Averaged Invariants in Storage Rings with Synchrotron Motion
PPrepared for submission to JINST
Stroboscopic Invariants in Storage Rings withSynchrotron Motion
S. Webb, π, N. Cook, π J. Eldred π π RadiaSoft LLC,3380 Mitchell Lane, Boulder, CO, USA π Fermi National Accelerator Laborator,Batavia, IL, USA
E-mail: [email protected]
Abstract: In an ideal accelerator, the single-particle dynamics can be decoupled into transverse mo-tion β the betatron oscillations β and longitudinal motion β the synchrotron oscillations. Chromaticand dispersive eο¬ects introduce a coupling between these dynamics, the so-called synchro-betatroncoupling. We present an analysis of the fully coupled dynamics over a single synchrotron oscil-lation that leads to a stroboscopic invariant with synchro-betatron coupling in a generic lattice.This invariant is correct to
O ( π π ) , where π π is the synchrotron tune. We apply this analysis to twoproblems: ο¬rst, a toy lattice where the computations are analytically tractable, then a design fora rapid cycling synchrotron built using the integrable optics described by Danilov and Nagaitsev,showing that although there is fairly complex behavior over the course of a synchrotron oscillation,the Danilov-Nagaitsev invariants are nevertheless periodic with the synchrotron motion.Keywords: Beam Dynamics, Beam OpticsArXiv ePrint: 2007.04935 Corresponding author. a r X i v : . [ phy s i c s . acc - ph ] O c t ontents Single-particle dynamics in particle accelerators can be broken into the fast transverse betatronoscillations, with tunes π π₯,π¦ (cid:29)
1, and the much slower synchrotron oscillations, with tunes π π (cid:28)
1. For a coasting beam, the momentum dependence of the focusing element strengths leadsto chromaticity, a momentum-dependent betatron tune, and dispersion, a momentum-dependentclosed orbit. When an rf cavity is added and synchrotron motion occurs, that synchrotron motioncouples to the betatron motion through the chromaticity and dispersion β so-called synchro-betatroncoupling.Synchro-betatron coupling can lead to complex coupled dynamics. The impact of synchro-betatron coupling has been well-studied for linear alternating gradient focusing lattices [1β3], buttheir inο¬uence on more novel lattice designs, such as nonlinear integrable optics [4β7], has yet tobe studied in detail.In this paper, we calculate a stroboscopic invariant of coupled synchro-betatron motion in thelimit of small synchrotron tune. This invariant is the Hamiltonian that generates an π -turn map,where π π π β
1. We show that this Hamiltonian is correct to
O ( π π ) , and that the perturbing termsdo not cause secular growth in the invariants. This Hamiltonian is a pure function of the transversecoordinates and the synchrotron action coordinate β thus if this Hamiltonian is integrable then theentire system is integrable over π turns. We demonstrate the preservation of integrable dynamics inthe context of an integrable rapid cycling synchrotron, designed to use nonlinear integrable opticsβ 1 βo mitigate beam loss due to coherent instabilities in high intensity proton beams. This result relieson the single-turn Lie map formalism of Dragt et al. [8β14], and therefore we give a brief survey ofkey results in Appendix A. The simplest model for single-particle dynamics in a particle accelerator is the uncoupled verticaland horizontal betatron oscillations, with independent synchrotron motion longitudinally. Disper-sion can complicate this picture, as each trajectoryβs momentum-dependent closed orbit oscillateswith the synchrotron motion. The