Non-linear dynamics! in particle accelerators!
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1 Non-linear dynamics! in particle accelerators! Yannis PAPAPHILIPPOU Accelerator and Beam Physics group Beams Department CERN" Cockroft Institute Lecture Courses Daresbury, UK September !
2 Contents of the 3 rd lecture! n Lie formalism and symplectic maps! q Hamiltonian generators and Lie operators! q Map for the quadrupole! q Map for the general multi-pole! q Map concatenation! n Symplectic integrators! q Taylor map for the quadrupole! q Restoration of symplecticity! q 3-kick symplectic integrator! q Higher order symplectic integrators! q Accurate symplectic integrators with positive kicks (SABA 2 C)! n Normal forms! q Effective Hamiltonian! q Normal form for a perturbation! q Example for a thin octupole! n Summary! n Appendix : SABA 2 C integrator for accelerator elements! 2!
3 Contents of the 3 rd lecture! n Lie formalism and symplectic maps! q Hamiltonian generators and Lie operators! q Map for the quadrupole! q Map for the general multi-pole! q Map concatenation! n Symplectic integrators! q Taylor map for the quadrupole! q Restoration of symplecticity! q 3-kick symplectic integrator! q Higher order symplectic integrators! q Accurate symplectic integrators with positive kicks (SABA 2 C)! n Normal forms! q Effective Hamiltonian! q Normal form for a perturbation! q Example for a thin octupole! n Summary! n Appendix : SABA 2 C integrator for accelerator elements! 3!
4 Symplectic maps! n Consider two sets of canonical variables!!which may be even considered as the evolution of the system between two points in phase space! n A transformation from the one to the other set can be constructed through a map! M : z 7! z z, z M = M(z,t) M z i n The Jacobian matrix of the map!! is composed by the j n The map is symplectic if M T JM! =! J where! J = n It can be shown that the variables defined through a symplectic map! [ z i, z! j ]=[z! i,z j ]=J! ij! which is a known relation satisfied by canonical variables! n In other words, symplectic maps preserve Poisson brackets! n Symplectic maps provide a very useful framework to represent and analyze motion through an accelerator! 0 I I 0 4!
5 Lie formalism! n The Poisson bracket properties satisfy what is mathematically called a Lie algebra! n They can be represented by (Lie) operators of the form : f : g =[f,g] : f : 2 g =[f,[f,g]]!!!and!!!!etc.! H(z,t) n For a Hamiltonian system!!there is a formal solution of the equations of motion!!! P dt written as! z(t) =! 1 t k :H:! k k! z 0 =! e t:h: z! 0 with a symplectic k=0 map! M = e :H: n The 1-turn accelerator map can be represented by the composition of the maps of each element!!! M! = e :f! 2: e :f3:! e :f 4:!...!where! f i (called the generator) is the Hamiltonian for each element, a polynomial of degree! in the variables! m dz =[H, z] =:H : z z 1,...,z n 5!
6 Magnetic element Hamiltonians! n Dipole:! n Quadrupole:! n Sextupole:! n Octupole:! H = x + x2 H = 1 2 k 1(x 2 H = 1 3 k 2(x 3 H = 1 4 k 3(x p2 x + p 2 y 2(1 + ) y 2 )+ p2 x + p 2 y 2(1 + ) 3xy 2 )+ p2 x + p 2 y 2(1 + ) 6x 2 y 2 + y 4 )+ p2 x + p 2 y 2(1 + ) 6!
7 Lie operators for simple elements! 7!
8 Formulas for Lie operators! 8!
9 Map for quadrupole! n Consider the 1D quadrupole Hamiltonian! H = 1 2 (k 1x 2 + p 2 ) n For a quadrupole of length!, the map is written as! n Its application to the transverse variables is! 1X e L 2 :(k 1x 2 +p 2 ( ): k1 L 2 ) n x = x + L ( k 1L 2 ) n (2n)! (2n + 1)! p n=0 1X e L 2 :(k 1x 2 +p 2 ( ): k1 L 2 ) n p ( k 1 L 2 ) n p = p k1 (2n)! (2n + 1)! p n This finally provides the usual quadrupole matrix!! e L 2 :(k 1x 2 +p 2 ): x = cos( p k 1 L)x + p 1 sin( p k 1 L)p k1 e L 2 :(k 1x 2 +p 2 ): p = L e L 2 :(k 1x 2 +p 2 ): n=0 p k 1 sin( p k 1 L)x + cos( p k 1 L)p 9!
