Hamilton-Jacobi Equation: an international process
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Additional resources for Hamilton-Jacobi Equation: A Global Approach
X"-l,O) = f(x1,. . ,x"- l ) 6. The Cauchy-Kowaleski Theorem 19 has an analytic solution, u(x), in a neighborhood of the initial manifold xn = 0. Outline ofproof First note that SF/@" # 0 is equivalent to the problem being noncharacteristic and implies that F = 0 may be solved for p": pn = G(x, u, p l . . , p"- Now, for a point y with y" = 1). 1) 0, consider the formal power series: u(x) = c C,(x - y)", where the summation is over all n-vectors o! whose components are nonnegative integers and z' is defined to be the product of all the zi raised to the corresponding power d.
Local variational maximums, with concavity in place of convexity, also solve the Hamilton-Jacobi initial value problem locally. Much more information is available from the classical theory. For example, in cases of strict convexity, the classical theory of conjugate points shows that the smooth solution is impossible only along lower dimensional manifolds where certain determinants vanish. However this topic would lead us too far astray, so we refer the reader to the classical texts. Bliss  has a readable account of conjugate points in the plane in his first chapter.
5. The Complete Integral 17 For a general solution, let b = g(a),and eliminate a from u = ax and 0 =X + g(a)y + h(a, + g’y + ha + hbg’. A singular integral is obtained by envelopes from + by + h(a, b), 0 = x + ha, u = ax and 0=y + hb. The singular integral may also be obtained directly from Clairaut’s equation, where now p = u,, q = up. by u = XP 0=x + yq + h(p, 41, + h,, and 0 = y f h,. Clearly both methods yield the same singular integral. See Courant [77, p. 271 for this and other examples. 4 The Hamilton-Jacobi equation has no singular integrals since differentiation of u, + H ( t , x, u,) = 0 with respect to u, yields 1, which clearly can never vanish.