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Title: Strong coupling expansions for nonintegrable hamiltonian systems

Abstract

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 | I00,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 | I00t0) 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.

Authors:
 [1];  [1]
  1. Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Univ. of California, Berkeley, CA (United States)
Publication Date:
Research Org.:
Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1109130
Report Number(s):
LBL-11887
Journal ID: ISSN 0167-2789
DOE Contract Number:  
W-7405-ENG-48
Resource Type:
Journal Article
Journal Name:
Physica. D, Nonlinear Phenomena
Additional Journal Information:
Journal Volume: 5; Journal Issue: 2-3; Journal ID: ISSN 0167-2789
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

Citation Formats

Abarbanel, Henry D. I., and Crawford, John David. Strong coupling expansions for nonintegrable hamiltonian systems. United States: N. p., 1982. Web. doi:10.1016/0167-2789(82)90025-2.
Abarbanel, Henry D. I., & Crawford, John David. Strong coupling expansions for nonintegrable hamiltonian systems. United States. https://doi.org/10.1016/0167-2789(82)90025-2
Abarbanel, Henry D. I., and Crawford, John David. 1982. "Strong coupling expansions for nonintegrable hamiltonian systems". United States. https://doi.org/10.1016/0167-2789(82)90025-2. https://www.osti.gov/servlets/purl/1109130.
@article{osti_1109130,
title = {Strong coupling expansions for nonintegrable hamiltonian systems},
author = {Abarbanel, Henry D. I. and Crawford, John David},
abstractNote = {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.},
doi = {10.1016/0167-2789(82)90025-2},
url = {https://www.osti.gov/biblio/1109130}, journal = {Physica. D, Nonlinear Phenomena},
issn = {0167-2789},
number = 2-3,
volume = 5,
place = {United States},
year = {1982},
month = {9}
}