Acknowledgements
Many thanks to Sasha Valishev (FNAL) for help and
discussions.
2
3
Report at HEAC 1971
How to make the beam stable?
Landau damping
–
the beam’s “immune system”. It is
related to the spread of betatron oscillation
frequencies. The larger the spread, the more stable the
beam is against collective instabilities.
External damping (feed

back) system
–
presently the
most commonly used mechanism to keep the beam
stable.
Can not be used for some instabilities (head

tail)
Noise
Difficult in linacs
4
Most accelerators rely on both
LHC
Has a transverse feedback system
Has 336 Landau Damping
Octupoles
Provide tune spread of
0.001
at 1

sigma at injection
Lyn Evans: “
The ultimate panacea for beam instabilities is
Landau damping where the tune spread in the beam is
large enough to stop it from oscillating coherently.
”
Tevatron
, Recycler, MI, RHIC etc.
In all machines there is a trade

off between Landau
damping and dynamic aperture.
5
Today’s talk will be about…
… How to improve beam’s immune system (Landau damping
through
betatron
frequency spread)
Tune spread not
~0.001
but
10

50%
What can be wrong with the immune
system?
The main feature of all present accelerators
–
particles have
nearly identical
betatron
frequencies (tunes) by design. This
results in two problems:
I.
Single particle motion can be unstable due to resonant
perturbations (magnet imperfections and non

linear elements);
II.
Landau damping of instabilities is suppressed because the
frequency spread is small.
6
Preliminaries
I will discuss the 2

D transverse beam dynamics. I’ll
ignore energy spread effects, but it can be included.
The longitudinal coordinate,
s,
is equivalent to the
time coordinate.
A 2

D Hamiltonian
will be called “
integrable
” if it has at least two
conserved functionally

independent quantities
(analytic functions of
x, y, p
x
, p
y
, s
) in involution.
A 1

D time

independent Hamiltonian is
integrable
.
7
s
p
p
y
x
H
H
y
x
,
,
,
,
A bit of history: single particle
stability
Strong focusing or alternating

gradient focusing was
first conceived by
Christofilos
in 1949 but not
published , and was later independently invented in
1952 at BNL (Courant, Livingston, Snyder).
they discovered that the frequency of the particle
oscillations about the central orbit was higher, and the
wavelengths were shorter than in the previous constant

gradient (weak) focusing magnets. The amplitude of
particle oscillations about the central orbit was thus
correspondingly smaller, and the magnets and the
synchrotron vacuum chambers could be made smaller
—
a savings in cost and accelerator size.
8
Strong focusing
)
(
)
(
0
)
(
0
)
(
,
,
s
K
C
s
K
y
s
K
y
x
s
K
x
y
x
y
x
y
x
s
is “time”
9

piecewise constant
alternating

sign functions
Also applicable
to Linacs
Courant

Snyder Invariant
Courant and Snyder found a conserved quantity:
10
y
x
z
z
s
K
z
or
,
0
)
(
"
Equation of motion for
betatron oscillations
3
1
)
(
where
s
K

auxiliary (Ermakov) equation
2
2
)
(
2
)
(
)
(
2
1
z
s
z
s
z
s
I
Normalized variables
Start with a time

dependent
Hamilatonian
:
Introduce new (canonical) variables:

new “time”
Time

independent Hamiltonian:
Thus,
betatron
oscillations are linear; all particles
oscillate with the same frequency!
,
)
(
2
)
(
)
(
,
)
(
s
z
s
s
p
p
s
z
z
N
N
2
2
2
1
N
N
z
p
H
1
11
2
2
)
(
2
1
z
s
K
p
H
First synchrotrons
In late 1953 R. Wilson has constructed the first electron AG
synchrotron at Cornell (by re

machining the pole pieces of
a weakly

focusing synchrotron).
In 1955 CERN and BNL started construction of PS and AGS.
1954: ITEP (Moscow) decides to build a strong

