Nonlinear Single-Particle Dynamics in High Energy Accelerators

Transcription

1 Nonlinear Single-Particle Dynamic in High Energy Accelerator Part 6: Canonical Perturbation Theory Nonlinear Single-Particle Dynamic in High Energy Accelerator Thi coure conit of eight lecture: 1. Introduction ome example of nonlinear dynamic 2. Baic mathematical tool and concept 3. Repreentation of dynamical map 4. Integrator I 5. Integrator II 6. Canonical Perturbation Theory 7. Normal form analyi 8. Some numerical technique Nonlinear Dynamic 1 Part 6: Canonical Perturbation Theory

2 In the previou lecture... We have een how nonlinear dynamic can play an important role in ome divere and common accelerator ytem. Nonlinear effect have to be taken into account when deigning uch ytem. A number of powerful tool for analyi of nonlinear ytem can be developed from Hamiltonian mechanic. Uing thee tool, the olution to the equation of motion for a particle moving through a component in an accelerator beamline may be repreented in variou way, including: (truncated) power erie; Lie tranform; (implicit) generating function. In the cae that the Hamiltonian can be written a a um of integrable term, the algebra aociated with Lie tranform allow contruction of an explicit ymplectic integrator that i accurate to ome pecified order. Nonlinear Dynamic 2 Part 6: Canonical Perturbation Theory In thi lecture... We now tart to conider what happen when nonlinear element are combined in an accelerator beam line. In particular, we hall aim to undertand feature of phae pace portrait that can be contructed by tracking multiple particle multiple time through a periodic beam line (e.g. a torage ring). Nonlinear Dynamic 3 Part 6: Canonical Perturbation Theory

3 In thi lecture... There are two common approache to analyi of nonlinear periodic ytem: 1. Canonical perturbation theory, baed on canonical tranformation of an -dependent Hamiltonian. 2. Normal form analyi, baed on Lie tranformation of a ingle-turn map. In fact, both thee approache are really attempting to do the ame thing: the goal i to find a tranformation that put the Hamiltonian, or the map, into the implet poible form. In thi lecture, we hall develop the firt approach, canonical perturbation theory. Nonlinear Dynamic 4 Part 6: Canonical Perturbation Theory Hamiltonian for a periodic beam line For clarity, we hall work in one degree of freedom. The extenion to multiple degree of freedom lengthen the algebra, but doe not add any ignificant new feature. In action-angle variable, the Hamiltonian for a periodic beam line can be written a: H = J β + ɛv (φ, J, ), (1) where the perturbation term V contain the nonlinear feature of the dynamic, and ɛ i a mall parameter. Note that V depend on the independent variable, (the ditance along the reference trajectory). Nonlinear Dynamic 5 Part 6: Canonical Perturbation Theory

4 Hamiltonian for a periodic beam line In general, the equation of motion following from (1) are difficult to olve. Our goal i to find canonical tranformation that remove the nonlinear term from the Hamiltonian, to progreively higher order in ɛ. The equation of motion in the new variable will be eay to olve, at leat in the approximation that term of higher order in ɛ may be neglected. In fact, ince we explicitly remove any angle-dependence from the Hamiltonian, the olution to the equation of motion will atify: J 1 = contant (2) where J 1 i the action variable in the tranformed coordinate. Expreing the original coordinate J and φ in term of J 1 allow u eaily to contruct a phae pace portrait. Nonlinear Dynamic 6 Part 6: Canonical Perturbation Theory Hamiltonian for a periodic beam line For reaon that will become clearer hortly, let u generalie the Hamiltonian (1), by replacing the linear term by a general function of the action variable: Then, we define ω β a: H = H 0 (J) + ɛv(φ, J, ). (3) ω β = dh 0 dj. (4) For purely linear motion, H 0 = J/β, o ω β = 1/β which i the betatron frequency (the betatron phae advance per metre of beam line). But in general, ω β i a function of the betatron amplitude (i.e. a function of the betatron action, J). Nonlinear Dynamic 7 Part 6: Canonical Perturbation Theory

