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Title: Rotation number of integrable symplectic mappings of the plane

Abstract

Symplectic mappings are discrete-time analogs of Hamiltonian systems. They appear in many areas of physics, including, for example, accelerators, plasma, and fluids. Integrable mappings, a subclass of symplectic mappings, are equivalent to a Twist map, with a rotation number, constant along the phase trajectory. In this letter, we propose a succinct expression to determine the rotation number and present two examples. Similar to the period of the bounded motion in Hamiltonian systems, the rotation number is the most fundamental property of integrable maps and it provides a way to analyze the phase-space dynamics.

Authors:
 [1];  [1];  [2]
  1. Fermilab
  2. Oak Ridge
Publication Date:
Research Org.:
Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States); Fermi National Accelerator Lab. (FNAL), Batavia, IL (United States)
Sponsoring Org.:
USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)
OSTI Identifier:
1418136
Report Number(s):
FERMILAB-PUB-17-305-AD-APC-CD
1649868
DOE Contract Number:
AC02-07CH11359
Resource Type:
Journal Article
Resource Relation:
Journal Name: TBD
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Arnold-Liouville theorem; KAM theory; discrete dynamical systems; integrability; McMillan map; symplectic topology; Poincare rotation number

Citation Formats

Zolkin, Timofey, Nagaitsev, Sergei, and Danilov, Viatcheslav. Rotation number of integrable symplectic mappings of the plane. United States: N. p., 2017. Web.
Zolkin, Timofey, Nagaitsev, Sergei, & Danilov, Viatcheslav. Rotation number of integrable symplectic mappings of the plane. United States.
Zolkin, Timofey, Nagaitsev, Sergei, and Danilov, Viatcheslav. Tue . "Rotation number of integrable symplectic mappings of the plane". United States. doi:. https://www.osti.gov/servlets/purl/1418136.
@article{osti_1418136,
title = {Rotation number of integrable symplectic mappings of the plane},
author = {Zolkin, Timofey and Nagaitsev, Sergei and Danilov, Viatcheslav},
abstractNote = {Symplectic mappings are discrete-time analogs of Hamiltonian systems. They appear in many areas of physics, including, for example, accelerators, plasma, and fluids. Integrable mappings, a subclass of symplectic mappings, are equivalent to a Twist map, with a rotation number, constant along the phase trajectory. In this letter, we propose a succinct expression to determine the rotation number and present two examples. Similar to the period of the bounded motion in Hamiltonian systems, the rotation number is the most fundamental property of integrable maps and it provides a way to analyze the phase-space dynamics.},
doi = {},
journal = {TBD},
number = ,
volume = ,
place = {United States},
year = {Tue Apr 11 00:00:00 EDT 2017},
month = {Tue Apr 11 00:00:00 EDT 2017}
}
  • We study the algebra Sp([ital n],[ital R]) of the symplectic model, in particular for the cases [ital n]=1,2,3, in a new way. Starting from the Poisson-bracket realization we derive a set of partial differential equations for the generators as functions of classical canonical variables. We obtain a solution to these equations that represents the classical limit of a boson mapping of the algebra. We show further that this mapping plays a fundamental role in the collective description of many-fermion systems whose Hamiltonian may be approximated by polynomials in the associated algebra, as is done in the simplest versions of themore » symplectic model. The relationship to the collective dynamics is formulated as a theorem that associates the mapping with an exact solution of the time-dependent Hartree approximation. This solution determines a decoupled classical symplectic manifold, thus satisfying the criteria that define an exactly solvable model in the theory of large amplitude collective motion. The models thus obtained also provide a test of methods for constructing an approximately decoupled manifold in fully realistic cases. We show that an algorithm developed in one of our earlier works reproduces the main results of the theorem.« less
  • Several sets of authors have recently developed a systematic method of obtaining boson realizations of the Lie algebras of the symplectic groups, utilizing the subgroup chain Sp(2..lambda..)containsU(..lambda..). Physical motivation stems both from the Sp(6,R-italic) collective model (noncompact algebra) and various shell-model applications (compact algebra). We present here a number of the simplest examples of interest for shell-model application. These include the L-italicS-italic coupled (identical nucleon) shell-model algebra (for application to the pseudo-SU(3) model) and the Sp(6) model of Ginocchio.
  • Cubic interactions are considered in 3 and 7 space dimensions, respectively, for bosonic membranes in Poisson Bracket form. Their symmetries and vacuum configurations are discussed. Their associated first order equations are transformed to Nahm's equations, and are hence seen to be integrable, for the 3-dimensional case, by virtue of the explicit Lax pair provided. The constructions introduced also apply to commutator or Moyal Bracket analogues.
  • It was found that any homogeneous polynomial can be written as a sum of integrable polynomials of the same degree, with which each associated Lie transformation can be evaluated exactly. An integrable polynomial factorization has thus been developed to convert an analytic symplectic map into a product of exactly evaluable Lie transformations associated with integrable polynomials. [copyright] 1995 [ital American] [ital Institute] [ital of] [ital Physics]
  • It has been shown that an analytic symplectic map can be directly converted into a product of Lie transformations in the form of integrable polynomial factorization with the desired accuracy. A map in the form of integrable polynomial factorization is exactly symplectic and easy to evaluate exactly. Error involved in the integrable polynomial factorization has been studied with the case of the Henon map. The results suggest that the map in the form of integrable polynomial factorization is a reliable and convenient model for the study of the long-term behavior of a symplectic system such as a large storage ring.