focusing
7

GeV proton synchrotron.
Yuri F.
Orlov
recalls:
“In 1954 G.
Budker
gave several seminars
there. At these seminars he predicted that the combination of
a big
betatron
frequency with even a small nonlinearity would
result in
stochasticity
of
betatron
oscillations.
”
12
Yuri Orlov
Professor of Physics, Cornell
In 1954 he was asked to check Budker’s serious
predictions.
He writes: “
… I analyzed all reasonable linear and
nonlinear resonances with tune

shifting nonlinearities
and obtained well

defined areas of stability between
and below resonances and the corresponding
tolerances.
”
Work published in 1955.
13
Concerns about resonances
So, at the time of AGS and CERN PS construction (1955

60) the danger of linear betatron oscillations were
appreciated but not yet fully understood.
Installed (~10) octupoles to “detune” particles from
resonances. Octupoles were never used for this purpose.
Initial research on
non

linear
resonances (Chirikov, 1959)
indicated that
non

linear
oscillations could remain stable
under the influence of periodic external force
perturbation:
14
)
sin(
)
sin(
0
2
0
t
a
z
z
First non

linear accelerator
proposals
In a series of reports 1962

65 Yuri Orlov has proposed
to use non

linear focusing as an alternative to strong
(linear) focusing.
Final report (1965):
15
Henon

Heiles paper (1964)
First general paper on appearance of chaos in a 2

d
Hamiltonian system.
16
Henon

Heiles model
Considered a simple 2

d potential (linear focusing plus
a
sextupole
):
There exists one conserved
quantity (the total energy):
For energies
E
> 0.125 trajectories
become chaotic
Same nature as Poincare’s
“
homoclinic
tangle”
17
3
2
2
2
3
1
2
1
)
,
(
y
y
x
y
x
y
x
U
)
,
(
)
(
2
1
2
2
y
x
U
p
p
E
y
x
B
KAM theory
Deals with the time evolution of a conservative dynamical system
under a small perturbation.
Developed by
Kolmogorov
, Arnold, Moser (1954

63).
Suppose one starts with an
integrable
2

d Hamiltonian,
eg
.:
It has two conserved quantities (integrals of motion),
E
x
and
E
y
.
The KAM theory states that if the system is subjected to a weak
nonlinear perturbation, some of periodic orbits survive, while
others are destroyed. The ones that survive are those that have
“sufficiently irrational” frequencies (this is known as the non

resonance condition). The KAM theory specifies quantitatively
what level of perturbation can be applied for this to be true. An
important consequence of the KAM theory is that for a large set of
initial conditions the motion remains perpetually
quasiperiodic
.
18
)
(
2
1
)
(
2
1
2
2
2
2
y
x
p
p
H
y
x
KAM theory
M. Henon writes (in 1988):
It explained the results Orlov obtained in 1955.
And it also explained why Orlov’s nonlinear focusing
can not work.
19
KAM for Henon

Heiles potential
This potential can be viewed as resulting from adding
a perturbation to the separable (
integrable
) harmonic
potential. It is non

integrable
.
All trajectories with total
energies
are bound.
However, for
E
>
0.125
trajectories become chaotic.
20
3
2
2
2
3
1
2
1
)
,
(
y
y
x
y
x
y
x
U
B
.1667
0
)
,
(
)
(
2
1
2
2
y
x
U
p
p
E
y
x
E = 0.113, trajectory projections
21
0.2
0.1
0
0.1
0.2
0.2
0.1
0
0.1
0.2
px
i
x
i
1
0.5
0
0.5
0.6
0.4
0.2
0
0.2
0.4
0.6
py
i
y
i
0.2
0.1
0
0.1
0.2
1
0.5
0
0.5
1
y
i
x
i
x

p
x
y

p
y
x
–
y
22
1
0.5
0
0.5
1
0.6
0.4
0.2
0
0.2
0.4
0.6
px
i
1
1
x
i
1
0.5
0
0.5
1
0.5
0
0.5
1
py
i
y
i
1
0.5
0
0.5
1
1
0.5
0
0.5
1
y
i
x
i
E = 0.144, trajectory projections
x

p
x
y

p
y
x
–
y
Accelerators and KAM theory
Unlike
Henon

Heiles
potential, the nonlinearities in
accelerators are not distributed uniformly around the
ring. They are s

