A pictorial view in one degree of freedom from the Hamiltonian to the map classification of one-turn maps from linear to nonlinear maps vector fields and canonical transformations the ring floquet rings a theoretical construct power series and analytic/symbolic calculations examples of the analytical normalization the layout in the laboratory frame symplectic integration "small" rings - using the correct Hamiltonian fringe effects in ring dynamics large ring approximations and the rest inclusion of radiation.
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