Euler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations is 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, 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. Statement. Let formula_1 be a mechanical system with formula_2 degrees of freedom. Here formula_3 is the configuration space and formula_4 the "Lagrangian", i.e. a smooth real-valued function such that formula_5 and formula_6 is an formula_2-dimensional "vector of speed". (For those familiar with differential geometry, formula_3 is a smooth manifold, and formula_9 where formula_10 is the tangent bundle of formula_11 Let formula_12 be the set of smooth paths formula_13 for which formula_14 and formula_15 The action functional formula_16 is defined via formula_17 A path formula_18 is a stationary point of formula_19 if and  only if \quad i = 1, \dots, n.</math> Here, formula_20 is the time derivative of formula_21 Using the boundary conditions formula_22, formula_23 Applying the fundamental lemma of calculus of variations now yields the Euler–Lagrange equation formula_24 Dividing the above equation by formula_25 gives formula_26 and taking the limit as formula_27 of the right-hand side of this expression yields formula_28 The left hand side of the previous equation is the functional derivative formula_29 of the functional formula_30. 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. Example. 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 . 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. Generalizations. 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 formula_36 derivatives (i.e. for all formula_37). The endpoint values of the highest derivative formula_38 remain flexible. Several functions of single variable with single derivative. If the problem involves finding several functions (formula_39) of a single independent variable (formula_40) 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 formula_43 is  some surface, then is extremized only if "f" satisfies the partial differential equation When "n" = 2 and functional formula_46 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 "x" and "x" 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 formula_52 are indices that span the number of variables, that is, here they go from 1 to 2. Here summation over the formula_52 indices is only over formula_54 in order to avoid counting the same partial derivative multiple times, for example formula_55 appears only once in the previous equation. Several functions of several variables with higher derivatives. If there are "p" unknown functions "f" to be determined that are dependent on "m" variables "x" ... "x" and if the functional depends on  higher derivatives of the "f" up to "n"-th order such that where formula_52 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 formula_52 is avoiding counting the same derivative formula_60 several times, just as in the previous subsection. This can be expressed more compactly as Generalization to manifolds. Let formula_62 be a smooth manifold, and let formula_63 denote the space of smooth functions formula_64. Then, for functionals formula_65 of the form where formula_67 is the Lagrangian, the statement formula_68 is equivalent to  the statement that, for all formula_69, each coordinate frame trivialization formula_70 of a neighborhood of formula_71 yields the following formula_72 equations: