Euler–Lagrange equation.doc

EulerLagrange equationFrom Wikipedia, the free encyclopediaJump to: navigation, search In calculus of variations, the EulerLagrange equation, or Lagranges 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 EulerLagrange equation is useful for solving optimization problems in which, given some functional, one seeks the function minimizing (or maximizing) it. This is analogous to Fermats theorem in calculus, stating that where a differentiable function attains its local extrema, its derivative is zero.In Lagrangian mechanics, because of Hamiltons principle of stationary action, the evolution of a physical system is described by the solutions to the EulerLagrange equation for the action of the system. In classical mechanics, it is equivalent to Newtons 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 (see, for example, the Field theory section below).Contents1 History 2 Statement 3 Examples o 3.1 Classical mechanics 3.1.1 Basic method 3.1.2 Particle in a conservative force field o 3.2 Field theory 4 Variations for several functions, several variables, and higher derivatives o 4.1 Single function of single variable with higher derivatives o 4.2 Several functions of one variable o 4.3 Single function of several variables o 4.4 Several functions of several variables o 4.5 Single function of two variables with higher derivatives 5 Notes 6 References 7 See also HistoryThe EulerLagrange 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. The two further developed Lagranges 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 1766.1StatementThe EulerLagrange equation is an equation satisfied by a function q of a real argument t which is a stationary point of the functionalwhere: q is the function to be found: such that q is differentiable, q(a) = xa, and q(b) = xb; 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 real-valued function with continuous first partial derivatives: The EulerLagrange equation, then, is the ordinary differential equationwhere Lx and Lv 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:Derivation of one-dimensional Euler-Lagrange equationThe derivation of the one-dimensional EulerLagrange 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 f which satisfies the boundary conditions f(a) = A, f(b) = B, and which extremizes the cost functionalWe assume that F has continuous first partial derivatives. A weaker assumption can be used, but the proof becomes more difficult.If f extremizes the cost functional subject to the boundary conditions, then any slight perturbation of f that preserves the boundary values must either increase J (if f is a minimizer) or decrease J (if f is a maximizer).Let g(x) = f(x)+(x) be such a perturbation of f, where (x) is a differentiable function satisfying (a) = (b) = 0. Then defineWe now wish to calculate the total derivative of J with respect to or the first variation of J.It follows from the total derivative thatSoWhen = 0 we have g = f and since f is an extreme value it follows that , i.e.The next crucial step is to use integration by parts on the second term, yieldingUsing the boundary conditions on , we get thatApplying the fundamental lemma of calculus of variations now yields the EulerLagrange equationAlternate derivation of one-dimensional Euler-Lagrange equationGiven a functionalon C1(a,b) with the boundary conditions y(a) = A and y(b) = B, we proceed by approximating the extremal curve by a polygonal line with n segments and passing to the limit as the number of segments grows arbitrarily large.Divide the interval a,b into n + 1 equal segments with endpoints and let t = tk tk 1. Rather than a smooth function y(t) we consider the polygonal line with vertices , where y0 = A and yn + 1 = B. Accordingly, our functional becomes a real function of n variables given byExtremals of this new functional defined on the discrete points correspond to points whereEvaluating this partial derivative gives thatDividing the above equation by t givesand taking the limit as of the right-hand side of this expression yieldsThe term denotes the variational derivative of the functional J, and a necessary condition for a differentiable functional to have an extremum on some function is that its variational derivative at that function vanishes.ExamplesA standard example is finding the real-valued 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 evaluated at (x, y, y) = (x, f(x), f(x).The partial derivatives of L are:By substituting these into the EulerLagrange equation, we obtainthat is, the function must have constant first derivative, and thus its graph is a straight line.Classical mechanicsBasic methodTo find the equations of motions for a given system, one only has to follow these steps: From the kinetic energy T, and the potential energy V, compute the Lagrangian L = T V. 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, of course, the EulerLagrange 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. Particle in a conservative force fieldThe 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 Hamiltons principle. The action for this system iswhere x(t) is the position of the particle at time t. The dot above is Newtons notation for the time derivative: thus (t) is the particle velocity, v(t). In the equation above, L is the Lagrangian (the kinetic energy minus the potential energy):where: m is the mass of the particle (assumed to be constant in classical physics); vi is the i-th 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 Noethers 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 EulerLagrange equation, we obtain a system of second-order differential equations for the coordinates on the particles trajectory,which can be solved on the interval t0, t1, given the boundary values xi(t0) and xi(t1). In vector notation, this system readsor, using the momentum,which is Newtons second law.Field theoryThis section contains too much jargon and may need simplification or further explanation. Please discuss this issue on the talk page, and/or remove or explain jargon terms used in the article. Editing help is available. (December 2009)Field theories, both classical field theory and quantum field theory, deal with continuous coordinates, and like classical mechanics, has its own EulerLagrange equation of motion for a field,where is the field, and is a vector differential operator: Note: Not all classical fields are assumed commuting/bosonic variables, (like the Dirac field, the Weyl field, the Rarita-Schwinger field) are fermionic and so, when trying to get the field equations from the Lagrangian density, one must choose whether to use the right or the left derivative of the Lagrangian density (which is a boson) with respect to the fields and their first space-time derivatives which are fermionic/anticommuting objects.There are several examples of applying the EulerLagrange equation to various Lagrangians: Dirac equation; Proca equation; electromagnetic tensor; Kortewegde Vries equation; quantum electrodynamics. Variations for several functions, several variables, and higher derivativesSingle function of single variable with higher derivativesThe stationary values of the functionalcan be obtained from the Euler-Lagrange equation2Several functions of one variableIf the problem involves finding several functions () of a single independent variable (x) that define an extremum of the functionalthen the corresponding Euler-Lagrange equations are2Single function of several variablesA multi-dimensional generalization comes from considering a function on n variables. If is some surface, thenis extremized only if f satisfies the partial differential equat