Representations of one-dimensional Hamiltonians in terms of their invariants

Journal Article · Sat Feb 01 04:00:00 EST 1992 · Journal of Mathematical Physics (New York); (United States)

DOI:https://doi.org/10.1063/1.529794· OSTI ID:5399540

Lewis, H R ^[1]; Leach, P G.L. ^[2]; Bouquet, S; Feix, M R ^[3]

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1. Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States)
2. CNLS-CAMS, University of the Witwatersrand, Johannesburg, Wits Post Office, 2050 (South Africa)
3. Physique Mathematique, Modelisation et Simulation, C.N.R.S., 45071 Orleans Cedex 2 (France)

A general formalism for representing the Hamiltonian of a system with one degree of freedom in terms of its invariants is developed. Those Hamiltonians {ital H}({ital q},{ital p},{ital t}) are derived for which any particular function {ital I}({ital q},{ital p},{ital t}) is an invariant. For each of those Hamiltonians, a function canonically conjugate to {ital I}({ital q},{ital p},{ital t}) is derived which is also an invariant of {ital H}({ital q},{ital p},{ital t}). The formalism is also presented for the case in which {ital I}({ital q},{ital p},{ital t}) is expressed as a function of two canonically conjugate functions. The formalism is illustrated by applying it to the case of a particle moving in a time-dependent potential. Some earlier results are recovered and an invariant is found for a new potential. Lines for further study are outlined that may be fruitful for finding more examples of integrable systems.

• OSTI ID: 5399540
• Journal Information: Journal of Mathematical Physics (New York); (United States), Journal Name: Journal of Mathematical Physics (New York); (United States) Vol. 33:2; ISSN 0022-2488; ISSN JMAPA
• Country of Publication: United States
• Language: English

Related Subjects

661100* -- Classical & Quantum Mechanics-- (1992-)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ANALYTICAL SOLUTION
CANONICAL TRANSFORMATIONS
CLASSICAL MECHANICS
CONSERVATION LAWS
DIFFERENTIAL EQUATIONS
EQUATIONS
EQUATIONS OF MOTION
HAMILTONIANS
MATHEMATICAL OPERATORS
MATHEMATICAL SPACE
MECHANICS
PARTIAL DIFFERENTIAL EQUATIONS
PHASE SPACE
QUADRATURES
QUANTUM OPERATORS
SPACE
TRAJECTORIES
TRANSFORMATIONS