Titlebook: Classical Mechanics and Electromagnetism in Accelerator Physics; Gennady Stupakov,Gregory Penn Textbook 2018 Springer International Publis

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DEMN

Canonical Transformations once we select a set of coordinates . we can define the generalized momenta . according to Eq. (.) and form a Hamiltonian (.). We could also have chosen a different set of generalized coordinates ., expressed the Lagrangian as a function of ., used Eqs. (.) and (.), and obtained a different set of

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Action-Angle Variables and Liouville’s Theoremnates which provide a simple description of the Hamiltonian motion, and are widely used in particle dynamics. A geometrical view of the Hamiltonian flow in phase space leads us to the formulation of Liouville’s theorem that is crucial for understanding the fundamental properties of large ensembles o

懦夫

戏服

Coordinate System and Hamiltonian for a Circular Accelerator, we assume that there is no electrostatic field, ., and the magnetic field does not vary with time. Second, the magnetic field is arranged in such a way that there is a . (or .) . for a particle with a nominal momentum .—this is achieved by a proper design of the magnetic lattice of the ring. We wi

冷淡一切

Equations of Motion in Acceleratorscle beam. To specify the Hamiltonian (.) we need to know the vector potential, ., for these magnets. We evaluate the fields and Hamiltonian for the major magnet types assuming that the field profiles are uniform over their length. Often in analysis and simulations, one has to take into account that

冷淡一切

Action-Angle Variables for Betatron Oscillationsthe action remains constant and the angle increases linearly with time. With minor modifications, the same transformation can be applied to the Hamiltonian (.) that describes betatron oscillations in an accelerator. This yields an invariant of the motion and is also a useful starting point for analy

神化怪物

glamor

forthy

The Kinetic Equationion function and describe the formalism of the kinetic equation for treating large ensembles of particles in a beam. While this chapter focuses on deterministic Hamiltonian motion, kinetic equations in general can also include stochastic motion and damping.