5 Generating function Let u conider a generating function of the econd kind: F 2 (φ, J 1, ) = φj 1 + ɛg(φ, J 1, ). (5) The new and old dynamical variable and Hamiltonian are related by: J = F 2 φ = J 1 + ɛ G φ, (6) φ 1 = F 2 J 1 = φ + ɛ G J 1, (7) H 1 = H + F 2 = H + ɛ G. (8) We ee that if ɛ i mall, then the tranformation i cloe to the identity. Nonlinear Dynamic 8 Part 6: Canonical Perturbation Theory Generating function The new Hamiltonian can be written a: ( H 1 = H 0 J 1 + ɛ G ) ( + ɛv φ, J 1 + ɛ G ) φ φ, + ɛ G. (9) Note that in thi form, the Hamiltonian i expreed in term of the mixed variable φ and J 1. Eventually, to olve the equation of motion, we will need to ubtitute for φ, o that the Hamiltonian i expreed purely in term of the new variable (φ 1, J 1 ). For now, however, it i convenient to leave the Hamiltonian in the mixed form. Nonlinear Dynamic 9 Part 6: Canonical Perturbation Theory

6 Generating function By adding and ubtracting appropriate term, we can rewrite the Hamiltonian a: ( H 1 = H 0 (J 1 ) + H 0 J 1 + ɛ G ) H 0 (J 1 ) φ ( +ɛv φ, J 1 + ɛ G ) φ, ɛv (φ, J 1, ) + ɛv (φ, J 1, ) +ɛ G, (10) [ H 0 (J 1 ) + ɛ ω β (J 1 ) G φ + G ] + V (φ, J 1, ) +ɛ 2 J 1 V (φ, J 1, ). (11) Note that in the lat tep, we have ued the definition ω β (J 1 ) = H 0 (J 1). Nonlinear Dynamic 10 Part 6: Canonical Perturbation Theory Generating function We ee that if we can find a generating function F 2 = φj 1 + ɛg(φ, J 1, ) where G atifie: ω β (J 1 ) G φ + G + V (φ, J 1, ) = 0, (12) then the term in ɛ in the Hamiltonian H 1 are econd-order and higher: H 1 H 0 (J 1 ) + ɛ 2 J 1 V (φ, J 1, ). (13) Nonlinear Dynamic 11 Part 6: Canonical Perturbation Theory

7 Generating function In an accelerator beam line, the perturbation V can be written in term of the coordinate x. Then, ince x i a periodic function of φ, it mut be the cae that V i alo a periodic function of φ. Therefore, we can write V a a um over mode: V (φ, J 1, ) = m Ṽ m (J 1, ) e imφ. (14) We aume that the function G appearing a a term in the generating function F 2 i alo periodic in φ: G(φ, J 1, ) = m G m (J 1, ) e imφ. (15) Then, equation (12) become: ( imω β (J 1 ) + ) G m = Ṽ m. (16) Nonlinear Dynamic 12 Part 6: Canonical Perturbation Theory Generating function By ubtitution into equation (16), we ee that the olution for G m (J 1, ) i: ( ) i +L G m (J 1, ) = 2in ( 1 2 mω β L ) e imω β L 2 Ṽ m (J 1, ) d, (17) where L i the length of one periodic ection of the beam line. Strictly peaking, thi olution aume that the betatron frequency ω β i contant along the beam line (i.e. i independent of ). However, we can generalie the reult. If we define the phae ψ() and the tune ν: ψ() = ω β d, ν = 1 ω β d, (18) 0 2π then we can write the expreion for G m (J 1, ): G m (J 1, ) = i 2in(πmν) e im[ψ( ) ψ() πν] Ṽm (J 1, ) d. (19) Nonlinear Dynamic 13 Part 6: Canonical Perturbation Theory

8 Generating function Finally, the function G m (φ, J 1, ) can be written: G(φ, J 1, ) = m i 2in(πmν) e im[φ+ψ( ) ψ() πν] Ṽm (J 1, ) d. (20) Equation (20) i an important reult: it tell u how to contruct a canonical tranformation that remove (to firt order) an -dependent perturbation from the Hamiltonian. We can already ee ome intereting propertie. For example, the expreion for G get large when mν i cloe to an integer. Even a mall perturbation can have a large effect when the lattice i tuned to a reonance: but the impact alo depend on the reonance trength, i.e. the integral in (20). If the reonance trength i zero, then we can it right on a reonance without advere effect. Nonlinear Dynamic 14 Part 6: Canonical Perturbation Theory Generating function Note that if V (φ, J 1, ) ha a non-zero average over φ, then Ṽ 0 (J 1, ) i non-zero. Thi implie there i a reonance trength for m = 0, given by: Ṽ 0 ( ) d. (21) But for m = 0, mν i an integer (zero) for all ν. Therefore, no matter what the tune of the lattice, we cannot contruct a generating function to remove any term in the perturbation that i independent of φ. However, uch term may be aborbed into H 0 (J): thi i the reaon why we introduced H 0 (J) (a a generaliation of J/β) in the firt place. Nonlinear Dynamic 15 Part 6: Canonical Perturbation Theory

