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[email protected] USPAS Advanced Accelerator Physics 12-23 June 2006
CHESS & LEPPCHESS & LEPP
87
),()( 0zsMsz =
The motion of particles can be represented by Generating Functions
Each flow or transport map:
With a Jacobi Matrix : izij MMj0
∂= ( )TTMM 0∂=or
That is Symplectic: JMJM T =
40404
30303
20202
10101
0
0
0
0
,with),,(
,with),,(
,with),,(
,with),,(
FqFqsppF
FqFpspqF
FpFqsqpF
FpFpsqqF
pq
pq
qp
∂−=∂=
∂−=∂−=
∂=∂=
∂=∂−=
Can be represented by a Generating Function:
6-dimensional motion needs only one function ! But to obtain the transport map this has to be inverted.
Generating Functions
[email protected] USPAS Advanced Accelerator Physics 12-23 June 2006
CHESS & LEPPCHESS & LEPP
88
Generating Functions produce symplectic tranport maps
10
1
01
0
0
00
01
0100101
)),,((
),(),,(
),(),,(
),,(,),,(with),,(
0
0
−
−
==
⎪⎪
⎭
⎪⎪
⎬
⎫
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂
=⎟⎟⎠
⎞⎜⎜⎝
⎛=
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂−
=⎟⎟⎠
⎞⎜⎜⎝
⎛=
∂=∂−=
gfMsszgfz
sQgsqqF
qpq
z
sQfsqqF
qpq
z
sqqFpsqqFpsqqF
q
q
(function concatenation)
Jacobi matrix of concatenated functions:
[ ] BBACzAzBCC
zBAzC
kzBzizkzijij kj
)()()(
)()(
)(0
00
00=⇒∂∂=∂=
=
∑ =
1)( −=⇒= GFgMfgM
F i SP(2N) [for notes]
[email protected] USPAS Advanced Accelerator Physics 12-23 June 2006
CHESS & LEPPCHESS & LEPP
89
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂∂∂
=⇒⎟⎟⎠
⎞⎜⎜⎝
⎛∂
=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂−∂∂−
=⇒⎟⎟⎠
⎞⎜⎜⎝
⎛∂−
=
1101
0
1101
0000
0
10),,(
),(
01),,(
),(
FFG
sqqFq
sQg
FFF
sqqFq
sQf
Tqq
Tqqq
Tqq
Tqqq
⎟⎟⎠
⎞⎜⎜⎝
⎛−=⇒⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛−−
=−−
−
0110
,01 1
21221
211
22211211
FFFG
FFG
FFF
⎟⎟⎠
⎞⎜⎜⎝
⎛
−−−
==−−
−−−
121111222
12111
12122
1211)(
FFFFFFFFF
FGgM
⎟⎟⎠
⎞⎜⎜⎝
⎛−
=⎟⎟⎠
⎞⎜⎜⎝
⎛
−−−
⎟⎟⎠
⎞⎜⎜⎝
⎛
−−−
=
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−−
−−
−−
−−
0110
0110
111
121
12
21111
12221
1222
12221
21111
2111
221
211
21
FFFFFFFFF
FFFFFFFFF
MM T The map from a generating function is symplectic.
F i SP(2N) [for notes]
[email protected] USPAS Advanced Accelerator Physics 12-23 June 2006
CHESS & LEPPCHESS & LEPP
90
Symplectic tranport maps have a Generating Functions
[ ] )(01002
000
0
01
0
0
0),()(
)(,)(
)(
)(
zlqqFJzhzM
ppp
zlq
zMqq
zMz
∂==⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛
=
FlhJF =−=∂ −11
Tijji FFFF =⇒∂=∂For F1 to exist it is necessary and sufficient that
llFhJlFhJ )(=−⇒=−Is J h l-1 symmetric ? Yes since:
⎟⎟⎠
⎞⎜⎜⎝
⎛ −=⎟⎟
⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛−
=
⎟⎟⎠
⎞⎜⎜⎝
⎛ ∂∂⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂⎟⎟
⎠
⎞⎜⎜⎝
⎛−
=
−−
−−
−−
−
−
111
121
12
111
1222211
1222
111
121
12
2221
1
11
22
1
1010
01
100110
00
00
MMMMMMMMM
MMMMM
MMMM
lhJTp
Tq
Tp
Tq
SP(2N) i F [for notes]
[email protected] USPAS Advanced Accelerator Physics 12-23 June 2006
CHESS & LEPPCHESS & LEPP
91
⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛ −=
−−
−−−
DCBA
MMMMMMMMM
lhJ11
112
112
111
1222211
12221
⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
2221
1211
002
0010 ,
),(),(
)(MMMM
MpqMpqM
zM
⎟⎟⎠
⎞⎜⎜⎝
⎛−
=⎟⎟⎠
⎞⎜⎜⎝
⎛− 01
100110 TMM ⎟⎟
⎠
⎞⎜⎜⎝
⎛−
=⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−−
0110
2212
2111
2122
1112TT
TT
MMMM
MMMM
( )
1
1
][
12211122
21122211
21222221
111
121212111
12111
1212111112
=−
=−
=
==⇒= −−−−
TT
TT
TT
TTTTT
MMMM
MMMM
MMMM
MMMMMMMMMMMM
( ) 11222212211
112222221121122
11222 ][][ −−−− =−=−= MMMMMMMMMMMMMM TTTTTT
TT MMMMMMMMM −−− =−=− 1212112221111
122221
TDD =TAA =
TCB =
SP(2N) i F [for notes]
[email protected] USPAS Advanced Accelerator Physics 12-23 June 2006
CHESS & LEPPCHESS & LEPP
92
Hamiltonian
Symplectic transport map
JMJM T =
Generating Functions
),,(),( 010 sqqFJpp ∂−=
),(' szHJz ∂=ODE
0,' =+= TFJJFFz
Symplectic Representations
[from notes]
[email protected] USPAS Advanced Accelerator Physics 12-23 June 2006
CHESS & LEPPCHESS & LEPP
93
Determinant of the transfer matrix of linear motion is 1:
One function suffices to compute the total nonlinear transfer map:
Therefore Taylor Expansion coefficients of the transport map are related.
