euler–lagrange_equation.doc

Euler–Lagrange equationJump to: navigation, search In calculus of variations, the Euler–Lagrange equation, 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, 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 (see, for example, the "Field theory" section below).HistoryThe 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. The two 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.[1]StatementThe Euler–Lagrange 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 Euler–Lagrange 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 equationAlternate derivation of one-dimensional Euler-Lagrange equationExamplesA 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, 2'1)',(yyxLy′) = ( x, f(x), f′( x)).The partial derivatives of L are:By substituting these into the Euler–Lagrange 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 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. 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 Hamilton's principle. The action for this system iswhere 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 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 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 subst。