10 Map for general monomial! n Consider a monomial in the positions and momenta! n The map is written as! n Its application to the transverse variables is!! q For! q For! n 6= m x n p m e a:xn p m : e : xn p m: x = x 1+ (n m)x n 1 p m 1 m m n e : xn p m: p = p 1+ (n m)x n 1 p m 1 n n m n = m e : xn p n: x = xe nxn 1 p n 1 e : xn p n: p = pe nxn 1 p n 1 10!
11 Map Concatenation! n For combining together the different maps, the Campbell- Baker-Hausdorff theorem can be used. It states that for sufficiently small, and A,! B!real matrices, there is a real matrix C!for which! n For map composition through Lie operators, this is translated to!!! with! or! e sa e tb = e C e :h: = e :f: e :g: h = f + g : f : g : f :2 g : g :2 f : f :: g :2 f i.e. a series of Poisson bracket operations.! : g :4 f n Note that the full map is by construction symplectic.! n By truncating the map to a certain order, symplecticity is lost.! t 1,t : f :4 g +... h =! f + g [f,g] [f,[f,g]] [g, [g, f]] [f,[g, [g, f]]] [g, [g, [g, f]]] 1 [f,[f,[f,g]]] !
12 Contents of the 3 rd lecture! n Lie formalism and symplectic maps! q Hamiltonian generators and Lie operators! q Map for the quadrupole! q Map for the general multi-pole! q Map concatenation! n Symplectic integrators! q Taylor map for the quadrupole! q Restoration of symplecticity! q 3-kick symplectic integrator! q Higher order symplectic integrators! q Accurate symplectic integrators with positive kicks (SABA 2 C)! n Normal forms! q Effective Hamiltonian! q Normal form for a perturbation! q Example for a thin octupole! n Summary! n Appendix : SABA 2 C integrator for accelerator elements! 12!
13 Why symplecticity is important! n Symplecticity guarantees that the transformations in phase space are area preserving! n To understand what deviation from symplecticity produces consider the simple case of the quadrupole with the general matrix written as! cos( p kl) pk 1 sin( p! kl) M Q = p p p k sin( kl) cos( kl) n Take the Taylor expansion for small lengths, up to first order! M Q = 1 L kl 1 n This is indeed not symplectic as the determinant of the 1+kL 2 matrix is equal to!!, i.e. there is a deviation from symplecticity at 2 nd order in the quadrupole length! + O(L 2 ) 13!
14 Phase portrait for non-symplectic matrix! n The iterated non-symplectic matrix does not provide the well-know elliptic trajectory in phase space! n Although the trajectory is very close to the original one, it spirals outwards towards infinity! 14!
15 Restoring symplecticity! n Symplecticity be can restored by adding artificially a correcting term to the matrix to become! M Q = 1 L kl 1 kl 2 n In fact, the matrix now!!!!! can be decomposed as a drift!!!! with a thin quadrupole!!!! L! at the end!! n This representation,!!!! although not exact!!!! produces an ellipse!!!!! in phase space! = 1 0 kl 1 1 L 0 1 kl 15!
16 Restoring symplecticity II! n The same approach can be continued to 2 nd order of the Taylor map, by adding a 3 rd order correction! 1 1 M Q = 2 kl2 L 1 4 kl3 1 L/ L/2 1 = kl 1 2 kl2 0 1 kl n The matrix now can be!!!! decomposed as two half!!!! drifts with a thin kick at the! L! kl!! center! 2! n This representation!!!! now is is even more exact as!!!!! the error now is at!!!!! 3 rd order in the!!!!!! length! L 2 16!
17 3-kick symplectic integrator! n The idea is to distribute three kicks with different strengths so as to get a final map which is more accurate then the previous ones! n For the quadrupole, one can write!!!!!! 1 d1 L d2 L/ d3 L/2! 1! 0! M Q = 0! 1! c 1 kl 1! 0 1! c 2 kl! 1 0! 1! c 3 kl! 1 which imposes!!!.! n A symmetry condition of this form can be added! m11 m n This provides the matrix! M Q =!! 12! with! m 22 m 11 = m 22 = 1 2 kl2 + c 1 d 2 (d 1 + c 2 2 )k2 L 4 d 1 d 2 2c 2 1c 2 k 3 L 6! X di = X c i =1 d 1 = d 4, d 2 = d 3, c 1 = c 3 m 21 1 d4 L/2 0 1 m 12 = L ( c d 1d 2 +2d 1 d 2 c 1 )kl 3 +2d 1 d 2 c 1 (d 1 d 2 + c 2 2 )k2 L 5 + d 2 1d 2 2c 2 1c 2 k 3 L 7 m 21 = kl + c 1 d 2 (1 + c 2 )k 2 L 3 d 2 2c 2 1c 2 k 3 L 5 17!