9 Generating function The canonical tranformation (5): F 2 (φ, J 1, ) = φj 1 + ɛg(φ, J 1, ) remove the perturbation term in the Hamiltonian, o that: H 1 = H 0 (J 1 ) + O(ɛ 2 ). (22) Since H 1 i independent of φ 1 (to firt order in ɛ), we can write: J 1 = contant + O(ɛ 2 ). (23) The original action variable i then given by (6): J = J 1 + ɛ G φ = J 0 + ɛ φ G(φ, J 0, ) + O(ɛ 2 ), (24) where J 0 i a contant. At a given location in the beam line (i.e. for a given ), we can contruct a phae pace portrait by plotting J a a function of φ (between 0 and 2π) for different value of J 0. Nonlinear Dynamic 16 Part 6: Canonical Perturbation Theory Firt-order perturbation theory: quadrupole example Equipped with equation (20), we are now ready to look at ome example. Before proceeding to a nonlinear cae (extupole perturbation), we will look at a linear perturbation, namely, quadrupole focuing. Thi will allow u to compare quantitie uch a the tune hift and phae pace (i.e. beta function) ditortion that we obtain from perturbation theory, with the expreion that we may obtain from a purely linear theory. In carteian variable, the Hamiltonian i: H = p2 2 + K()x2 2 + ɛk()x2 2. (25) The firt two term can be written in action-angle variable a J/β(). The perturbation term (in ɛ) can be tranformed into action-angle variable uing: x = 2β()J co φ. (26) Thu, we have: H = J β() + 1 ɛk()β()j(1 + co2φ). (27) 2 Nonlinear Dynamic 17 Part 6: Canonical Perturbation Theory

10 Firt-order perturbation theory: quadrupole example We ee immediately that the perturbation introduce a term independent of φ. We therefore define: H 0 (J) = J β() + 1 ɛk()β()j, (28) 2 o that the Hamiltonian i written: H = H 0 (J) + 1 ɛk()β()j co2φ. (29) 2 The tune of the lattice (i.e. the phae advance acro one periodic ection of length L) i given by: ν = 1 2π ω β d = 1 2π dh 0 dj From equation (28), we ee that the tune with the perturbation i: ν = 1 2π d. (30) d β() + ɛ k()β() d. (31) 4π Nonlinear Dynamic 18 Part 6: Canonical Perturbation Theory Firt-order perturbation theory: quadrupole example The change in tune reulting from the perturbation i: ν = 1 d 2π β() + ɛ k()β() d. (32) 4π Thi i the ame expreion we would have obtained uing tandard linear (matrix) theory. Nonlinear Dynamic 19 Part 6: Canonical Perturbation Theory

11 Firt-order perturbation theory: quadrupole example To find the change in beta function, we need to ue equation (20). Thi will give u a (canonical) tranformation to new variable J 1, φ 1 ; to firt order in the perturbation (ɛ), J 1 i contant. Firt, we need the Fourier tranform of the (φ dependent part of the) perturbation. From equation (14) where: Thu: Ṽ m (J 1, ) = 1 2π e imφ V (φ, J 1, ) dφ, (33) 2π 0 with all other Ṽ m equal to zero. V (φ, J 1, ) = 1 2 k()β()j 1co2φ. (34) Ṽ ±2 (J 1, ) = 1 4 k()β()j 1, (35) Nonlinear Dynamic 20 Part 6: Canonical Perturbation Theory Firt-order perturbation theory: quadrupole example Subtituting for Ṽ ±2 (J 1, ) into equation (20) give: J 1 G(φ, J 1, ) = 4in(2πν) in2 ( φ + ψ( ) ψ() πν ) k( )β( ) d. (36) Note that the phae advance ψ and the tune ν include the phae hift reulting from the perturbation, i.e. d ψ() = 0 β() + ɛ k()β() d, (37) 2 0 and (32): ν = 1 d 2π β() + ɛ k()β() d. (38) 4π Nonlinear Dynamic 21 Part 6: Canonical Perturbation Theory