simplest linear cases create normal modes that couple transverseand longitudinal motion, and the eigenemittances and tunes can computed with a generic symplecticmatrix formulation [15]. The next-leading-order dynamics result from chromaticity, the momentumdependence of the betatron tune. Because the system is Hamiltonian, a betatron tune that dependson the momentum implies a betatron amplitude dependence in the synchrotron tune β so-called synchro-betatron coupling .This coupling has a number of implications. The slow change in the momentum oο¬set ofa particleβs trajectory suggests that there will also be a slow change in the chromaticity β thesynchrotron motion will modulate the betatron oscillations with a frequency of the betatron phase.Synchro-betatron coupling therefore can lead to sidebands in the betatron motion located at π β₯ Β± ππ π for betatron tune π β₯ and synchrotron tune π π [1, 2]. Because of the coupling in the system, thesynchrotron motion modiο¬es the usual transverse action-angle variables [3]. This can lead tosynchro-betatron coupling induced parametric resonances when the betatron tune is a harmonic ofthe synchrotron tune.Synchro-betatron coupling is conceptually similar to adiabatic analysis in that a quantity thataο¬ects the transverse motion is changing slowly compared to the transverse oscillations. Becausethe synchrotron oscillations are slow and periodic, we expect to be able to ο¬nd a period-averagedHamiltonian treatment of synchro-betatron coupling. However, because the slowly changing quan-tity is a dynamical quantity in a Hamiltonian system, the analysis is more subtle β we must makesure any treatment of synchro-betatron coupling reο¬ects the Hamiltonian nature of the dynamics. Synchrotron motion in the absence of coupling is a periodic system, slowly varying compared to themuch faster betatron oscillations. The periodicity of the synchrotron motion suggests examiningthe total dynamics stroboscopically, looking every π turns where π Γ π π β M = M πΏ M π (3.1)where the Hamiltonian that generates M πΏ is of the form π» πΏ = π» β₯ ((cid:174) π§ β₯ , πΏ ) + πΌ π ( πΏ ) (3.2)β 2 βnd the rf potential that generates the thin cavity map M π is generated by π ( π ) . Here πΏ and π arecanonically conjugate, πΌ π captures the momentum compaction of the ring, and π» β₯ describes thetransverse motion with chromatic and dispersive eο¬ects. Because πΌ π commutes with π» β₯ , we canfactor this into three maps: M β₯ M π M π , where π» β₯ generates M β₯ , πΌ π generates M π , and M π is asbefore. In the absence of synchro-betatron coupling, the transverse dynamics are speciο¬ed entirelyby M β₯ and the longitudinal dynamics are speciο¬ed entirely by the synchrotron map M π = M π M π .We assume the synchrotron motion is integrable, so the dynamics can be speciο¬ed in action-angle coordinates ( π΄, π ) , and M π = exp {β : β π ( π΄ ) : } (3.3)with amplitude-dependent synchrotron phase advance π π ( π΄ ) . Because the synchrotron motion ispresumed integrable, we know that πΏ must be a periodic function with the synchrotron phase, andcan be written as a Fourier series πΏ = βοΈ π πΏ π ( π΄ ) π πππ . (3.4)The full single-turn map is therefore M = exp {β : π» β₯ ((cid:174) π§ β₯ ; π΄, π ) : } exp {β : β π ( π΄ π ) : } . (3.5)Once again, because πΏ is periodic with the synchrotron phase, so too is π» β₯ , and we can rewrite π» β₯ = β βοΈ π = ββ β π ((cid:174) π§ β₯ , π΄ ) π ππ π = π» ((cid:174) π§ β₯ , π΄ ) + βοΈ π β β π ((cid:174) π§ β₯ , π΄ ) π ππ π (3.6)where we have called out π» = (cid:104) π» β₯ (cid:105) π π , the