Computer codes can numerically approximate with exact
symplectic symmetry.
Liouville’s Theorem for phase space densities holds.
1))(det(with)()( 0 +=⋅= sMzsMsz
10
1
01
0
0
00
01
0100101
)),,((
),(),,(
),(),,(
),,(,),,(with),,(
0
0
−
−
==
⎪⎪
⎭
⎪⎪
⎬
⎫
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂
=⎟⎟⎠
⎞⎜⎜⎝
⎛=
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂−
=⎟⎟⎠
⎞⎜⎜⎝
⎛=
∂=∂−=
gfMsszgfz
sQgsqqF
qpq
z
sQfsqqF
qpq
z
sqqFpsqqFpsqqF
q
q
),( 0zsM
Advantages of Symplecticity
[email protected] USPAS Advanced Accelerator Physics 12-23 June 2006
CHESS & LEPPCHESS & LEPP
94 11/15/03
For matrices with real coefficients:If there is an eigenvector and eigenvalue:then the complex conjugates are also eigenvector and eigenvalue:
⇒== jTijij
TTij
Ti vJvvMJMvvJv λλ 0)1( =−jij
Ti vJv λλ
iii vvM λ=***iii vvM λ=
For symplectic matrices:If there are eigenvectors and eigenvalues: with
then
Therefore is orthogonal to all eigenvectors with eigenvalues that are not . Since it cannot be orthogonal to all eigenvectors, there is at least one eigenvector with eigenvalue
iii vvM λ= MJMJ T=
jvJjλ/1
jλ/1Two dimensions: is eigenvalueThen and are eigenvalues
jλjλ/1 *
jλ1||/1 *
112 =⇒== jλλλλ1* −= jj λλ
jλjλ
1−jλ
*jλ
1*−jλ
Four dimensions:
Eigenvalues of a Symplectic Matrix
*212 /1 λλλ ==
[email protected] USPAS Advanced Accelerator Physics 12-23 June 2006
CHESS & LEPPCHESS & LEPP
95
ρκκ κ
1with =−= eesdsd
dsRd =
κ
κ
eeeeee
sRe
sb
sdsd
sdsd
dsd
s
×≡
−≡
≡ )(
bsdsd eTee '+= κκ
( ) κκκ −⋅=⋅= eeee dsd
ssdsd0
κeTebdsd '−=
( ) '0 Teeee bdsd
bdsd +⋅=⋅= κκ
)(sR
)()()( seysexsRr yx ++=
sb exeTxyeTyxr )1()''()''(' κκ ++++−=
Accumulated torsion angle T
The Frenet Coordinate System
[email protected] USPAS Advanced Accelerator Physics 12-23 June 2006
CHESS & LEPPCHESS & LEPP
96
yyxxsdsd eee κκ −−=
)cos()sin()sin()cos(
TeTeeTeTee
by
bx
+≡−≡
κ
κ
sysydsd
sxsxdsd
eeTeeeTe
κκκκ
==
==
)sin()cos(
)(sR
)()()( seysexsRr yx ++=
syxbdsd eyxeyexr )1('' κκκ ++++=
The Curvi-linerar System
[email protected] USPAS Advanced Accelerator Physics 12-23 June 2006
CHESS & LEPPCHESS & LEPP
97
ppsrsrsp
hmdt
ddsd === −−
γ111
Frdtd =2
2
FFsps
epppeppeppp
sphm
dtd
syyxxsyysyxxsxdsd
γ
κκκκ
===
+++−+−=−− 11
''' )()()(
s
h
yxyxdsd eyxeyexr )1('' κκ ++++=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
++
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
ysyphm
xsxphm
yph
xph
y
x
pFpF
pp
ppyx
s
s
s
s
κκ
γ
γ
'
'
''
sphmst γ== −1'
FppEpcpmcpcE
sph
dsd
dsd
dpd ⋅==+= 2222 )()('
222 )()( mcpcE +=
Phase Space ODE
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CHESS & LEPPCHESS & LEPP
98
),,,,,( δτbyaxz =
0
00
0
2
00
0
00
)()(,,,pEtt
vctt
EEE
pp
bppa yx −=−=
−=== τδ
01
00
00
00'
0
2
0
0
2
''
''
','',''
pKKKE
pKEKKt
pKbpKypKapKx
KpKx
vc
Et
vc
E
yb
xa
xx
px
ττ
δδ
δ
τ
−∂=∂−=⇒∂−=
∂=∂=⇒∂=−⎩⎨⎧
−∂=∂=−∂=∂=
⇒⎭⎬⎫
−∂=∂=
−
New Hamiltonian:
0~ pKH =
Using a reference momentum p0 and a reference time t0:
Usually p0 is the design momentum of the beamAnd t0 is the time at which the bunch center is at “s”.