(time)

dependent and periodic (in
rings)!
… And non

integrable
(in general).
Luckily, an ideal accelerator is an
integrable
system
and small enough non

linearities
still leave
enough tune space to operate it.
However, it was still not fully
understood at the time of the first
colliders (1960)…
23
octupole
First storage ring colliders
First 3 colliders, AdA (1960), Princeton

Stanford CBX
(1962) and VEP

1 (1963), were all weakly

focusing
machines.
This might reflect the concern designers had for the
long

term particle stability in a strongly

focusing
storage ring.
24
CBX layout
Octupoles and sextupoles enter
1965,
Priceton

Stanford CBX: First mention of an 8

pole
magnet
Observed vertical resistive wall instability
With
octupoles
, increased beam current from
~
5
mA
to 500
mA
CERN PS: In 1959 had 10
octupoles
; not used until 1968
At
10
12
protons/pulse
observed (1
st
time) head

tail
instability.
Octupoles
helped.
Once understood, chromaticity jump at transition was
developed using
sextupoles
.
More instabilities were discovered; helped by
octupoles
and
by feedback.
25
Tune spread from an
octupole
potential
In a 1

D system:
Tune spread is unlimited

In a 2

D system:
Tune spread (in both x and y) is
limited to
~
12%
26
4
2
2
4
1
2
1
2
1
x
x
p
H
x
2
2
4
4
2
2
2
2
6
4
1
)
(
2
1
)
(
2
1
y
x
y
x
y
x
p
p
H
y
x
0
1
2
3
4
5
0
0.5
1
1.5
Energy
Freq.
f
W
(
)
W
B
1

D freq.
Tune spread from a single
octupole
in a linear
latice
Tune spread depends on a linear tune location
1

D system:
Theoretical max. spread is
0.125
2

D system:
Max. spread <
0.05
27
octupole
1 octupole in a linear 2

D lattice
28
Typical phase space portrait:
1. Regular orbits at small amplitudes
2. Resonant islands + chaos at larger
amplitudes;
Are there “magic” nonlinearities that
create large spread and zero resonance
strength?
The answer is
–
yes
(we call them “integrable”)
McMillan nonlinear optics
In 1967 E. McMillan published a paper
Final report in 1971. This is what later became known
as the “McMillan mapping”:
29
)
(
1
1
i
i
i
i
i
x
f
x
p
p
x
C
Bx
Ax
Dx
Bx
x
f
2
2
)
(
const
2
2
2
2
2
2
Dxp
p
x
C
xp
p
x
B
p
Ax
If
A = B
=
0
one obtains the Courant

Snyder invariant
McMillan 1D mapping
At small
x
:
Linear matrix: Bare tune:
At large
x:
Linear matrix: Tune:
0.25
Thus, a tune spread of 50

100% is
possible!
30
C
Bx
Ax
Dx
Bx
x
f
2
2
)
(
x
C
D
x
f
)
(
C
D
1
1
0
C
D
2
acos
2
1
0
)
(
x
f
0
1
1
0
4
2
0
2
4
4
2
0
2
4
p
n
k
x
n
k
A
=1,
B
= 0,
C
= 1
,
D = 2
What about 2D optics?
How to extend McMillan mapping into 2

D?
Danilov
,
Perevedentsev
found two 2

D examples:
Round beam:
xp
y

yp
x
= const
1.
Radial McMillan kick:
r
/(1 +
r
2
)

Can be realized with
an “Electron lens” or in beam

beam head

on collisions
2.
Radial McMillan kick:
r
/(1

r
2
)