average of π» β₯ over the synchrotron phase, as it will beimportant later. That π» β₯ is a real function requires that π» be real, and that β β π = β β π .For stroboscopic dynamics, we want to look at an π -turn map, given by M π = (M β₯ M π ) π . (3.7)Through a judicious insertion of an identity operator, we can move all of the synchrotron motionmaps to the left, and leave only the transverse dynamics to the right. This can be accomplished bynoting that M β₯ M π M β₯ M π = M β₯ M π M π M β π M β₯ M π = M β₯ M π Λ M ( )β₯ (3.8)where Λ M ( π )β₯ = M β ππ M β₯ M ππ . It is straightforward to show that, by moving each successivesynchrotron map to the left in this process, we get the π -turn map M π = (M π ) π (cid:32) (cid:214) π = π Λ M ( π )β₯ (cid:33) (3.9)where we are counting the index down from left to right.β 3 βrom the similarity transformation identity described in Appendix A, it is straightforward tocompute Λ M ( π )β₯ . The similarity transformation moves the synchrotron motion into the argument ofthe M β₯ exponential, thus:Λ M ( π )β₯ = M β ππ M β₯ M ππ = M β ππ exp (cid:40) β : β βοΈ π = ββ β π ((cid:174) π§ β₯ , π΄ ) π ππ π : (cid:41) M ππ = exp (cid:40) β : M β ππ β βοΈ π = ββ β π ((cid:174) π§ β₯ , π΄ ) π ππ π : (cid:41) = exp (cid:40) β : β βοΈ π = ββ β π ((cid:174) π§ β₯ , π΄ ) π ππ ( π β ππ ( π΄ )) : (cid:41) (3.10)where π ( π΄ ) is the amplitude-dependent synchrotron phase advance, and we have used the factthat M π β¦ ( π΄, π ) = ( π΄, π + π ( π΄ )) . We now need to compute the product in eqn. (3.9) as asingle exponential operator to ο¬rst order using the BCH formula from Appendix A to compute thestroboscopic Hamiltonian and its ο¬rst order correction.To construct the single exponential operator, we will rely on the BCH formula and a recursionrelation deο¬ned by concatenating the ο¬rst π terms from the right of the product in eqn. (3.9) withthe next map to its left. Speciο¬cally, let us write the product as (cid:214) π = π Λ M ( π )β₯ = (cid:32) π + (cid:214) π = π Λ M ( π )β₯ (cid:33) Λ M ( π )β₯ (3.11)where Λ M ( )β₯ = Λ M ( )β₯ (3.12)and Λ M ( π + )β₯ = Λ M ( π )β₯ Λ M ( π )β₯ (3.13)so that we end with Λ M ( π )β₯ = (cid:214) π = π Λ M ( π )β₯ . (3.14)The goal is therefore to write Λ M ( π )β₯ = exp (cid:110) β : Λ π» ( π ) : (cid:111) (3.15)and compute Λ π» ( π ) perturbatively using the BCH series.From the BCH series, we can derive a recursion relation for Λ π» to leading order in the Poissonbrackets asΛ π» ( π + ) = Λ π» ( π ) + β βοΈ π = ββ β π ((cid:174) π§ β₯ , π΄ ) π ππ ( π β π π ( π΄ )) + π (cid:34) Λ π» ( π ) , β βοΈ π = ββ β π ((cid:174) π§ β₯ , π΄ ) π ππ ( π β π π ( π΄ )) (cid:35) +O ( π ) (3.16)where we have included π to bookkeep the order in Poisson brackets.β 4 βhus to order π =
0, the stroboscopic Hamiltonian for the π -turn map isΛ π» ( π ) = π βοΈ π = β βοΈ π = ββ β π ((cid:174) π§ β₯ , π΄ ) π ππ ( π β ππ ( π΄ )) = β βοΈ π = ββ π βοΈ π = β π ((cid:174) π§ β₯ , π΄ ) π ππ ( π β ππ ( π΄ )) = β βοΈ π = ββ β π ((cid:174) π§ β₯ , π΄ ) π ππ π (cid:32) π βοΈ π = π β ππππ ( π΄ ) (cid:33) = β βοΈ π = ββ β π ((cid:174) π§ β₯ , π΄ ) π ππ π π ππ π ( π΄ ) β π ππ π π ( π΄ ) β π ππ π ( π΄ ) (3.17)Now, we consider the π -turn map following this relation. If π π ( π΄ ) = ππΎ for some integer πΎ β i.e. the synchrotron tune is given by π π = πΎ / π for the amplitude π΄ β then all but the π = π» ( π ) = π Γ π» ((cid:174) π§ β₯ , π΄ ) (3.18)and the π -turn map is equivalent to π turns of synchrotron motion and π turns of the period-averaged Hamiltonian π» ((cid:174) π§ β₯ , π΄ ) . In the general case of nonlinear synchrotron motion, π may be afunction of the synchrotron amplitude π΄ . Therefore, π» will be stroboscopically invariant.If, however, π π ( π΄ ) = π + π , we are left with a term for π β β π ππ π π ( π΄ ) β π ππ π ( π΄ ) = β π ππ π β π ππ π ( π΄ ) β π (3.19)for π (cid:28)
