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!

Poisson Brackets and Lie Operators

Poisson Brackets and Lie Operators T. Satogata January 22, 2008 1 Symplecticity and Poisson Brackets 1.1 Symplecticity Consider an n-dimensional 2n-dimensional phase space) linear system. Let the canonical