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Alternating Gradient Focusing
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action-angle variables amplitude approximation area-preserving maps beam behavior betatron betatron motion betatron oscillations bucket bunch calculated canonical transformation cavity chromaticity closed orbit coherent collider components constant coordinates coupling curve defined degrees of freedom dipole discussed distortion dynamic aperture effect electron elements equations of motion example Fermilab Figure fixed point focusing Fourier frequency given Hamilton's equations Hamiltonian harmonic horizontal impedance instability integral islands Landau damping lattice linear longitudinal machine magnetic field matrix method mode modulation momentum multipole nonlinear obtain parameters Particle Accelerators particle motion periodic orbits perturbation phase space Physics plane problem programs proton quadrupole quadrupole magnets residue resonance rotation number rotational Invariant circles sextupole sideband SLAC solution space charge stable storage ring surface of section symplectic synchrotron radiation theorem theory trajectory transverse tune shift twist maps unstable fixed point variables vector vertical voltage winding number zero