1. Thus, the perturbation to the stroboscopic Hamiltonian is proportional to the irrationalpart of the synchrotron tune, which can be made arbitrarily small.We have thus shown that π» ((cid:174) π§ β₯ , π΄ ) is a stroboscopic invariant with synchrotron motion solong as the synchrotron tune is rational, and the correction is proportional to the arbitrarily smallirrational part of the synchrotron tune. Therefore, if π» ((cid:174) π§ β₯ , π΄ ) is integrable, then the stroboscopicdynamics will be integrable to leading order.Furthermore, the leading order term π Γ π» is O ( π β π ) , since we deο¬ned π such that π Γ π π β πΎ and, since π π (cid:28) πΎ (cid:28) π . In Appendix B we show that the next leading orderterm is O ( π π ) and oscillatory, and therefore the secular synchro-betatron dynamics is dominatedby this stroboscopic Hamiltonian. To see this, we can cast the π -turn map as the π π‘β power of asingle turn map: Λ M ( π )β₯ βΌ (cid:16) Λ M β₯ (cid:17) π (3.20)with Λ M β₯ = exp (cid:18) β : π» + π π β : (cid:19) (3.21)where β is a bounded periodic function of the phase space variables. Thus, in the limit of large π , this term becomes perturbatively small compared to π» . This argument, that the oscillatoryterm does not lead to secular growth in the action, is analogous to the arguments for averagingtime-continuous Hamiltonian systems described in Β§19 of Arnold [16].β 5 β Octupole Ring with Chromaticity: A Toy Model
To verify our predictions of stroboscopic invariance, we consider a toy ring, comprised of a linearfocusing elements with a small octupolar contribution along with a thin linear RF kick. The resultingmodel produces nonlinear transverse dynamics, including nonlinear chromaticity and synchrotronmotion that is analytically tractable.
We model this ring with the product of two symplectic maps: M = M πΏ M π (4.1)where the Hamiltonians that generate these maps are given by π» πΏ = (cid:18) π π₯ + π πΏ + π πΏ (cid:19) (cid:16) π + π₯ (cid:17) + π π + πΌ π πΏ (4.2)and π ( π ) = π π π π . (4.3)Breaking this into M β₯ and M π , as we do above, gives the two Hamiltonians π» β₯ = (cid:18) π π₯ + π πΏ + π πΏ (cid:19) (cid:16) π + π₯ (cid:17) + π π₯ (4.4)and π» π = π π (cid:16) Λ πΏ + Λ π (cid:17) (4.5)where Λ πΏ = πΏ βοΈ π½ π + πΌ β π½ π π (4.6a)Λ π = π β π½ π (4.6b)with the synchrotron Twiss parameters π½ π = πΌ π βοΈ β πΌ π π π π and πΌ π = πΌ π π π π βοΈ β πΌ π π π π , and synchrotronphase advance π π = arccos (cid:16) β πΌ π π π π (cid:17) . The synchrotron action is then π΄ π = ( Λ πΏ + Λ π )/ π» = (cid:20) π π₯ + π Γ (cid:18) + πΌ π π½ π π΄ π (cid:19) (cid:21) (cid:16) π + π₯ (cid:17) + π π₯ (4.7)We note that the linear chromatic term does not contribute in this example, because for linearsynchrotron motion (cid:104) πΏ (cid:105) π =
0. This is speciο¬c to linear synchrotron motion, and does not holdwith nonlinear rf eο¬ects. To demonstrate that this is the correct stroboscopic Hamiltonian, wewill compare particle trajectories from integrating the full model to trajectories computed usingthe stroboscopic Hamiltonian β as noted in eqn. (3.20) the stroboscopic Hamiltonian generates thetransverse dynamics with a period of a synchrotron oscillation.β 6 β .2 Numerical Model
To generate the numerical data, we have to integrate the individual maps. The synchrotron mapand the transverse map, in the limit of π β