18 3-kick symplectic integrator! n By imposing that the determinant is 1, the following additional relations are obtained! c 1 d 2 (d 1 + c 2 2 )= 1 24 n Although these are 5 equations!!! with 4 unknowns, solutions exist! d 1 = d 4 = c 1 = c 3 = c d 1d 2 +2d 1 d 2 c 1 = 1 6 c 1 d 2 (1 + c 2 )= (2 2 1/3 ), d 2 = d 3 = 1 21/3 2(2 2 1/3 ) 1 2 2, c 1/3 2 = 2 1/ /3 n This is actually the famous 7 step!!!! 4 th order symplectic integrator of Forest, Ruth and Yoshida (1990). It can be generalized for any non-linear element! n It imposes negative drifts! 18!
19 Higher order integrators! n Yoshida has proved that a general integrator map of order 2k!can be used to built a map of order 2k +2 S 2k+2! (t) =S! 2k (x! 1 t) S! 2k (x 0 t)! S 2k (x! 1 t)!!!! with! x 0 = 2 1 2k k+1, x 1 = n For example the 4 th order scheme!!! can be considered as a composition!!! of three 2 nd order ones (single kicks)!!!!!!!!!!!!!!!!! with! x 0 =, x = n A 6 th order integrator can be!!! produced by three interleaved 4 th order ones (9 kicks) S 6 (t)! =S! 4 (x 1 t)! S 4 (x 0! t) S 4!(x 1 t)!!!! with! x 0 =, x = k+1 S 4 (t) =S 2 (x 1 t) S 2 (x 0 t) S 2 (x 1 t) 19!
20 Modern symplectic integration schemes n Symplectic integrators with positive steps for Hamiltonian systems with both A and B integrable were proposed by McLachan (1995). n Laskar and Robutel (2001) derived all orders of such integrators n Consider the formal solution of the Hamiltonian system written in the Lie representation n A symplectic integrator of order n from to consists of approximating the Lie map by products of and which integrate exactly A and B over the time-spans and n The constants and are chosen to reduce the error 20!
21 SABA 2 integrator n The SABA 2 integrator is written as with n When is integrable, e.g. when A is quadratic in momenta and B depends only in positions, the accuracy of the integrator is improved by two small negative steps with n The accuracy of SABA 2 C is one order of magnitude higher than then than the Forest-Ruth 4 th order scheme n The usual drift-kick scheme corresponds to the 2 nd order integrator of this class 21!
22 Contents of the 3 rd lecture! n Lie formalism and symplectic maps! q Hamiltonian generators and Lie operators! q Map for the quadrupole! q Map for the general multi-pole! q Map concatenation! n Symplectic integrators! q Taylor map for the quadrupole! q Restoration of symplecticity! q 3-kick symplectic integrator! q Higher order symplectic integrators! q Accurate symplectic integrators with positive kicks (SABA 2 C)! n Normal forms! q Effective Hamiltonian! q Normal form for a perturbation! q Example for a thin octupole! n Summary! n Appendix : SABA 2 C integrator for accelerator elements! 22!
23 Normal forms! n Normal forms consists of finding a canonical transformation of the 1-turn map, so that it becomes simpler to analyze! n In the linear case, the Floquet transformation is a kind a normal form as it turns ellipses into circles" n The transformation can be written formally as M z!! z 0!!!!!!!!!! with the original map!! 1?? M = N! 1y!! y 1! and its normal form! u! N u 0 n The transformation! is better suited in action angle = e :F r: = e :F : N = M 1 = e :h eff : h variables, i.e.!!!taking the system from the original action-angle h ± x,y! = p 2J! x,y e! i x,y!to a new set! ± x,y(n) = p 2I x,y e i x,y(n)!!!! with the angles being just simple rotations, " x,y(n) =2 N x,y + x,y0 23!
24 Effective Hamiltonian! n The generating function can be written as a polynomial in the new actions, i.e.! F r = X f jklm x + j k x + l m y y = fjklm (2I x ) j+k 2 (2Iy ) l+m 2 e i jklm jklm n There are software tools that built this transformation" n Once the new effective Hamiltonian is known, all interesting quantities can be derived! n This Hamiltonian is a function only of the new actions, and to 3 rd order it is obtained as! h eff = x I x + y I y c 2 + c x1 I x + c y1 I y + c c xx I 2 x + c xy I x I y + c yy I 2 y + c x2 I x 2 + c y2 I y 2 c !
