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

  • Author(s): Mitchell, Chad E;
  • Ryne, Robert D;
  • Hwang, Kilean;
  • Nagaitsev, Sergei;
  • Zolkin, Timofey
  • et al.
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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