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Calculus of Variations

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Calculus of Variations Calculus of Variations In this chapter, we discuss the basics of Calculus of Variations. This will provide us with the mathe- matical language|and the key tools|necessary for introducing and utilizing Lagrangian formalizm. Functionals The central notion of the Calculus of Variations is a functional, a real-valued function defined for a certain class of real- or complex-valued functions of one or many variables. The functional assigns a real value to one or more functions of the given class. We will be interested in the functionals of the following form Z xb F [f] = g(f; fx; x) dx : (1) xa Here f|the argument of the functional F |is a differentiable function, x is the argument of the 0 function f, fx ≡ f (x) is the derivative of the function f, and g is a certain fixed differentiable function of three variables. For the sake of briefness, we consider the case of one real function of one variable. Apart from being differentiable, the function f(x) is typically subject to certain constraints. For example, f(xa) = ya ; f(xb) = yb ; (2) with ya and yb certain fixed numbers. Our definition (1) is straightforwardly generalized to the cases of more than one function f, more than one variable x, and higher-order (partial) derivatives. Example 1. Length of a line in a plane. Suppose we have a smooth line in the xy-plane passing through two given points, (xa; ya) and (xb; yb), and specified by the equation y = f(x) : (3) The length of the line is a functional of f of the form (1). Indeed, Z Z q Z q Z xb q 2 2 2 2 2 l[f] = dl = (dx) + (dy) = (dx) + (fxdx) = 1 + (fx) dx : (4) xa We see that l[f] has the form (1) with q 2 g(f; fx; x) = 1 + (fx) : (5) In this example, the function g does not depend explicitly on f and x, which reflects two translational symmetries: f ! f + const and x ! x + const. Example 2. Potential energy of a flexible cable suspended from two fixed points. Choose the y axis perpendicular to the surface of the Earth, the cable being represented by a curve (3) in the xy plane, with the constraint (2). The potential energy of the element dl is proportional to its hight y and its length dl. Hence, the total potential energy is given by the integral Z Z xb q 2 U[f] = y dl = f 1 + (fx) dx : (6) xa 1 We do not care about the units, and thus set the pre-factor equal to unity. We see that U[f] has the form (1) with q 2 g(f; fx; x) = f 1 + (fx) : (7) Here the function g explicitly depends on f and fx, but not on x (reflecting the translational symmetry x ! x + const. Example 3. Length of a line in a surface. Suppose we have a smooth line L in the smooth surface S, passing through two given points, (xa; ya; za) and (xb; yb; zb). Let the surface S and the line L be specified by the equations z = s(x; y) (the surface) ; (8) y = f(x) ; z = s(x; f(x)) (the line in the surface) : (9) Let us show that the length of the line is the functional of the form (1). Introducing convenient notation @s(x; y) @s(x; y) s = ; s = ; (10) x @x y @y and using dy = fx dx ; dz = sx dx + sy dy = sx dx + sy fx dx ; (11) we get Z Z q Z xb 2 2 2 l[f] = dl = (dx) + (dy) + (dz) = g(f; fx; x) dx ; (12) xa where q 2 2 g(f; fx; x) = 1 + fx + [sx(x; f) + sy(x; f) fx] : (13) Extrema. Euler's equation A typical problem arising in connection with a functional F [f] is the problem of finding a function f that minimizes (or maximizes) the functional. With our examples, this problem comes from natural questions: What is the shortest distance between two fixed points in a given surface? What is the shape of the suspended cable? Calculus of Variations provides mathematical tools for solving the problem. Suppose the function f is a (local) minimum/maximum of the functional F . Then, for any small variation of the function f, the variation of the functional has to be sign-definite. Let us see what are the implications of this fact. We introduce the symbol δf(x) for an infinitesimal variation of the function f and the symbol δF for corresponding variation of the functional F . That is δF = F [f + δf] − F [f] : (14) With a simple observation that 0 f ! f + δf ) fx ! fx + (δf) ; (15) we have Z xb Z xb 0 δF = g( f + δf; fx + (δf) ; x) dx − g(f; fx; x) dx : (16) xa xa Then we use the infinitesimal smallness of δf: 0 @g @g 0 g( f + δf; fx + (δf) ; x) ! g(f; fx; x) + δf + (δf) : (17) @f @fx 2 Z xb @g Z xb @g δF = δf dx + (δf)0 dx : (18) xa @f xa @fx In the second integral, we integrate by parts, with Eq. (2) taken into account [implying δf(xa) = δf(xb) = 0]: Z xb @g Z xb d @g (δf)0 dx = − dx (δf) : (19) xa @fx xa dx @fx We get Z xb δF = A(x) δf(x) dx ; (20) xa where @g d @g A(x) = − : (21) @f dx @fx Equation (20) generalizes the notion of the differential of a multivariable function. Indeed, if we discretize the variable x|so that it becomes a label for discrete variables f(x), then δf(x) and A(x) acquire (respectively) the meaning of the differential of the variable f(x) and corresponding partial derivative. In view of this close analogy with the multivariable calculus, A(x) is called variational derivative of F with respect to f at the point x, with a convenient symbolic notation (note the importance of keeping the \label" x in the denominator; the symbolic character of the notation is already clearly from the dimensionality of the variational derivative: [A] = [F ][f]−1[x]−1 6= [F ][f]−1) δF A(x) ≡ (symbolic notation for the variational derivative): (22) δf(x) What really distinguishes the notion of variational derivative from the generic notion of partial deriva- tive is the relation (21) playing the key part in the Calculus of Variations. According to Eq. (20), the variation δF is a linear functional of δf. The linearity means that the sign of δF changes upon the transformation δf ! −δf. Meanwhile, the condition of mini- mum/maximum requires that the sign of variation not change. This is only possible if A(x) ≡ 0, and we arrive at the celebrated Euler's equation, the central result of the Calculus of Variations: @g d @g − = 0 : (23) @f dx @fx As a simple illustration, let us make sure that the shortest distance between two points is a straight line. Applying Eq. (23) to the function g of the Example 1, we get 00 f (x) = 0 ) f(x) = C1x + C2 : (24) [The constants C1 and C2 are then fixed by the boundary conditions (2).] Alternate form of Euler's equation If fx 6= 0, then Euler's equation is equivalent to d @g @g g − fx − = 0 : (25) dx @fx @x Indeed, by the chain rule we have dg @g @g @g = fx + fxx + : (26) dx @f @fx @x 3 Taking into account that d @g @g d @g fx = fxx + fx ; (27) dx @fx @fx dx @fx we then get d @g @g @g d @g g − fx = + fx − ; (28) dx @fx @x @f dx @fx and see that Eq. (25) is equivalent to @g d @g fx − = 0 ; (29) @f dx @fx which, in its turn, is equivalent to the original Euler's equation (23) if fx 6= 0. To put it differently, any solution of Eq. (23) is simultaneously a solution of Eq. (25), but, in contrast to Eq. (23), equation (25) has also a trivial solution f = const. The alternate form of Euler's equation is very important in the case when g does not depend explicitly on x: g = g(f; fx). Here Eq. (25) simplifies to d @g g − fx = 0 ; (30) dx @fx thus yielding the first integral of the Euler's equation: @g g − fx = const : (31) @fx Soap film. The equilibrium shape of a soap film corresponds to a (local) minimum of its surface energy, the latter being directly proportional to the surface area. Hence, to find the equilibrium shape of a soap film one has to find the (local) minimum of the surface area under the appropriate boundary conditions. Here we confine ourselves with the simplest example when the film is axially symmetric being stretched between two parallel coaxial wire circles. Let the center of the first circle be at x = 0 and the center of the second circle be at x = h; the two radii are Ra and Rb, respectively; x is the symmetry axis. The surface is parameterized by the radius r at a given x: r = f(x) ; f(0) = Ra ; f(h) = Rb : (32) The surface area, A, is then a functional of f: Z Z q Z h q 2 2 2 A[f] = 2πr dl = 2πr (dx) + (dr) = 2π f 1 + fx dx : (33) 0 We see that q 2 g = 2π f 1 + fx (34) does not depend explicitly on x (which reflects the translational symmetry of the problem) and we can enjoy Euler's equation in the form of its first integral (31).
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