1. Hamiltonian formulation of single particle dynamics. 1.1. Introduction. 1.2. Hamiltonian formalism. 1.3. Canonical transformations. 1.4. Electric and magnetic fields. 1.5. Synchro-Betatron formalism in beam dynamics -- 2. Linear betatron motion. 2.1. Introduction. 2.2. The transfer matrix. 2.3. Hill's equation and Floquet's theorem. 2.4. Twiss parameters and courant-snyder invariant. 2.5. Action-angle variables and beam emittance. 2.6. Adiabatic damping of betatron oscillations -- 3. Nonlinear resonances of betatron oscillations. 3.1. Introduction. 3.2. General description and basic properties of a nonlinear resonance. 3.3. The method of effective potential. 3.4. Stability analysis of third and fourth order resonances. 3.5. The method of successive linearization. 3.6. Adiabatic crossing of a nonlinear resonance. 3.7. Periodic crossing of a nonlinear resonance -- 4. Canonical perturbation theory. 4.1. Introduction. 4.2. Classical perturbation theory. 4.3. Effect of linear and nonlinear perturbations in one dimension. 4.4. Secular perturbation theory. 4.5. The method of formal series. 4.6. Renormalization transformation for two resonances -- 5. Special methods in accelerator theory. 5.1. Introduction. 5.2. Renormalization group method. 5.3. The method of multiple scales. 5.4. Renormalization group analysis of Hill's equation. 5.5. Renormalization group reduction of nonlinear resonances. 5.6. Reduction of nonlinear resonances using the method of multiple scales. 5.7. Renormalization group reduction of Hamilton's equations of motion -- 6. Transfer maps. 6.1. Introduction. 6.2. Nonlinear transfer maps of betatron motion. 6.3. Linear transfer maps. 6.4. The Henon map. 6.5. Resonance structure of the Henon map. 6.6. Renormalization group reduction of a generic transfer map. 6.7. The standard Chirikov-Taylor map
7. Statistical description of charged particle beams. 7.1. Introduction. 7.2. The Liouville theorem and the Liouville equation. 7.3. Ensemble of identical macroscopic systems. 7.4. The method of microscopic phase space density. 7.5. The equation for the microscopic phase space density with a small source. 7.6. The generalized kinetic equation. 7.7. The Balescu-Lenard kinetic equation. 7.8. The Landau kinetic equation. 7.9. The approximate collision integral and the generalized kinetic equation -- 8. Statistical description of non integrable Hamiltonian systems. 8.1. Introduction. 8.2. Projection operator method. 8.3. Renormalization group reduction of the Liouville equation. 8.4. Modulational diffusion. 8.5. The Liouville operator and the Frobenius-Perron operator -- 9. The Vlasov equation. 9.1. Introduction. 9.2. The Vlasov equation for collisionless beams. 9.3. The Hamiltonian formalism for solving the Vlasov equation. 9.4. Propagation of an intense beam through a periodic focusing lattice. 9.5. Propagation of an intense beam with a uniform phase-space density. 9.6. Dynamical equations for the beam envelope and for the mean emittance. 9.7. Solution of the equations for the boundary curves. 9.8. Description of beam dynamics in terms of Lagrangian variables. 9.9. Landau damping -- 10. Nonlinear waves and turbulence in intense beams. 10.1. Introduction. 10.2. Renormalization group reduction of the hydrodynamic equations. 10.3. The Parametric wave-particle resonance. 10.4. The nonlinear Schrodinger equation for a single mode. 10.5. Nonlinear damped waves in intense beams. 10.6. Fluctuation spectrum and turbulence
Summary
This book adopts a non-traditional approach to accelerator theory. The exposition starts with the synchro-betatron formalism and continues with the linear and nonlinear theories of transverse betatron motion. Various methods of studying nonlinear dynamical systems (the canonical theory of perturbations and the methods of multiple scales and formal series) are explained through examples. The renormalization group approach to studying nonlinear (continuous and discrete) dynamical systems as applied to accelerators and storage rings is used throughout the book. The statistical description of charged particle beams (the Balescu-Lenard and Landau kinetic equations as well as the Vlasov equation) is dealt with in the second part of the book. The processes of pattern formation and formation of coherent structures (solitons) are also described
Bibliography
Includes bibliographical references (pages 301-305) and index
