Three Beam Dynamics Problems Kwang-Je Kim University of Chicago Three examples of beam dynamics problems will be discussed in order to illustrate some terminologies and methods used in accelerator physicists. We hope that additional examples and details from other speakers will complement this presentation. Example 1: Various collective phenomena occur in intense beams that are usually harmful but can occasionally be beneficial. For a proper understanding of these phenomena, it is necessary to pay attention to both the individual particle and the collective aspects. A “physical” approach to such a problem is to start from the Klimontovich density function, specifying the locations of discrete particles in 6D phase space by delta functions, and to develop a suitable perturbation scheme to solve the coupled Maxwell-Klimontovich equations. As a simple example, we discuss the 1D plasma oscillation here. Example 2: The phase space profile of beams needs to be “conditioned” to bring about an optimal performance for the intended application. In addition to providing special accelerator elements, it may also be necessary to tailor the beam profile at the gun. The task can be expedited if the simulation program tracking the beam evolution can be run in reverse—from the desired final beam profile toward the gun. An interesting question is how much approximation would be necessary to allow a reverse tracking—obviously any dissipative interactions will preclude the reverse tracking. We discuss the beam conditioning for an X-ray free-electron laser oscillator. Example 3: Propagation of partially coherent X-ray beams through synchrotron radiation beamlines containing apertures and grazing incidence mirrors is an important but numerically intensive problem. We will speculate whether the Lie algebraic method could offer a practically useful approach here.