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184, p. 1910 (1989). C. Lejeune and J. Aubert, "Emittance and Brightness: Definition and Measurements," in Adv. Elec. and Elec. , Supp. 13a, 159 (1980). 6. T. Weng, "Fundamentals Longitudinal Motion," in AIP Conf. Proc. 184, pp. 243-287 (1989); BNL 42195 (1988). G. DSme, "Theory of RF Acceleration," in 1985 CERN Accelerator School, CERN 87-03, pp. 110-158. D. Holmes, "Low Energy Aspects of Circular Accelerators," FNAL-Conf-90/275 (1990).

There axe several caveats to this, however. The emittance can be changed through the operation of non-Hamiltonian processes. Liouville's theorem applies only if the motion in the phase space is described by a Hamiltonian. Examples of processes that are not Hamiltonian occurring in an accelerator are residual gas scattering (with gas molecules or ions in the beam pipe), space-charge forces (mutual repulsion of the particles because they have the same sign of charge), power supply noises (leading to random fluctuations in the magnetic fields), etc.

We see that the definition of the normalized emittance is deliberately chosen so that it will remain constant during acceleration to high energy. 1 Longitudinal Oscillations Equations Motion Synchrotron of Motion For thistopic,see alsoRef. 7. In a synchrotron,radio-frequency longitudinalelectric fieldsare created locallyacrossrf cavitiesto provide an acceleratingfieldto the beam. 1) and is an integer. In that case, the circumference will be divided into h sections. Within each section will be created and populated by particles.

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