Can be realized with
solenoids (may be useful for
linacs
)
In general, the problem is that the Laplace equation
couples
x
and
y
fields of the non

linear thin lens
31
Danilov’s
approximate solution
McMillan 1

D kick can be obtained by using
Then,
Make beam size small in one
direction in the non

linear lens
(by making large ratio of beta

functions)
32
1
ln
Re
)
,
(
2
iy
x
y
x
V
B
1
2
)
0
,
(
2
x
x
x
f
Danilov’s approximate solution
33
FODO lattice,
0.25,0.75 bare tunes
2 nonlinear, 4 linear
lenses.
For beta ratio > 50,
nearly regular
decoupled motion
Tune spread is
around 30% .
Summary thus far
In all present machines there is a trade

off between Landau
damping and dynamic aperture.
J. Cary et al. has studied how to increase dynamic
aperture by eliminating resonances.
The problem in 2

D is that the fields of non

linear
elements are coupled by the Laplace equation.
There exist exact 1

D and approximate 2

D non

linear
accelerator lattices with 30

50%
betatron
tune spreads.
34
New approach
See: http://arxiv.org/abs/1003.0644
The new approach is based on using the time

independent potentials.
Start with a round axially

symmetric beam (FOFO)
35
40
0
Sun Apr 25 20:48:31 2010 OptiM  MAIN:  C:\Documents and Settings\nsergei\My Documents\Papers\Invariants\Round
20
0
5
0
BETA_X&Y[m]
DISP_X&Y[m]
BETA_X
BETA_Y
DISP_X
DISP_Y
V(x,y,s)
V(x,y,s)
V(x,y,s)
V(x,y,s)
V(x,y,s)
Special time

dependent potential
Let’s consider a Hamiltonian
where
V(x,y,s)
satisfies the Laplace equation in 2d:
In normalized variables we will have:
)
,
,
(
2
2
2
2
2
2
2
2
s
y
x
V
y
x
s
K
p
p
H
y
x
)
(
,
)
(
,
)
(
)
(
2
2
2
2
2
2
s
y
x
V
y
x
p
p
H
N
N
N
N
yN
xN
N
36
0
)
,
(
)
,
,
(
y
x
V
s
y
x
V
s
0
)
s
(
d
)
(
s
s
Where new “time” variable is
Three main ideas
1.
Chose the potential to be time

independent in new
variables
2.
Element of periodicity
3.
Find potentials
U(x, y)
with the second integral of
motion
)
,
(
2
2
2
2
2
2
N
N
N
N
yN
xN
N
y
x
U
y
x
p
p
H
s
L
β
(
s
)
T
insert
1
0
0
0
1
0
0
0
0
1
0
0
0
1
k
k
2
2
1
1
)
(
Lk
s
L
sk
L
s
37
Test lattice (for an 8

GeV beam)
38
Six quadrupoles
provide a phase advance
of
π
Linear tune: 0.5
–
1.0
1
0
1
k
Examples of time

independent
Hamiltonians
Quadrupole
2
2
2
,
,
y
x
s
q
s
y
x
V
2
2
)
,
(
N
N
N
N
y
x
q
y
x
U
39
0
0.2
0.4
0.6
0.8
0
0.5
1
1.5
2
s
(
)
q
s
(
)
s
β
(s)
quadrupole
amplitude
2
2
2
2
2
2
2
2
N
N
N
N
yN
xN
N
y
x
q
y
x
p
p
H
L
Tunes:
)
1
(
)
1
(
2
0
2
2
0
2
q
q
y
x
Tune spread: zero
Examples of time