0, can be solved analytically. To include the octupoleterm, we use a fourth order splitting of M β₯ using the identity described by Yoshida [17]. As abenchmark, we look at variations in π» β₯ for an on-momentum particle, which should be conservedexactly should given an exact solution to π» β₯ .The operator splitting requires a second-order integration, which splits π» β₯ into the transverselinear component, and the octupole kick. Using the Yoshida identity transforms this to a fourthorder operation, reducing the numerically induced error in π» β₯ conservation. As we can see inο¬g. (1), π» β₯ is conserved on the order of .2%. This provides a baseline for our assessment, and anyvariation in π» β₯ varying above this level should indicate a physical outcome rather than a numericalartifact of our integration scheme. Figure 1 : On-momentum π» β₯ invariant calculated using the fourth order scheme for a particle withno longitudinal dynamics. For the following simulations, we use parameters designed to emulate the tunes of the iRCS latticedescribed in Section 5 below. In particular, we use the numerical values in Table (1). For theseparameters, the lattices have the same linear betatron tune, synchrotron tune, and linear chromaticity.The resulting evolution in π» β₯ and π» are shown as a function of πΏ in ο¬gure (2). The near-closed-loop periodicity in π» β₯ with πΏ suggests the existence of some invariant of the motion. π» shows much smaller variation with πΏ , suggesting that it is an approximation to that invariant β thescale of variation of π» β₯ is an order of magnitude larger than the scale of variation of π» . It isfurthermore worth noting that π΄ π is not a turn-by-turn invariant due to the chromatic coupling, soall of the quantities used to compute π» are varying turn-by-turn.Comparing π» to π» β₯ with the synchrotron motion, as we do in ο¬gure (3), we can see that thestroboscopic Hamiltonian is very well-conserved throughout the synchrotron oscillation, while π» β₯ ,β 7 βarameter Value π π₯ . Γ ππ -79. π . π . π π π .42 πΌ π . π½ π . Γ β πΌ π . π π .08 Table 1 : Parameters of the toy octupole ring with synchrotron motion.
Figure 2 : Comparison of the coasting beam Hamiltonian π» β₯ and stroboscopic Hamiltonian π» asthe momentum varies over 500 turns, or approximately 40 synchrotron oscillations.which generates the turn-by-turn variation, varies periodically with the synchrotron motion. Thisperiodicity with the synchrotron motion is suggestive of underlying integrable motion β- if a systemis integrable, any dynamical quantity will exhibit some periodicity with the dynamics [18]. In ourcase, π» β₯ is periodic with the synchrotron motion, which suggests the existence of an underlyingintegrable Hamiltonian which has as one set of action-angle variables the synchrotron action andphase. That π» is extremely well-conserved suggests that π» is that Hamiltonian.This picture is unchanged with nonlinear rf eο¬ects, such as amplitude-dependent synchrotrontune. In ο¬gure (4), we see the particle motion at a larger initial π and with the rf potential π π π π replaced with π π π cos π in eqn. (4.3). We can see a stronger second harmonic oscillation in both π» β₯ and π» , but as with the case in ο¬gure (3), π» is very nearly conserved, with variations at the βΌ igure 3 : Turn-by-turn comparison of π» β₯ and π» with linear synchrotron motion. Figure 4 : Turn-by-turn comparison of π» β₯ and π» with nonlinear synchrotron motion.level, while π» β₯ has oscillations at the βΌ β
50% level.