12 Firt-order perturbation theory: quadrupole example At a given location in the beam line, the invariant curve in phae pace (the contour line in a phae pace portrait) are given by (24): J = J 0 + ɛ φ G(φ, J 0, ) + O(ɛ 2 ), (39) where J 0 i a contant, and φ varie from 0 to 2π. Uing (36), we find: ɛj J = J 0 0 2in(2πν) co2 ( φ + ψ( ) ψ() πν ) k( )β( ) d +O(ɛ 2 ). (40) Nonlinear Dynamic 22 Part 6: Canonical Perturbation Theory Firt-order perturbation theory: quadrupole example The effect of a quadrupole perturbation may be repreented a a ditortion of the beta function. Thi can be calculated by conidering a particle with a given x coordinate, and with φ = 0. In term of the unperturbed action J and beta function β, we would write: x = 2βJ. (41) In term of the perturbed action J 0 and beta function β 0 we have: x = 2β 0 J 0. (42) Therefore, ince βj = β 0 J 0, we have: β β = J J = J 0 J. (43) J So, from (40), the change in the beta function reulting from a mall quadrupole perturbation i: β β = ɛ 2in(2πν) co2 ( φ + ψ( ) ψ() πν ) k( )β( ) d. (44) Nonlinear Dynamic 23 Part 6: Canonical Perturbation Theory

13 Firt-order perturbation theory: extupole example A a econd example, let u conider the perturbation introduced by extupole component in the lattice. Again, we follow cloely the analyi of Ruth. And, to keep thing imple, we again conider only one degree of freedom. The Hamiltonian i given by: H = p2 2 + K()x2 2 + ɛk 2() x3 6. (45) In action-angle variable, the Hamiltonian become: where: H = J β() + ɛk 2() (2β()J) co 3 φ, (46) = H 0 (J, ) + ɛv (φ, J, ), (47) H 0 (J, ) = J β(), and V (φ, J, ) = k 2() (2β()J) co 3 φ. Nonlinear Dynamic 24 Part 6: Canonical Perturbation Theory Firt-order perturbation theory: extupole example We can write the perturbation term a: V (φ, J, ) = 1 24 k 2()(2β()J) 3 2 (co3φ + 3co φ). (48) Uing equation (20), the generating function that will remove the perturbation to firt order in ɛ i F = φj 1 + ɛg, where: G = (2J 1) 3 { in πν 1 + 3in3πν where: k 2 ( )β( ) 3 2 in [ φ + ψ( ) ψ() πν ] d k 2 ( )β( ) 3 2 in3 [ φ + ψ( ) ψ() πν ] d } ψ() = (49) d β( ). (50) Nonlinear Dynamic 25 Part 6: Canonical Perturbation Theory

14 Firt-order perturbation theory: extupole example The firt-order invariant i given by equation (24): J = J 0 + ɛ φ G(φ, J 0, ) + O(ɛ 2 ). If we know the extupole ditribution along the beamline, we can evaluate G from (49). Equation (24) then give u the invariant, which allow u to plot the phae pace. A a pecific example, conider a ingle thin extupole located at = 0. The extupole trength k 2 () can be repreented by a Dirac delta function: k 2 () = k 2 l δ(0). (51) Nonlinear Dynamic 26 Part 6: Canonical Perturbation Theory Firt-order perturbation theory: extupole example We hall plot the phae pace at = 0. From equation (49), we find that: G = (2J 1) 2 3 [ ] 16 k 2l β(0) 2 3 in(φ πν) in3(φ πν) +. (52) in πν 3in3πν From equation (24) the action variable J at = 0 i related to the (firt order) invariant J 1 by: J = J 1 + ɛ G φ = J 1 ɛ (2J 1) k 2l β(0) 3 2 [ ] co(φ πν) co3(φ πν) +. in πν in 3πν (53) Nonlinear Dynamic 27 Part 6: Canonical Perturbation Theory