independent
Hamiltonians
Octupole
40
2
3
4
4
,
,
2
2
4
4
3
y
x
y
x
s
s
y
x
V
2
3
4
4
2
2
4
4
N
N
N
N
x
y
y
x
U
This Hamiltonian is NOT
integrable
Tune spread (in both x and y) is
limited to
~
12%
2
2
4
4
2
2
2
2
6
4
)
(
2
1
)
(
2
1
y
x
y
x
k
y
x
p
p
H
y
x
B
Tracking with octupoles
41
Tracking with a test lattice
10,000 particles for 10,000 turns
No lost particles.
50 octupoles in a 10

m long drift space
Integrable 2

D Hamiltonians
Look for second integrals quadratic in momentum
All such potentials are separable in some variables
(cartesian, polar, elliptic, parabolic)
First comprehensive study by Gaston Darboux (1901)
So, we are looking for integrable potentials such that
)
,
(
2
2
2
2
2
2
y
x
U
y
x
p
p
H
y
x
)
,
(
2
2
y
x
D
Cp
p
Bp
Ap
I
y
y
x
x
,
,
2
,
2
2
2
ax
C
axy
B
c
ay
A
42
Second integral:
Darboux equation (1901)
Let
a
≠
0
and
c
≠
0
, then we will take
a
=
1
General solution
ξ
: [1, ∞],
η
: [

1, 1],
f
and
g
arbitrary functions
0
3
3
2
2
2
y
x
xy
yy
xx
xU
yU
U
c
x
y
U
U
xy
2
2
)
(
)
(
)
,
(
g
f
y
x
U
c
y
c
x
y
c
x
c
y
c
x
y
c
x
2
2
2
2
2
2
2
2
2
2
43
The second integral
The 2
nd
integral
Example:
2
2
2
2
2
2
2
2
)
(
)
(
2
,
,
,
g
f
c
p
c
yp
xp
p
p
y
x
I
x
x
y
y
x
2
2
2
1
)
,
(
y
x
y
x
U
1
2
)
(
2
2
2
1
c
f
2
2
2
1
1
2
)
(
c
g
2
2
2
2
2
,
,
,
x
c
p
c
yp
xp
p
p
y
x
I
x
x
y
y
x
44
Laplace equation
Now we look for potentials that also satisfy the Laplace
equation (in addition to the Darboux equation):
We found a family with 4 free parameters (b, c, d, t):
0
yy
xx
U
U
acos
1
)
(
acosh
1
)
(
2
2
2
2
t
b
g
t
d
f
2
2
)
(
)
(
)
,
(
g
f
y
x
U
45
The Hamiltonian
So, we found the integrable Laplace potentials…
2
2
2
2
2
2
2
2
)
(
)
(
2
2
2
2
,
,
,
g
f
y
x
p
p
p
p
y
x
H
y
x
y
x
F
F
F
c
=1,
d
=1,
b=t
=0
c
=1,
b
=1,
d=t
=0
c
=1,
t
=1,
d=b
=0
→ 1 at
x,y
→ ∞
→ 0 at
x,y
→ ∞
→ ln(
r
) at
x,y
→ ∞
46
The
integrable
Hamiltonian (elliptic
coordinates)
47
B
U(x,y):
c
=1,
t
=1,
d=
0,
)
,
(
2
2
2
2
2
2
y
x
kU
y
x
p
p
H
y
x
...
315
128
35
16
15
8
3
2
Re
)
,
(
10
8
6
4
2
iy
x
iy
x
iy
x
iy
x
iy
x
y
x
U
Multipole expansion
k <
0.5 to provide linear stability for small
amplitudes
Tune spread:
)
1
(
)
1
(
2
0
2
2
0
2
k
k
y
x
at small amplitudes
2
0
2
2
0
2
y
x
at large amplitudes
Max. ~100% in
y
and ~40% in
x
2
b
Example of trajectories
25 nonlinear lenses in a drift space
Small amplitude trajectories
48
0.04
0.02
0
0.02
0.04
0.04
0.02
0
0.02
0.04
px
j j
x
j j
0.04
0.02
0
0.02
0.04
0.04
0.02
0
0.02
0.04
py
j j
y
j j
0.5
0.6
0.7
0.8
0.9
1
0
0.2
0.4
0.6
f1
j
f2
j
f3
j
j
bare tune
y
x
Large amplitudes
49
2
1
0
1
2
2
1
0
1
2
px
j j
x
j j
2
1
0
1
2
2
1
0
1
2
py
j j
y
j j
2
1
0
1
2
2
1
0
1
2
y
j j
x
j j
0.5
0.6
0.7
0.8
0.9
1
0
10
20
30
40
f1
j
f2
j
f3
j
j
y
x
bare tune
More examples of trajectories
Trajectory encircles the singularities (
y
=0
, x=
±
1)
2
)
(
and
,
1
2
)
(
4
2
4
g
f
c
=1,
d
=1,
b=t
=0
50
Normalized coordinates:
Polar coordinates
Let
a
≠
0
and
c
=
0
2
2
2
2
)
(
)
(
2
2
2
r
g
r
f
r
p
p
H
y
x
2
cos
2
sin
)
(
ln
)
(
t
b
g
r
d
r
f
)
(
2
2
g
yp
xp
I
x
y
51
Two types of trajectories
d =

