Calculus of Variations
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second variation δ2J[h] be nonnegative. In the case of a functional of the form (4), we can use formula (10) to establish a necessary condition for the second variation to be nonnegative. The argument goes as follows: Consider the quadratic functional (10) for functions h(x) satisfying the condition h(a) = 0. With this condition, the function h(x) will be small in the interval [a, b] if its derivative h′(x) is small in [a, b]. However, the converse is not true, i.e., we can construct a function
order n, and the vectors h(i)′(a) are the rows of the unit matrix of order n. 21 It can be shown that this is compatible with W being symmetric, even when Fyy′ fails to be symmetric and (62) is replaced by a more general equation (H. Niemeyer, private communication). 22 The fact that det P does not vanish in [a, b] is tacitly assumed, but this is guaranteed by the positive definiteness of P (cf. footnote 9, p 108). 23 Equations (70)–(73) closely resemble equations (39)–(42) of Sec. 27, except
respect to x, we obtain i.e., Thus, the system (7) is a consequence of the system (6) if and only if (8) holds. Example 1. Consider a single linear differential equation The corresponding Hamilton-Jacobi system reduces to a single equation i.e., The set of solutions of (10) depends on an arbitrary function, and according to the theorem, each of these solutions is a field for equation (9). The simplest solutions of (10) are those that are linear in y: Substituting (11) into
symbol ∼ denotes equality except for terms of order higher than 1 relative to ε. But Δu ∼ Δu, since δu is the principal part of Δu, and hence Moreover, since (80) also implies Example. Let u be a function of a single independent variable x, and let (71) be the transformation i.e., a counterclockwise rotation of the xu-plane about the small angle α = ε. As shown in Figure 10, (82) carries the point (x,u(x)) on the curve Γ with equation u = u(x) into the point (x*, u*(x*)) on its image
boundary conditions y(a) = A, y(b) = B. The problem usually considered in the theory of differential equations is that of finding a solution which is defined in the neighborhood of some point and satisfies given initial conditions (Cauchy’s problem). However, in solving Euler’s equation, we are looking for a solution which is defined over all of some fixed region and satisfies given boundary conditions. Therefore, the question of whether or not a certain variational problem has a solution does