J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
1
UW Spring 2008
Accelerator Physics
Accelerator PhysicsTopic III
Perturbations and Nonlinear Dynamics
Joseph BisognanoSynchrotron Radiation Center
University of Wisconsin
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
2
UW Spring 2008
Accelerator Physics
Chromaticity
•
pppxyKxKH
etcdsdxp and pp-(p with and p,by Dividing
xpp2pp
2pp
yKxKpH
is equationsorder first gives that nHamiltonia The
yx
x0
2y
2x
2ˆ
2ˆ]
22)1)[(1(
.,/ˆ/)
)(
]22
)1[(
2222
2
0
0
22
20
From form, it’s clear tune will depend on momentum
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
3
UW Spring 2008
Accelerator Physics
Sextupoles
• A sextupole field can remove much of this
))(()1()1(
)33(6
)()1()1(
2ˆ
2ˆ)3(
6)()1(
)1(]22
)1)[(1(
)3(6
)(
22
23
22
2
230
xysSyKy
yxsSxKx
pppxxyxsS
yKxKHH
xyxsSpc
eA
y
x
yx
s
s
Tune change
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
4
UW Spring 2008
Accelerator Physics
Natural Chromaticity
][
)()(
)()(
minmax41
,1,1,41
,
1141
ffC
lattice FODO a For
CsKs
dssks :Recall
yxyxyxyx
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
5
UW Spring 2008
Accelerator Physics
Dispersion to the Rescue
dependence weak withfunction, periodic a is Dρ
)(DSD)(D)K--(D
satisfiesD dispersion Recall
x
12
11 2
We can move to orbit at energy offset by canonical transform
))(ˆ))((()ˆ,(2 sDpsDxpxFLet
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
6
UW Spring 2008
Accelerator Physics
Chromaticity Correction
)ˆˆ(2
)ˆˆ3ˆ(6
)(2ˆ)(
2ˆ)[(
)ˆˆ3ˆ)((2ˆ
2ˆ
2ˆ
2ˆˆ
ˆ
)(ˆ);(ˆ
ˆ
2222322
232222
yxSDyxxsSyKSDxKSD
yxxsSyKxKppH
sFHH
sDpxFpsDx
pFx
yx
yxyx
s
ss
Judicious choice of SD vs K’s can cancel chromaticityPrice: NONLINEARITY
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
7
UW Spring 2008
Accelerator Physics
Linear Coupling
])[()(
))(())(([21~
)(;)()(
222222
xzzxsgxzsp
ysgKxsgKyxH
xsBgAzsBgA :SolenoidsxzsBpA :quads Skew
Potentials Vector
yx
zx
s
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
8
UW Spring 2008
Accelerator Physics
Linear Coupling with Skew Quads
1,
)(
))()(sin())()(cos()(2)(
))()(cos()(2)(
,
))()((21
yx
ll
lliyxyx 1
1
uuuu
uu
ll1- where
eaaspH
have weterm, quad skewthe beH Lettingcanonical"" be to out turns this y; or xu where
sssssasu
sssasuconstants of variationtry wesolutions,
uncoupled the of form the Following
yx
yyyxxx
See Wiedemann II
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
9
UW Spring 2008
Accelerator Physics
Linear Coupling/cont.
rNlterm slow
qNl where
laaH
dsesp
with
eaa
espsH
length) period lattice Ls/L;2 (withterms varying slowout separatetoStrategy
yx
yx
yxyxq
ql
LqNlllli
yxql
lllliyx
lllli
llyx1
yyxxyyxx
yyxxyyxx
yyxxyyxx
yx
00
00
0
)(
)(
)(
,21
)cos()(~
)(21
)()(
:
00
00
00
periodic
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
10
UW Spring 2008
Accelerator Physics
Linear Coupling/cont.
)~~cos(~
);~~sin(
)~~cos(~
);~~sin(
)~~cos()(
~~~~
~~)(~)(~
21
21
21
21
21
21
21
21
21
21
21
yxy
xrlr
yyxyxrl
y
yxx
yrlr
xyxyxrl
x
yxyxrlyrxr
yrxrr
yyryy
xxrxx
ryyrxx
laallaal
a
laa
laaa
laaala
alaHH
aa l
aa
laaG Using
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
11
UW Spring 2008
Accelerator Physics
Difference Resonance l=-1
;sin2
)(;sin2
cos)(
)(2
);(2
;
0)(
000
~~
iwviwww
w with startsnoscillatio ssumingA
vkwiddvwkvi
ddw
eavea wLet
widthStopband
constant or
STABLEaadd
r
0
rr
iy
ix
yx
yx
yx
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
12
UW Spring 2008
Accelerator Physics
Difference Resonance/cont.
