Editing Accelerator physics (section)

==Beam dynamics==
{{See also|Particle beam|Strong focusing|Beam emittance|Radiation damping|Collective effects (accelerator physics)}}

Due to the high velocity of the particles, and the resulting [[Lorentz force]] for magnetic fields, adjustments to the beam direction are mainly controlled by [[magnetostatics|magnetostatic]] fields that deflect particles. In most accelerator concepts (excluding compact structures like the [[cyclotron]] or [[betatron]]), these are applied by dedicated [[electromagnets]] with different properties and functions.  An important step in the development of these types of accelerators was the understanding of [[strong focusing]].<ref>{{cite journal | last1 = Courant | first1 = E. D. | author-link1=Ernest Courant | last2 = Snyder |first2= H. S. | author-link2=Hartland Sweet Snyder | date=Jan 1958 | title = Theory of the alternating-gradient synchrotron | journal = Annals of Physics | volume = 3 | issue = 1 | pages = 360–408 | doi = 10.1006/aphy.2000.6012 | url = http://ab-abp-rlc.web.cern.ch/ab-abp-rlc/AP-literature/Courant-Snyder-1958.pdf|bibcode = 2000AnPhy.281..360C }}</ref>  [[Dipole magnet]]s are used to guide the beam through the structure, while [[quadrupole magnet]]s are used for beam focusing, and [[sextupole magnet]]s are used for correction of [[Dispersion (optics)|dispersion]] effects.

A particle on the exact design trajectory (or design ''orbit'') of the accelerator only experiences dipole field components, while particles with transverse position deviation <math>x(s)</math> are re-focused to the design orbit. For preliminary calculations, neglecting all fields components higher than quadrupolar, an inhomogenic [[Hill differential equation]]

:<math> \frac{d^2}{ds^2}\,x(s) + k(s)\,x(s) = \frac{1}{\rho} \, \frac{\Delta p}{p} </math>

can be used as an approximation,<ref>{{cite book
 | last = Wille
 | first = Klaus
 | title = Particle Accelerator Physics: An Introduction
 | publisher = [[Oxford University Press]]
 | year = 2001
 | isbn = 978-0-19-850549-5
}} (slightly different notation)</ref> with
:a non-constant focusing force <math>k(s)</math>, including strong focusing and [[weak focusing]] effects
:the relative deviation from the design beam impulse <math>\Delta p / p</math>
:the trajectory [[radius of curvature (mathematics)|radius of curvature]] <math>\rho</math>, and
:the design path length <math>s</math>,
thus identifying the system as a [[parametric oscillator]]. Beam parameters for the accelerator can then be calculated using [[Ray transfer matrix analysis]]; e.g., a quadrupolar field is analogous to a lens in geometrical optics, having similar properties regarding beam focusing (but obeying [[Earnshaw's theorem]]).

The general equations of motion originate from [[Theory of relativity|relativistic]] [[Hamiltonian mechanics]], in almost all cases using the [[Paraxial approximation]]. Even in the cases of strongly nonlinear magnetic fields, and without the paraxial approximation, a [[Lie group|Lie transform]] may be used to construct an integrator with a high degree of accuracy.{{Citation needed|date=December 2011}}