To observe the existence of stroboscopic invariants in a complex system, we consider an integrablerapid-cycling synchrotron [19] (iRCS). This lattice design includes nonlinear integrable dynamics [4,5], which intrinsically includes a tune spread with transverse amplitude designed to Landau dampβ 9 βoherent instabilities. Because our prior analysis is independent of the nature of the transverseHamiltonian dynamics in the lattice, we expect to see two stroboscopic invariants of the motion.The iRCS transverse dynamics with synchrotron motion is quite complicated. As described byWebb et al. [20], we expect to see integrable behavior for momenta in ranges where the vertical andhorizontal chromaticity are equal. In these regions we would therefore expect the beam dynamicsto be much better behaved in general. Computing the stroboscopic Hamiltonian is diο¬cult for thefull lattice, as it must include a nonlinear normal form analysis of the transverse dynamics withmomentum spread as well as the longitudinal dynamics.We must therefore look for properties we would expect from the existence of an underlyingconstant of the motion, such as periodicity in the behavior of dynamical variables commensuratewith the periodicity we consider for the stroboscopic Hamiltonian, i.e. the synchrotron period.To test this periodicity prediction, we computed the on-momentum ( π΄ =
0) Danilov-Nagaitsevinvariants from [4] through many synchrotron oscillations. For small-amplitude synchrotron oscil-lations, we expect the eο¬ects of ο¬nite π΄ to be perturbative, and we will see a synchrotron motionperiodicity with the Danilov-Nagaitsev invariants.Table 2 shows the key parameters for this lattice design. The phase advance through thenonlinear insert of π = .
3, the nonlinear strength parameter is π‘ = .
3, and elliptic potentialparameter is π = .
14 m / . (see [4, 5]).The iRCS is designed with 1.680 MV total RF voltage to provide to achieve a 20 Hz ramprate and a 8 GeV extraction energy. In application, every other cell of the iRCS would containRF cavities and the harmonic number for the ring would be 113. For modeling purposes, each ofthe twelve periodic cells has RF cavities providing 140 kV and the harmonic number for the ringis 9 Γ = . Γ β . At the injection energy 0.8 GeV, the synchrotron tune for the ring is 0.08(and 0.007 per periodic cell).The iRCS lattice was optimized to control the discrepancy between the horizontal and verticaltune across the momentum span Β± .
5% without the use of sextupoles. The iRCS lattice also hasthe ο¬exibility to ο¬nely adjust the betatron tune-matching and chromaticity matching independently.Figure 6 shows the tune dependence on momentum, measured by tracking the small-amplitudebetatron oscillation of oο¬-momentum particles, with the strength of the elliptic element set to zero.The chromaticity combined with the nonlinear integrable optics makes the iRCS lattice a fairlycomplex example of synchro-betatron coupling.In ο¬g. (7) we plot the particle momentum oο¬set and on-momentum Danilov-Nagaitsev invari-ants against the turn number π times the zero-amplitude synchrotron tune π π . As we can see, thereis oscillatory behavior in the invariants periodic with the synchrotron oscillation, indicating theexistence of a stroboscopic invariant. We can also see that this is in a regime where there is aο¬nite amplitude-dependent synchrotron tune depression, as the successive minima in the top plotof ο¬g. (7) are slightly greater than π Γ π π = π π ( π΄ ) > π π ( ) .This periodicity is consistent over many hundreds of synchrotron periods, and across manyinitial particle trajectories. That persistence without secular growth indicates the presence of stro-boscopic invariants in the Danilov-Nagaitsev Hamiltonian with synchrotron motion and chromaticeο¬ects, as