Tree graphs and the solution to the Hamilton--Jacobi equation

Journal Article · Wed Jan 01 04:00:00 EST 1986 · J. Math. Phys. (N.Y.); (United States)

DOI:https://doi.org/10.1063/1.527305· OSTI ID:6384378

Molzahn, F H; Osborn, T A

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)p/sup 2/+v(x,t) have complete integrals S that are analytic functions of the inverse mass parameter m/sup -1/ in a punctured disk about the origin. If v(x,t) is bounded, C/sup infinity/ in the x variable, and has controlled x-derivative growth, then the coefficients of the Laurent expansion of S about m/sup -1/ = 0 may be expressed in terms of gradient structures associated with tree graphs. This series expansion for S(x,t; y,t/sub 0/) converges absolutely, and uniformly for all x, y for time displacements Vertical Bart-t/sub 0/Vertical Bar

• Research Organization: Department of Physics and Astronomy, University of Maryland, College Park, Maryland 20742 and Department of Physics, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada
• OSTI ID: 6384378
• Journal Information: J. Math. Phys. (N.Y.); (United States), Journal Name: J. Math. Phys. (N.Y.); (United States) Vol. 27:1; ISSN JMAPA
• Country of Publication: United States
• Language: English

Related Subjects

657002* -- Theoretical & Mathematical Physics-- Classical & Quantum Mechanics
658000 -- Mathematical Physics-- (-1987)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ANALYTICAL SOLUTION
DIFFERENTIAL EQUATIONS
EQUATIONS
HAMILTON-JACOBI EQUATIONS
HAMILTONIANS
INTEGRALS
INTERACTIONS
MATHEMATICAL OPERATORS
PARTIAL DIFFERENTIAL EQUATIONS
PARTICLE INTERACTIONS
POTENTIALS
QUANTUM OPERATORS
SEMICLASSICAL APPROXIMATION
SERIES EXPANSION
SYMMETRY
VARIATIONAL METHODS