Orbit Hamiltonian, Synchrotron Oscillations and Synchro-betatron Coupling

Crowley-Milling and Rabinowitz [2] first discovered the synchro-betatron resonances driven by dispersion in rf cavities in NINA. They explained the mechanism of the resonances by considering the sudden change of energy by rf cavities that causes a sudden changes of the equilibrium orbit. Thus, betatron oscillations are excited. These are forced betatron oscillations driven by synchrotron oscillations. Piwinski and Wrulich [3] studied the counter effect of betatron oscillations on synchrotron oscillations due to the path lengthening by betatron oscillations. Then,the oscillations become coupled oscillations and the mechanism become symplectic. They described a complete theory of this effect. Morton and Chao [4] and later Corsten and Hagedoorn [5] developed a Hamiltonian formalism for this this effect and derived a formula for the path lengthening by a simple canonical transformation. This term is also known as the CP (Central Position) phase in the theory of cyclotrons. (See the references in [5]) As an independent variable, Morton and Chao used the orbit length s and Corsten and Hagedoorn used the time t. The former authors treated a static case and the latter included acceleration. The use of s (s-description) is more suitable than the use of t (t-description) in accelerator theories. All the devices are placed at fixed positions and observations are also done at fixed positions along the circumference of the accelerator. This s-description is necessary when we take into account the effects of localized objects. In the s-description, the effects can be described by a periodic δ-function. There is no easy way in the t-description. The localized nature of rf cavities is very important in synchro-betatron resonances. In this case, the resonances νx = n + mνs are excited, where νx and νs are betatron and synchrotron tunes, and n,m are arbitrary integers. In a smooth, travelling wave approximation for rf, only the resonances νx = mνs can be excited. The t-description is usually used for synchrotron oscillations in an accelerated case [6]. This is certainly an approximatio. The concept of the rf bucket is also an approximation [7]. The present author [1] presented a symplectic theory of synchrotron oscillations and synchro-betatron coupling in the s-description. He sticked to a standing wave picture for rf and made a travelling wave approximation for analytical treatents only in the final stage. He followed a formalism by McMillan [8] and proved McMillan’s equations of synchrotron motion. This s-description is important in the theory of synchro-betatron coupling because this description is usually used for betatron oscillations. In this paper, we add a Hamiltonian formalism to Ref. [1] because only equations of motion are used there though the author used only canonical variables.