220,0,
22
2
22
2
0222
20 )sin4
);cos(41
rryxIII
ry
x
xyrxx aaaa
Implies measurement scheme for
tunes
quad
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
13
UW Spring 2008
Accelerator Physics
Sum Resonance
2221
;0)(
r
iy
ix
yx
with
eaieauform the of are Solutions
YINSTABILIT ofy possibilit constant, remains
differenceonly sinceaadd
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
14
UW Spring 2008
Accelerator Physics
Action Angle Variables
222
1
2
1
1
22
tan22
/
tan2
),(
/;tan)sin()();cos()(
)sin();cos(2
xxFJ
xxF
xFpx-p SinceJapJax
for Looktaptax
x2pH
Oscillator Harmonic2
Ruth/Wiedemann
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
15
UW Spring 2008
Accelerator Physics
Action Angle Variables/cont.
]2
[tan2
),(
]cos2
[sin)(
cos)(
)])(cos(2
))([sin(
))(cos(2)(
sin2;cos/2
)(21
2
1
2/1
2/1
2/1
2/1
2
222
zzF
Japp
JazWant
ssap
saz
zsK2pH
have weproblem, raccelerato for NowJK
JpJx
pxJ
2
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
16
UW Spring 2008
Accelerator Physics
Action Angle Variables/cont.
JRsFHH
JJssd
Cs
Jssd
CsJJF withFinally
JpJxssds
sJsFHK
zpzFJ
s
s
s
ˆ)/(/ˆ
ˆ;)(
2ˆ
ˆ])(
2[ˆ)ˆ,(,
)cos(sin/2;cos2)(
)(
)(//2
)2
(2
2
0
02
2
00
22
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
17
UW Spring 2008
Accelerator Physics
Canonical Perturbation Theory
θG/HH
ΦG/JJ
JG/ΦΦ
θ)J,ΦG(JΦθ),J(ΦΦF
tiontransforma canonical Consider
Φθ, over 0V of average Assume2π period has H Assume
JH(J) nonlinear bemay (J)Hθ)J;V(Φ((J)HHan of Form HamiltoniStart with
2
00
0
ˆ
ˆˆ
ˆˆ
ˆˆˆ
/
Following R. Ruth
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
18
UW Spring 2008
Accelerator Physics
Canonical Perturbation Theory/cont.
, of tindependen left everythingGJVGJ
IfGJVGJ
JVGJV
GJJHGJHJHH
GGJVGJHH
0),ˆ,()(
),ˆ,()(
)],ˆ,(),ˆ,([
])()ˆ()ˆ([)ˆ(ˆ),ˆ,()ˆ(ˆ
000
0
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
19
UW Spring 2008
Accelerator Physics
Canonical Perturbation Theory/cont.
nm
inimnm
m
nmnm
inim
mnm
inim
mnm
nmeV
iG
VGinimThen
eJGJG
eJVJV
solutionsperiodic for Look
,
,
,,
,
,
][
)ˆ(),ˆ,(
)ˆ(),ˆ,(
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
20
UW Spring 2008
Accelerator Physics
Canonical Perturbation Theory/cont.