15 Firt-order perturbation theory: extupole example We hall chooe value k 2 l = 600m 2, and β(0) = 1m, and plot the contour in phae pace obtained from: 2J x = 2βJ co φ, p = in φ, (54) β for 0 < φ < 2π, and for a et of value of J 1. The phae pace depend on the tune, ν. For a range of value of the tune, we hall compare the phae pace plot obtained from perturbation theory, with the plot obtained by tracking particle, applying extupole kick with a phae advance between one kick and the next. Nonlinear Dynamic 28 Part 6: Canonical Perturbation Theory Firt-order perturbation theory: extupole example Nonlinear Dynamic 29 Part 6: Canonical Perturbation Theory

16 Firt-order perturbation theory: extupole example Nonlinear Dynamic 30 Part 6: Canonical Perturbation Theory Firt-order perturbation theory: extupole example Nonlinear Dynamic 31 Part 6: Canonical Perturbation Theory

17 Firt-order perturbation theory: extupole example Nonlinear Dynamic 32 Part 6: Canonical Perturbation Theory Firt-order perturbation theory: extupole example Nonlinear Dynamic 33 Part 6: Canonical Perturbation Theory

18 Firt-order perturbation theory: extupole example Nonlinear Dynamic 34 Part 6: Canonical Perturbation Theory Firt-order perturbation theory: extupole example There doe eem to be ome reemblance between the phae pace plot obtained from perturbation theory, and the plot obtained from tracking: there i the ame general kind of ditortion, and clearly dramatic effect around the third integer reonance. However, perturbation theory mie many of the detail revealed by tracking, and alo ome nonenical feature: the contour hould not cro in the way they appear to do for particular tune. We can hope that applying perturbation theory to progreively higher order improve the ituation... but that i beyond the cope of thi coure. Nonlinear Dynamic 35 Part 6: Canonical Perturbation Theory

19 Motion near a reonance To finih thi lecture, we conider in a little more detail the motion near a reonance. We have een how, if we are not cloe to a reonance, we can apply perturbation theory to find a canonical tranformation to variable in which the Hamiltonian take the form: H 1 = H 0 (J 1 ), (55) i.e. the Hamiltonian i purely a function of the (new) action variable. Recall that we could not ue a generating function to remove term from the Hamiltonian independent of the angle variable, φ. But uch term imply lead to a tune hift with amplitude: and uch effect are fairly innocuou. In particular, they do not ditort the phae pace, or limit the tability of the particle motion. Nonlinear Dynamic 36 Part 6: Canonical Perturbation Theory Motion near a reonance We alo aw, from equation (20), that to remove a term Ṽ m co mφ from the Hamiltonian, we need a term in the generating function given by: G = 1 2in(πmν) Ṽ m in m [ φ + ψ( ) ψ() πν ] d. (56) If mν approache an integer value, the generating function diverge, and we cannot hope for an accurate reult. We aw thi in the extupole (m = 3) example, above, where we tarted eeing trange reult cloe to ν = 1/3. Generally, therefore, after applying perturbation theory, we are left with a Hamiltonian of the form: H = H 0 (J, ) + f(j, )co mφ. (57) Nonlinear Dynamic 37 Part 6: Canonical Perturbation Theory

20 Motion near a reonance Let u aume that, by making an -dependent canonical tranformation, we can remove the -dependence of the Hamiltonian, and put it into the form: H = H 0 (J) + f(j)co(mφ). (58) For the pecial cae, the generating function: H = J β(), (59) F 2 = φj 1 + J 1 2πν L J 1 lead to the -independent Hamiltonian: where J 1 = J. 0 d β( ), (60) H 1 = 2πν L J 1, (61) Nonlinear Dynamic 38 Part 6: Canonical Perturbation Theory Motion near a reonance Since, in the new variable, the Hamiltonian i independent of, the Hamiltonian give u a contant of the motion. That i, the evolution of a particle in phae pace mut follow a contour on which the value of H i contant. A a pecific example, conider the Hamiltonian: H = 1.6J 4J 2 + J 3 co6φ. (62) Thi repreent dynamic with a firt-order tune hift (with repect to J), and a ixth-order reonance driving term. We can eaily plot the countour of contant value for H... Nonlinear Dynamic 39 Part 6: Canonical Perturbation Theory

