Action-Angle Variables for Betatron Oscillations

  • Gennady StupakovEmail author
  • Gregory Penn
Part of the Graduate Texts in Physics book series (GTP)


In Chap.  3 we showed that choosing the action-angle canonical variables in a one-dimensional Hamiltonian system dramatically simplifies the dynamics: the action remains constant and the angle increases linearly with time. With minor modifications, the same transformation can be applied to the Hamiltonian ( 6.14) that describes betatron oscillations in an accelerator. This yields an invariant of the motion and is also a useful starting point for analyzing more complicated dynamics.


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    E.D. Courant, H.S. Snyder, Theory of the alternating-gradient synchrotron. Ann. Phys. 3, 1–48 (1958)ADSCrossRefGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.SLAC National Accelerator LaboratoryStanford UniversityMenlo ParkUSA
  2. 2.Lawrence Berkeley National LaboratoryBerkeleyUSA

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