applied to the iRCS lattice. β 10 β igure 5 : (top) Twiss parameters for one of the twelve periodic cells. (bottom) Beamline layoutwhere dipoles are shown as short blue rectangles and quadrupoles as tall orange rectangles. Fractional Momentum Deviation p/p T un e D e v i a t i o n Q Q y Q x Figure 6 : Vertical and horizontal chromaticities are plotted for a single cell of the 12-cell iRCSdesign. β 11 β able 2 : Parameters of iRCSv3 LatticeParameter ValueCircumference 636 mPeriodicity 12Bend Radius 15.4 mMax Beta Function 30 mMax Dispersion 0.22 mBetatron Tune 21.6Linear Chromaticity -79Momentum Compaction 5.9 Γ β Insertion lengths per cell 7.2 m, 4 Γ . Γ π Nonlinear Strength t-value 0.3Elliptic Distance c-value 0.14 m
95% Transverse Emittance 20 mm mrad95% Longitudinal Emittance 0.09 eV Β· sVertical Lattice Tune Spread 0.52Horizontal Lattice Tune Spread 0.34Chromatic Tune spread 0.52 Figure 7 : Sample trajectory showing oscillations in the on-momentum invariants in an integrableRCS. β 12 β
Discussion
We have presented an approach to computing the π -turn map for an entire synchrotron period,and derived a stroboscopic Hamiltonian π» which deο¬nes the secular Hamiltonian dynamics ofthe full synchro-betatron coupling. The stroboscopic Hamiltonian is the average of the transverseHamiltonian over a synchrotron period. This Hamiltonian is O ( π β π ) , with π π (cid:28) O ( ) . Therefore, this holds well for small synchrotrontune. Furthermore, for multi-synchrotron-period maps, these correction terms will oscillate withthe number of periods, while the stroboscopic Hamiltonian term will grow linearly, suggesting thatit dominates the long-term dynamics. We presented evidence of this stroboscopic Hamiltonianin the context of an integrable optics rapid cycling synchrotron, showing that the on-momentumDanilov-Nagaitsev invariants vary with momentum oο¬set, but are periodic with the synchrotronperiod. The result, however, is generic to any Hamiltonian for the transverse dynamics, so long asa single Hamiltonian which generates the single-turn map for the transverse dynamics exists, i.e. inthe absence of chaos. A Symplectic Maps, Lie Algebras, and the Baker-Campbell-Hausdorο¬ Formula
In this Appendix we will overview the mathematics of symplectic maps and Lie operators, high-lighting key mathematical identities that we will use in this paper. Much of this is a survey of priorwork by Dragt and others [8β14] as it pertains to the work presented here. We omit proofs for thesake of brevity, opting to state the relevant identities.Given a Hamiltonian π» , the equations of motion for a particleβs phase space trajectory willsatisfy the Poisson bracket diο¬erential equation π (cid:174) π§ππ‘ = β[ π», (cid:174) π§ ] (A.1)We can interpret [ π», β ] as a Lie operator that acts on (cid:174) π§ , denoted by : π» :. This implies that theevolution of (cid:174) π§ can be cast as an operator diο¬erential equation, with the ο¬ow (cid:174) π§ π = M π β π (cid:174) π§ π . Thisleads to the operator diο¬erential equation for the symplectic map M which describes the ο¬ow forthe Hamiltonian π» : πππ‘ M π‘ π β π‘ = β : π» : M π‘ π β π‘ (A.2)with the initial condition M π‘ π β π‘ π = I , the identity. M contains all of the dynamics for theHamiltonian π» . We can solve this operator equation by iterative integration, i.e. M π‘ π β π‘ = I β β« π‘π‘ π ππ‘ (cid:48) : π» : M π‘ π β π‘ (cid:48) (A.3)Assuming π» is independent of time, the solution can be written as the exponential operator M π‘ π β π‘ = β βοΈ π = (β ) π π ! ( π‘ β π‘ π ) π : π» : π β‘ exp {β : π» : ( π‘ β π‘ π )} (A.4)β 13 βhere : π» : π is deο¬ned as repeated application of the operator : π» :. Operator exponentials play animportant role in Lie algebraic treatments of symplectic maps.In a particle accelerator, a symplectic