Tree graphs and the solution to the Hamilton–Jacobi equation

Abstract

A combinatorial method is used to construct solutions of the Hamilton–Jacobi equation. An exact expression for Hamilton’s principal function S is obtained for classical systems of finitely many particles interacting via a certain class of time-dependent potentials. If x, p, and t are the position, momentum, and time variables for N point particles of mass m, it is shown that Hamiltonians of the form H(x,p,t)=(1/2m)p2+v(x,t) have complete integrals S that are analytic functions of the inverse mass parameter m−1 in a punctured disk about the origin. If v(x,t) is bounded, C∞ in the x variable, and has controlled x-derivative growth, then the coefficients of the Laurent expansion of S about m−1=0 may be expressed in terms of gradient structures associated with tree graphs. This series expansion for S(x,t; y,t0) converges absolutely, and uniformly for all x, y for time displacements ‖t−t0‖<T≡2K−1(m/eU)1/2, where K and U are bounds associated with the space derivatives of the potential. For ‖t−t0‖<T, the classical path (from any initial space-time configuration y,t0 to any final configuration x,t) induced by S is unique, passes through no conjugate points, and furnishes the action functional with a strong minimum. The local solution S given above may be used to obtain the classical trajectories for arbitrarily large times.