x'xa ≈
s∆≈τ
6 Dimensional Phase Space
[email protected] USPAS Advanced Accelerator Physics 12-23 June 2006
CHESS & LEPPCHESS & LEPP
99
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
++
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
ysyphm
xsxphm
yph
xph
y
x
pFpF
pp
ppyx
s
s
s
s
κκ
γ
γ
'
'
''
sphmst γ== −1'
FpEsp
h ⋅='
⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
++−
+−+
+−+
=
⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
⋅
−
+
+
=
⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
)(
)(
)(
)(''''''
0
0
2
20
200
0
00
0
0
0
0
0
0
00
0
00
0
ssyyxxpEh
vvc
vc
ypp
sxxsypph
pp
xpp
yssyxpph
pp
pEh
pm
pm
pE
xpp
ypphm
pp
xpp
xpphm
pp
EpEpEpqh
BpBpEmqbh
BpBpEmqah
Fph
Fbh
Fah
byax
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
κγ
κγ
κ
κ
δτ γγ
γ
γ
0
00
0
0
00
)(,,,pEtt
EEE
pp
bppa yx −=
−=== τδ
The Equation of Motion [for notes]
[email protected] USPAS Advanced Accelerator Physics 12-23 June 2006
CHESS & LEPPCHESS & LEPP
100
One expands around the reference trajectory:Condition: The reference or design trajectory can be the path of a particle.
The particle transport is then origin preserving.
0),0(0),0(with),(' =⇒== sMsFszFz
0th order:
00
000
00
000
00
=
−−=
−=
s
yxy
xyx
E
Evp
qBpq
Evp
qBpq
κ
κ
(No acceleration on the design trajectory)
If the energy E changes on the reference trajectory thenδ = E-E0 does not stay 0. One then works with px, py, and E rather than
with a, b, and δ.
…++= 10 EEE
Note: called magnetic rigiditycalled electric rigidity
0pq)( 00vpq
The 0th Order Equation of Motion [for notes]
[email protected] USPAS Advanced Accelerator Physics 12-23 June 2006
CHESS & LEPPCHESS & LEPP
101
)()(''
][)()(
)('
1001
20
10
20101
1
00
20
00
00
syxEssyyxxpEh
mcqE
xysxvpq
xyx
xpp
yssyxpph
EbEaEqEpEpEpqb
BbBEyx
BpBpEmqa
s
x
s
s
++=++=
=
−+−+++−=
+−+=−−−
δ
βγκδβκκκ
κγ
…
1st order:
2200
222 ]1[ Oppppp yxs ++=−−= − δβ
( )vpc
EdEdpmcE
cp 11
2
222 ==⇒−=
δβκκβτγ
40
1201
11
200
2
20
2
00
)('
','−− ++−=−=
====
yxvvc
vc
pp
pp
yxh
bbhyaahx
s
ss
=++=== − 2200 ]1[2 Ovppv sE
csp
vs δβ
The Linear Equation of Motion [for notes]
[email protected] USPAS Advanced Accelerator Physics 12-23 June 2006
CHESS & LEPPCHESS & LEPP
102
Only bend in the horizontal plane: ρκκκ 1,0 === xy
Only magnetic fields: 0=EMid-plane symmetry: ),,(),,(,),,(),,( syxBsyxBsyxBsyxB yyxx −=−−=
0'
'
'''
)('''
40
120
20
220
2
20
0
0
=
+−=
=⇒∂=
++−=⇒+∂−−=
−−
−−
δ
δβκβτ
κβδκκβδκ
γx
ykyyBb
kxxxBxa
xypq
yxpq
Mid-plane symmetry:
240
1212
022
2122
212
212
21
20
)( δβδκβκγ
−− +−+−++= xxyxkbaHHamiltonian:
Simplified Equation of Motion