21 Motion near a reonance Nonlinear Dynamic 40 Part 6: Canonical Perturbation Theory Motion near a reonance The phae pace clearly reemble that contructed from repeated application of a map repreenting a periodic ection of beam line near a ixth-order reonance. Although there i a fundamental difference between the two cae (in one cae the Hamiltonian i time-dependent, and in the other cae it i time-independent) tudying the continuou cae can help in undertanding feature of the dicrete cae. The width of the iland i related to both the tune hift with amplitude, and the trength of the driving term. Without a reonance term, the iland would vanih. The centre of the iland repreent table fixed point. There are alo untable fixed point, where the contour line appear to cro. The line paing through the untable fixed point divide the iland from the ret of phae pace: uch a line i known a a eparatrix. Nonlinear Dynamic 41 Part 6: Canonical Perturbation Theory

22 Motion near a reonance It i poible to carry the analyi of reonant ytem much further. In particular, it i poible to derive expreion for uch thing a the width of the iland, in term of the tune hift with amplitude, and the trength of the reonant driving term. For more information, ee Ruth. In general, there will be more than one reonance preent in a ytem. For example, extupole in a lattice can combine to drive reonance of any order. Perturbation theory can help to reveal the trength of the driving term of the different reonance. The onet of chaotic motion can be aociated with two et of reonant iland that overlap. Thi condition for chaotic motion i known a the Chirikov criterion. Nonlinear Dynamic 42 Part 6: Canonical Perturbation Theory Illutrating reonance: frequency map analyi In two degree of freedom, a reonance i pecified by integer (m, n) for which the following condition i atified: where l i an integer. mν x + nν y = l, (63) Frequency map analyi (ee lecture 8), baed on tracking tudie, can indicate the trength of different reonance: not all reonance may be harmful. Nonlinear Dynamic 43 Part 6: Canonical Perturbation Theory

23 The Kolmogorov-Arnold-Moer Theorem ( ) The KAM theorem tate that if [an integrable Hamiltonian] ytem i ubjected to a weak nonlinear perturbation, ome of the invariant tori are deformed and urvive, while other are detroyed. The one that urvive are thoe that have ufficiently irrational frequencie (thi i known a the non-reonance condition). Thi implie that the motion continue to be quaiperiodic, with the independent period changed... The KAM theorem pecifie quantitatively what level of perturbation can be applied for thi to be true. An important conequence of the KAM theorem i that for a large et of initial condition the motion remain perpetually quaiperiodic. Wikipedia. Nonlinear Dynamic 44 Part 6: Canonical Perturbation Theory Summary In applying perturbation theory, we contruct a canonical tranformation that put the Hamiltonian into a imple a form a poible. The dynamic in the new Hamiltonian are (in principle) impler to olve. Then, the dynamic with the original Hamiltonian are obtained from the relationhip between the old and new variable, defined by the canonical tranformation. Term purely dependent on J cannot be removed from the Hamiltonian; nor can reonant term, uch a co mφ, if the tune i cloe to a reonance, i.e. if mν i cloe to an integer. Some of the ignificant feature of dynamic near a reonance can be undertood in term of a Hamiltonian in which non-reonant term have been removed (by applying perturbation theory). Nonlinear Dynamic 45 Part 6: Canonical Perturbation Theory

24 Further reading Much of the material in thi lecture follow cloely the report by Ruth: R. Ruth, Single particle dynamic in circular accelerator, SLAC-PUB-4103 (1986). Thi report give a very clear explanation of the ubject. It alo goe ome way beyond the material in thi lecture, with ome dicuion on Hamilton-Jacobi theory, and the Chirikov criterion for the onet of chaotic motion. Nonlinear Dynamic 46 Part 6: Canonical Perturbation Theory Coming next... Perturbation theory can alo be carried out in the framework of Lie tranformation. In that context, it i generally known a normal form analyi. In the next lecture, we hall develop the theory of normal form analyi, and how how it can be applied in ome imple cae. Nonlinear Dynamic 47 Part 6: Canonical Perturbation Theory

25 Exercie Apply perturbation theory to an octupole perturbation. Hence, find the tune hift with amplitude generated by an octupole. Nonlinear Dynamic 48 Part 6: Canonical Perturbation Theory