1
, b =
0
,
and
t
= 0.1
2
cos
2
sin
)
(
ln
)
(
t
b
g
r
d
r
f
52
Parabolic coordinates
a = 0
Not considered by Darboux (but considered by Landau)
Equation for potential:
)
,
(
2
2
y
x
D
Cp
p
Bp
Ap
I
y
y
x
x
x
C
y
B
A
0
0
3
2
xx
yy
y
xy
U
U
y
U
xU
53
Parabolic coordinates
Solution
The only parabolic solution is
y
2
+ 4
x
2
Potentials that satisfy the Laplace equation
.
2
)
(
)
(
)
(
)
(
,
2
)
(
)
(
)
,
(
r
x
r
g
x
r
x
r
f
x
r
p
xp
yp
I
r
x
r
g
x
r
f
y
x
U
y
y
x
r
x
r
g
x
r
f
y
x
r
p
p
H
y
x
2
)
(
)
(
)
(
10
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
)
(
)
(
x
r
t
x
r
d
x
r
g
x
r
t
x
r
b
x
r
f
54
Example of trajectories
Trajectories never encircle the singularity
F
r
x
r
y
x
y
x
U
2
1
.
0
)
(
10
3
,
2
2
55
Summary
The lattice can be realized by both
and
Found nonlinear lattices (e.g.
octupoles
) with one integral
of motion and no resonances. Tune spread 5

10% possible.
Found first examples of completely
integrable
non

linear
optics.
Betatron
tune spreads of 50% are possible.
We used quadratic integrals but there are might other
functions.
Beyond
integrable
optics: possibly there exist realizable
lattices with bound but
ergodic
trajectories
1
0
1
,
k
M
y
x
1
0
1
,
k
M
y
x
56
Conclusions
Nonlinear “integrable” accelerator optics has
advanced to possible practical implementations
Provides “infinite” Landau damping
Potential to make an order of magnitude jump in beam
brightness and intensity
Fermilab is in a good position to use of all these
developments for next accelerator projects
Rings or linacs
Could be retrofitted into existing machines
57
Extra slides
58
Vasily P. Ermakov
(1845

1922)
Professor of Mathematics, Kiev university
“Second order diﬀerential equations: Conditions of
complete integrability”, Universita Izvestia Kiev, Series
III 9 (1880) 1
–
25.
Found a solution to the equation:
Ermakov invariant is what we now call Courant

Snyder
invariant in accelerator physics
59
0
)
(
z
s
K
z
Joseph Liouville
(1809

1882)
Liouville
(1846) observed that if the
Hamiltonian function admitted a
special form
in some system of coordinates (u, v),
the Hamiltonian system could be
solved in
quadratures
. The form of the
Hamiltonian function also implies
additive separation of variables
for the associated Hamilton

Jacobi
equation.
60
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