87 generating functions - cornell universityhoff/lectures/06uspas/notes4.pdf · the motion of...

[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

qq

∂−=∂=

∂−=∂−=

∂=∂=

∂=∂−=

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



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

qq

(function concatenation)

Jacobi matrix of concatenated functions:

[ ] BBACzAzBCC

zBAzC

kzBzizkzijij kj

)()()(

)()(

)(0

00

00=⇒∂∂=∂=

=

∑ =

1)( −=⇒= GFgMfgM

F i SP(2N) [for notes]



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]



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]



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]



92

Hamiltonian

Symplectic transport map

JMJM T =

Generating Functions

),,(),( 010 sqqFJpp ∂−=

),(' szHJz ∂=ODE

0,' =+= TFJJFFz

Symplectic Representations

[from notes]



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

qq

),( 0zsM

Advantages of Symplecticity



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 ==



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



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



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



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



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]



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]



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]



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

87 generating functions - cornell universityhoff/lectures/06uspas/notes4.pdf · the motion of...

Documents