Action-Angle Variables and Liouville’s Theorem

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


One of the most powerful uses of canonical transformations is to express the dynamics in terms of action-angle variables. These are phase space coordinates which provide a simple description of the Hamiltonian motion, and are widely used in particle dynamics. A geometrical view of the Hamiltonian flow in phase space leads us to the formulation of Liouville’s theorem that is crucial for understanding the fundamental properties of large ensembles of beam particles in accelerators.


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Copyright information

© 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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