Abstract
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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Reference
E.D. Courant, H.S. Snyder, Theory of the alternating-gradient synchrotron. Ann. Phys. 3, 1–48 (1958)
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Stupakov, G., Penn, G. (2018). Action-Angle Variables for Betatron Oscillations. In: Classical Mechanics and Electromagnetism in Accelerator Physics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-90188-6_7
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DOI: https://doi.org/10.1007/978-3-319-90188-6_7
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