JV)J(ν)Jν(
spreadfrequency gives Vagain over game repeat can We
order secondV whereVJHH
tiontransforma this With
0
ˆ/ˆˆ
)ˆ(ˆ
0
,
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
21
UW Spring 2008
Accelerator Physics
Octopole
dependent amplitude JdsBchangetune
JB1//dsdhave weterm constant forSo
BJsJHJBsJH
JBsJH
BxxsKpH
C
]23[
3,
)32cos44(cos)(/)cos4()(/
)cos2()(/
\2
)(2
0
2
2
22
4422
4
422
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
22
UW Spring 2008
Accelerator Physics
Isolated Resonance
mn and mJfJJJ mnHH
JJmn
JmnF
)n-f(J)cos(m(J)JHE.g.,
tionstransforma canonical more Rather,apply tdoesn'theory onperturbati
singular,getting rsdenominato Sincenm whenoccur can Resonances
2
/)cos()ˆ()ˆ(ˆˆ/ˆ
ˆ;/
ˆ)/(
1
1
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
23
UW Spring 2008
Accelerator Physics
Fixed Points
stabilitydetermines m of Sign
mJfJ
m
HJH
ˆcos
0ˆcos)ˆ()ˆ(
0ˆsin
ˆ/ˆ0ˆ/ˆ
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
24
UW Spring 2008
Accelerator Physics
Island Structure
From Ruth
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
25
UW Spring 2008
Accelerator Physics
Resonance Widths
• Expanding around unstable fixed point at a resonance action Jr yields an equation for the separatrix, and, on expanding, a “bucket height” or width
small assuming,)()(2)(
cos)()(cos)()(
2 fJJfJJ
mJfJJmJfJJ
r
rr
rrrr
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
26
UW Spring 2008
Accelerator Physics
R Ruth
Avoiding Low Order Resonances
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
27
UW Spring 2008
Accelerator Physics
Dynamic Aperture
resonances sumfor saw weas , stabilityfor ntinsufficie
is that but integrals, Poincare volumes, spacephase preserves Map
0000
0000010
J
e.g., is, J where,JJM MThen,
jacobian be MLetpq
MPQ
mapping a yield equations sHamilton
T
0110
1
;~
'
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
28
UW Spring 2008
Accelerator Physics
Eigenvalues
buys condition symplecticthe whatsthat' *;*,1/,,1/
quadruple in comethey thatcondition satisfy Mof seigenvalue The
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
29
UW Spring 2008
Accelerator Physics
Surface of Section• For an nD time independent Hamiltonian, energy is
conserved, and motion is on shell, a (2n-1)D set• Condition qn=constant gives (2n-2) surface, a surface
of section• Let’s take a look at Henon map, with the Hamiltonian
having a cubic nonlinearity, sort of sextupole like
)2()( 323
22
21
22
212
121
212
1 qqqqqppH
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
30
UW Spring 2008
Accelerator Physics
Position Plot of Henon Map
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
31
UW Spring 2008
Accelerator Physics
E=1/12
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
32
UW Spring 2008
Accelerator Physics
E=1/8
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
33
UW Spring 2008
Accelerator Physics
E Almost 1/6
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
34
UW Spring 2008
Accelerator Physics
Dynamic Aperture
• Determines usable aperture of accelerator, which must be consistent with emittance, injection gymnastics
• Determines whether intrabeam scattered particles survive and be damped in electron machines
• Definition: Region in phase space where particles have stable motion, will be stored indefinitely
• More practically, will particles remain in the machine for the planned storage time; e.g., 107-109 turns in proton accelerators, or synchrotron damping times (104 turns in electron storage rings
• For higher dimensional systems Arnold diffusion adds further complications, but we will take a practical approach
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
35
UW Spring 2008
Accelerator Physics
Tools
• Tracking (approximate computer mapping) is primary game
• But tracking for “storage time” is still beyond computational limits, so some “numerically derived” criteria to extrapolate are essential
• Since systems are “chaotic,” they are very sensitive to initial conditions and numerical error, so one has to be careful
Scandale, et al.
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
36
UW Spring 2008
Accelerator Physics
Tracking Tools• Work-horse programs such as MAD, SIXTRACK use
transfer maps for linear part of mapping, but “thin lens” approximation for nonlinearities. This maintains symplecticity of transforms
• Extensions of transfer maps of finite length (or turn) for nonlinearities using differential algebra techniques with Taylor expansions, etc. used for “analysis.”
• “Symplectification” is issue that limits initially perceived advantages of maps over element by element approach
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
37
UW Spring 2008
Accelerator Physics
Indicators of Chaos
)()()(
)0()(log1
21
0
tdtdtd
dtd
tlim
Exponent Lyapunov
td(0)
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
38
UW Spring 2008
Accelerator Physics
Survival Plots
• Plot maximum number of turns that survive as function of starting amplitude
• Plots are interpolated with fitting on functional form
4(1997))-65 Accel Part. al. et zi,(Giovannozanalysisby suggesteddwith
NNbDND
2/)1(
))/(log
1()(0
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
39
UW Spring 2008
Accelerator Physics
A Survival Plot
Scandale, Todesco
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
40
UW Spring 2008
Accelerator Physics
Implications of Dynamic Aperture Studies• Sources of nonlinearities: chromatic sextupoles,
multipoles in dipoles, multipoles in lattice quads, multipoles in low- quads, long-range beam-beam kicks
• For hadron colliders, multipoles of dipoles can dominate at injection; at collision, low- quads can dominate
• Target aperture roughly 12 at 105, which implies a 6 with safety margins
• Yields limits on multipole content, suggests multipole correction schemes, optimized optics, beam separation
J. J. Bisognano
Topic Three: Perturbations & Nonlinear Dynamics
41
UW Spring 2008
Accelerator Physics
Homework for Topic III
• From S.Y. Lee– 2.5.1– 2.5.3– 2.5.8– 2.6.1– 2.6.2– 2.7.3
