
[edit]See also
Look up Euler–Lagrange equation in Wiktionary, the free dictionary. 