map describes the change of phase space coordinates atthe exit of the element given the coordinates at the entrance of the element: (cid:174) π§ out = M π β¦ (cid:174) π§ in (A.5)In a ring, the product of all of these symplectic maps forms the single-turn map M = (cid:214) π M π (A.6)which contains the full dynamics of the ring. Computing this single-turn map is the subject ofnormal form analysis and Taylor and Cremona polynomials, as well as, indirectly, the goal oftracking codes. For the purposes of this paper, we assume that we have already calculated thesingle-turn map, and that it is of the form M = π β : π» : (A.7)where β π» is the generator of the map. This Hamiltonian is related to the invariants of motion, suchas the Courant-Snyder invariants or the Danilov-Nagaitsev Hamiltonian.The computation in this paper relies on two identities for these maps: the similarity transfor-mation, and the Baker-Campbell-Hausdorο¬ formula.The similarity transform states that G β : π : G = : G π : . (A.8)This identity frequently appears in the context of coordinate transformations, but in our case arisesas we move all the synchrotron motion maps to the left. It is straightforward to show that G β : π : π G = : G π : π . (A.9)by judicious insertion of GG β between each instance of : π :, and we can therefore see that G β π : π : G = π : G π : (A.10)The Baker-Campbell-Hausdorο¬ (BCH) formula provides a procedure for combining two non-commuting exponential Lie maps into one exponential Lie map, by deο¬ning a series for the generatorof that combined Lie map. If we want to write the product of two exponential Lie maps as a singleexponential Lie map, π : π : π : π : = π : β : , then the BCH formula tells us the series for β in terms of π and π : β = π + π + [ π , π ] + ( [ π , [ π , π ]] β [ π, [ π , π ]]) + . . . (A.11)Although this is a formal power series, it may be asymptotic and indeed may not converge at all.We therefore need [ π , π ] to be in some sense βsmallβ. This can mean multiple things, and the BCHseries can be a perturbation series in, for example, powers of (cid:174) π§ in the multipole picture of particleaccelerators, or in this case the synchrotron tune, as we discuss in Appendix B.β 14 β Leading Order Correction
To compute the next-leading order term for ο¬nite synchrotron tune, we need to go to the next orderin the BCH series. We will truncate the series at
O ( π ) , so that we only consider single pairwisePoisson brackets. From eqn. (3.16), we can add a term so that we are computing Λ π» ( π ) + ππ ( π ) ,where π is the next order Poisson bracket term. This immediately gives the recursion relation: ππ ( π + ) = ππ ( π ) + π (cid:34) Λ π» ( π ) , βοΈ π β π π ππ ( π β π π ) (cid:35) (B.1)with the initial condition that π ( ) =
0. Therefore, we have that π ( π ) = π βοΈ π = (cid:34) Λ π» ( π ) , βοΈ π β π π ππ ( π β ππ ) (cid:35) (B.2)and furthermore, from our solution of the leading order Hamiltonian Λ π» ( π ) , we have π ( π ) = π βοΈ π = π βοΈ π = βοΈ π,π (cid:48) (cid:104) β π (cid:48) π ππ (cid:48) π π β ππ (cid:48) ππ , β π π ππ π π β ππππ ) (cid:105) . (B.3)For clarity, deο¬ne π π = β π π ππ π and get that π ( π ) = π βοΈ π = π βοΈ π = βοΈ π,π (cid:48) (cid:104) π π (cid:48) π β ππ (cid:48) ππ , π π π β ππππ (cid:105) (B.4)The ( π, π (cid:48) ) = π πππ -type terms in the series. This means that this perturbationremains O ( π π ) , compared to the O ( π β π ) of the stroboscopic Hamiltonian. Acknowledgments
This work was supported in part by the United States Department of Energy, Oο¬ce of Science,Oο¬ce of High Energy Physics under contract no. DE-SC0011340 and in part by Fermi ResearchAlliance, LLC under Contract No. DE-AC02-07CH11359 with the United States Department ofEnergy.
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