25 Effective Hamiltonian! n The correction of the tunes is given by!! Q x = eff = 1 x 2 x +2c xx I x + c xy I y + c x1 + c x2 2 Q y = eff = 1 y 2 y +2c yy I y + c xy I x + c y1 + c y2 2 tunes tune-shift 1 st and 2 nd order with amplitude chromaticity n The correction to the path length is! s eff = c + c c c x1 I x + c y1 I y +2c x2 I x +2c y2 I 1 st, 2 nd and 3 rd momentum compaction 25!
26 Normal form for perturbation! n Using the BCH formula, one can prove that the composition of two maps with! small can be written as! g n Consider a linear map (rotation) followed by a small perturbation! n We are seeking for transformation such that! n This can be written as! M = e :f 2: e :f 3: N = M 1 = e :F : e :f 2: e :f 3: e : F : N = e :f 2: e :f 2: e :F : e :f 2: e :f 3: e : F : = e :f 2: e :e :f 2 : F +f 3 F : +... = e :f 2: e :(e :f 2 : 1)F +f 3 : +... F = f 3 1 e :f 2: n This will transform the new map to a rotation to leading order! 26!
27 Example: Octupole! n Consider a linear map followed by an octupole!! n The generating function has to be chosen such as to make the following expression simpler!! M = e 2 :x2 +p 2: e : x4 4 : = e :f 2: e : x4 (e :f 2: n The simplest expression is the one that the angles are eliminated and there is only dependence on the action! n We pass to the resonance basis variables! n The perturbation is! 1)F + x4 4 h ± = p 2Je i = x ip 4 : x 4 =(h + + h ) 4 = h ± = h h 3 +h +6h 2 +h 2 +4h + h 3 + h 4 27!
28 Example: Octupole! n The term 6h! 2 +h 2! = 24J! 2 is independent on the angles. Thus we may choose the generating functions such that the other terms are eliminated. It takes the form! F = 1 16 h e 4i + 4h3 +h 1 e 2i + 4h +h 3 1 e n The map is now written as! M = e :F : e : J+ 3 8 J 2: e :F :! n The new effective Hamiltonian is depending only on the actions and contains the tune-shift terms! n The generator in the original variables is written as! F = e 4i 2i + h4 5x 4 +3p 4 +6x 2 p 2 +4x 3 p(2 cot( ) + cot(2 )) + 4xp 3 (2 cot( ) cot(2 )) 28!
29 Graphical resonance representation! n It is possible by constructing the one turn map to built the generating (sometimes called distortion ) function! F r X n For any resonance!!!, and setting!, the associated part of the functions is! jklm =0 jklm f jklm J j+k 2 x a x + bq y = c F (a,b) X jklm j+k+l+mapplen j+k=a,l+m=b J l+m 2 y e i jklm f jklm J a 2 x J b 2 y 29!
30 Normal forms for LHC models! YP and F. Schmidt, AIP 468, 1999 o With warm quad. errors o Without warm quad. errors n In the LHC at injection (450 GeV), beam stability is necessary over a very large number of turns (10 7 )! n Stability is reduced from random multi-pole imperfections mainly in the super-conducting magnets! n Area of stability (Dynamic aperture - DA) computed with particle tracking for a large number of random magnet error distributions! n Numerical tool based on normal form analysis (GRR) permitted identification of DA reduction reason (errors in the warm quadrupoles)! 30!
31 Contents of the 3 rd lecture! n Lie formalism and symplectic maps! q Hamiltonian generators and Lie operators! q Map for the quadrupole! q Map for the general multi-pole! q Map concatenation! n Symplectic integrators! q Taylor map for the quadrupole! q Restoration of symplecticity! q 3-kick symplectic integrator! q Higher order symplectic integrators! q Accurate symplectic integrators with positive kicks (SABA 2 C)! n Normal forms! q Effective Hamiltonian! q Normal form for a perturbation! q Example for a thin octupole! n Summary! n Appendix : SABA 2 C integrator for accelerator elements! 31!
32 Summary! n Symplectic maps are the natural way to represent accelerator dynamics! n They are obtained through Lie transformations! n Truncation of the map makes it deviate from symplecticity! n Symplecticity essential for preserving Hamiltonian structure of system (area preservation)! n Use symplectic integrators for tracking! n Even high order integrators with positive steps exist! n Normal form construction on the 1-turn map makes non-linear dynamics analysis straightforward! 32!
33 SABA 2 C for accelerators n The accelerator Hamiltonian in the small angle, hard-edge approximation is written as with the unperturbed part and the perturbation n The unperturbed part of the Hamiltonian can be integrated with 33!
34 SABA 2 C for accelerators II n The perturbation part of the Hamiltonian can be integrated n The corrector is expressed as with and the operator for the corrector is written as 34!
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