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Journal of Mathematical Physics / Volume 27 / Issue 1

J. Math. Phys. 27, 88 (1986); http://dx.doi.org/10.1063/1.527305 (12 pages)

Tree graphs and the solution to the Hamilton–Jacobi equation

F. H. Molzahn and T. A. Osborn

Department of Physics and Astronomy, University of Maryland, College Park, Maryland 20742 and Department of Physics, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada

(Received 21 February 1985; accepted 23 August 1985)

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.

KEYWORDS and PACS

Keywords

HAMILTON−JACOBI EQUATIONS, ANALYTICAL SOLUTION, POTENTIALS, PARTICLE INTERACTIONS, HAMILTONIANS, SERIES EXPANSION, SYMMETRY, SEMICLASSICAL APPROXIMATION, INTEGRALS, VARIATIONAL METHODS

PACS

  • 02.30.Jr

    Partial differential equations

  • 45.05.+x

    General theory of classical mechanics of discrete systems

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.527305

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

For access to fully linked references, you need to log in.
    T. A. Osborn, R. A. Corns, and Y. Fujiwara, J. Math. Phys. 26, 453 (1985JMAPAQ000026000003000453000001).

    E. P. Wigner, Phys. Rev. 40, 749 (1932).

    J. G. Kirkwood, Phys. Rev. 44, 31 (1933).

    Y. Fujiwara, T. A. Osborn, and S. F. J. Wilk, Phys. Rev. A 25, 14 (1982).



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