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Extracting dynamical frequencies from invariants of motion in finite-dimensional nonlinear integrable systems

Chad E. Mitchell, Robert D. Ryne, Kilean Hwang, Sergei Nagaitsev, and Timofey Zolkin
Phys. Rev. E 103, 062216 – Published 25 June 2021
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  • INTRODUCTION
  • INTEGRABLE MAPS AND FLOWS
  • TUNES OF AN INTEGRABLE MAP
  • TUNES FROM INVARIANTS
  • PROPERTIES OF THE SOLUTION
  • FREQUENCIES OF HAMILTONIAN FLOWS
  • NUMERICAL EXAMPLES
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Integrable dynamical systems play an important role in many areas of science, including accelerator and plasma physics. An integrable dynamical system with n degrees of freedom possesses n nontrivial integrals of motion, and can be solved, in principle, by covering the phase space with one or more charts in which the dynamics can be described using action-angle coordinates. To obtain the frequencies of motion, both the transformation to action-angle coordinates and its inverse must be known in explicit form. However, no general algorithm exists for constructing this transformation explicitly from a set of n known (and generally coupled) integrals of motion. In this paper we describe how one can determine the dynamical frequencies of the motion as functions of these n integrals in the absence of explicitly known action-angle variables, and we provide several examples.

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    • Received 15 March 2021
    • Accepted 2 June 2021

    DOI:https://doi.org/10.1103/PhysRevE.103.062216

    ©2021 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Classical mechanicsNonlinear beam dynamicsSingle-particle dynamics
    Accelerators & BeamsNonlinear Dynamics

    Authors & Affiliations

    Chad E. Mitchell*, Robert D. Ryne, and Kilean Hwang

    • Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

    Sergei Nagaitsev† and Timofey Zolkin

    • Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA

    • *ChadMitchell@lbl.gov
    • †Also at the University of Chicago, Chicago, Illinois 60637, USA.

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    Vol. 103, Iss. 6 — June 2021

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    Images

    • Figure 1
      Figure 1

      Illustration of the map (18), showing one of the level sets Mc(c>0) of the invariant f in Eq. (19) and the two curves γ (red) and γ1 (black) used to evaluate (16). Although not shown here, each curve is allowed to change direction during transit. The curve γ may wind around the origin multiple times.

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    • Figure 2
      Figure 2

      Frequencies of the Hamiltonian (82) with λ=1, shown for the level set Mc0 defined by (f1,f2)=(c1,c2). Dots correspond to the analytical expression given in Eq. (93), while solid curves correspond to the result obtained using (16). (a) The value ω1 is shown for 6λ2(c1−c2)=1/2. (b) The value ω2 is shown for 6λ2(c1+c2)=1/2. In both cases, a separatrix is approached as the horizontal axis approaches 1.

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    • Figure 3
      Figure 3

      Demonstration that the frequencies of the Hamiltonian (82) (λ=1) obtained using (75) are unchanged under deformation of the cycles γ1 and γ2. These are defined by intersection of the level set Mc0 with the hyperplanes y=kx and y=−kx, respectively. The results are shown for the case c1=0.1, c2=0.03.

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    • Figure 4
      Figure 4

      Orange: Level set (f1,f2)=(2,0.5) of the 4D McMillan map (95) with a=1.6, b=1. The apparent self-intersections of the 2D surface are an artifact of projection into R3. This is shown together with examples of basis cycles γ1 and γ2 and the path γ used to evaluate the tunes ν1, ν2 from (16).

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    • Figure 5
      Figure 5

      Tunes ν1, ν2 of the 4D McMillan map (95) with a=1.6, b=1, shown for the invariant level set defined by (f1,f2)=(c1,c2). Dots correspond to the analytical expression given in Ref. [7], while solid curves correspond to the result obtained using (16). Compare Fig. 5 of Ref. [7]. The curves from uppermost to lowermost correspond to (in the upper figure) c2=1.0,0.5,0.01 and (in the lower figure) c2=0.01,0.5,1.0.

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    • Figure 6
      Figure 6

      Examples of 1-cycles on the torus T2. One of the two holes has been made larger than the other, in order to embed the torus in R3 without self-intersection. (a) Two basis cycles with [γ1]=(1,0) and [γ2]=(0,1). (b) Two cycles that do not form a basis, with [γ1]=(1,0), [γ2]=(−1,0). (c) Two basis cycles with [γ1]=(1,−1) and [γ2]=(0,1).

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