Strong coupling expansions for nonintegrable hamiltonian systems

Journal Article · Wed Sep 01 00:00:00 EDT 1982 · Physica. D, Nonlinear Phenomena

DOI:https://doi.org/10.1016/0167-2789(82)90025-2· OSTI ID:1109130

Abarbanel, Henry D. I. ^[1]; Crawford, John David ^[1]

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1. Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Univ. of California, Berkeley, CA (United States)

In this paper, we present a method for studying nonintegrable Hamiltonian systems H(I,θ) = H₀(I) + kH₁(I,θ) (I, θ are action-angle variables) in the regime of large k. Our central tool is the conditional probability P(I,θ,t | I₀,θ₀,t₀) that the system is at I. θ at time t given that it resided at I₀, θ₀ at t₀. An integral representation is given for this conditional probability. By discretizing the Hamiltonian equations of motion in small time steps, ϵ, we arrive at a phase volume-preserving mapping which replaces the actual flow. When the motion on the energy surface E = H(I,θ) is bounded we are able to evaluate physical quantities of interest for large k and fixed ϵ. We also discuss the representation of P (I,θ,t | I₀,θ₀t₀) when an external random forcing is added in order to smooth the singular functions associated with the deterministic flow. Explicit calculations of a “diffusion” coefficient are given for a non-integrable system with two degrees of freedom. Finally, the limit ϵ → 0, which returns us to the actual flow, is subtle and is discussed.

• Research Organization: Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
• Sponsoring Organization: USDOE
• DOE Contract Number: W-7405-ENG-48
• OSTI ID: 1109130
• Report Number(s): LBL-11887
• Journal Information: Physica. D, Nonlinear Phenomena, Vol. 5, Issue 2-3; ISSN 0167-2789
• Publisher: Elsevier
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS