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Integrable Particle Dynamics in Accelerators

Tuesday: Integrable systems with static electromagnetic fields

Sergei Nagaitsev

Jan 29, 2019

Examples of Integrable systems

▪ Electric fields➢ Coulomb potential

➢ Two Fixed Coulomb Centers

➢ Vinti potential

▪ Magnetic fields➢ Uniform magnetic field

➢ Magnetic monopole

▪ Electric and magnetic fields➢ Penning trap and its modifications

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Kepler problem – a nonlinear integrable system

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▪ Kepler problem (Coulomb potential):

▪ Separable in spherical coordinates:➢ For bounded motion

▪ “Nonlinear” means that

▪ Example of this system: the Solar system

kV

r= −

( )

2

2

2 r

mkH

J J J

= −+ +

1 1 2 2 3 3H J J J + +

Electric Charge in the Field of a Magnetic Pole

▪ Magnetic pole – “end” of a semi-infinite solenoid

▪ In 1896, Birkeland reported studies of cathode rays in a Crookes tube when a strong, straight electromagnet was placed outside and to the left.

▪ The nature of cathode rays was not yet understood

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Kristian Birkeland

▪ Birkeland’s scientific efforts are honored on the 200-kroner Norwegian banknote.➢ In 1896 his major interest was Aurora Borealis.

▪ He was one of Poincare’s students in 1892

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Magnetic monopole

▪ The nature of cathode rays was not understood in 1896, which were “discovered” to be electrons by J.J. Thomson in 1897 (in experiments with Crookes tubes and magnets).

▪ In 1896, before the Thomson’s discovery, Poincare has suggested that Birkeland’s experiment can be explained by “cathode rays being charges moving in the field of a magnetic monopole”➢ He wrote a brilliant paper in (1896), proving that charge

motion in the field of magnetic monopole is fully integrable (but unbounded).

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3

k

r=

rB

Motion is on the cone surface

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http://physics.princeton.edu/~mcdonald/examples/birkeland.pdf

Aurora Borealis

▪ Beginning in 1904, a younger colleague, C. Størmer, was inspired by Birkeland to make extensive modeling of the trajectories of electrons in the Earth’s magnetic field, approximated as that of a magnetic “bottle”.➢ Størmer studied under Darboux and Poincare in 1898-1900

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Størmer and Birkeland in 1910

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Two magnetic monopoles

▪ One can imagine the motion of an electric charge between two magnetic monopoles (of opposite polarity) would be integrable, but it is not.➢ Only the “adiabatic” integrals exist, when poles are far

apart (compared to the Larmour radius)

➢ This is the principle of the magnetic bottle; also, the principle of “weak focusing in accelerators”.

▪ The non-integrability in this case is somewhat surprising because the motion in the field of two Coulomb centers is integrable.➢ This has been know since Euler and was Poincare’s

starting point for the 3-body problem quest.

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Weak focusing

▪ The magnetic fields can be approximated by the fields of two magnetic monopoles of opposite polarity

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The race for highest beam energy

➢ Cosmotron (BNL, 1953-66): 3.3 GeV• Produced “cosmic rays” in the lab

• Diam: 22.5 m, 2,000 ton

➢ Bevatron (Berkeley, 1954): 6.3 GeV• Discovery of antiprotons

and antineutrons: 1955

• Magnet: 10,000 ton

➢ Synchrophasatron (Dubna,1957): 10 GeV• Diam: 60 m, 36,000 ton

• Highest beam energy until 1959

Strong Focusing

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CERN PS

▪ In Nov 1959 a 28-GeV Proton Synchrotron started to operate at CERN➢ 3 times longer than the Synchrophasatron but its

magnets (together) are 10 times smaller (by weight)

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Particle motion in static magnetic fields

▪ For accelerators, there are no useful exactly integrable systems for axially symmetric magnetic fields in vacuum:

▪ Until 1959, all circular accelerators relied on approximate (adiabatic) integrability.➢ These are the so-called weakly-focusing accelerators

➢ Required large magnets and vacuum chambers to confine particles;

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22 2 ( , )1

2 2

z rp eA r zp p

Hm m r c

+ = + −

Charge in a uniform magnetic field, B

▪ Let’s use cylindrical coordinates:

▪ This Hamiltonian is separable, however, the motion in z is unbounded (thus the action variable is undefined).

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1;

2zA Br B B = =

, , zr

22 2 1

2 2

z rp eAp p

Hm m r c

+ = + −

2 2 22 2

22 2 2 8

c cz rp p m rp p

Hm mr

+= + − + c

eB

mc =

Penning trap

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▪ An ideal Penning trap is a LINEAR and integrable system➢ It is a harmonic 3-d oscillator

1 1 2 2 3 3H J J J = + +

Penning trap: equations of motion

▪ The system is bounded if eV0 > 0 and

After some math, we obtain:

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1;

2zA Br B B = =

( )22 2

2 20

2

12

2 2 4

z rp eA eVp p

H z rm m r c d

+ = + − + −

2 2 2 2 22 2

0 0

2 2 22 2 2 8 4 2

c cz reV z p m r eV r pp p

Hm d mr d

+= + + + − −

c

eB

mc =

2 2

02

cmdeV

p const = Integral of motion

2

0 0

2 2;

4 2

cz r

eV eV

md md

= = −

After variable separation

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( )2 2 2

22 2 2

2

1

2

1

2

z z z

r r r

J p const

H p z

JH p r

r

= =

= +

= + +

z rH H H= +

2sin

2 cos

zz

z

z z z z

Jz

p J

=

=

( ) ( )1/2

2

2

1, 2 2 sin

2 cos

( , )

r r r r r r

r

r r r

r

r r

r J J J J J J

J J Jp

r J

= + + +

+=

2z z r r rH J J J = + +

Perturbation theory (introduction)

▪ Let’s add a weak octupole nonlinearity to the Penning trap

▪ This is a non-integrable nonlinearity (…unlike a 1D non-linear oscillator).

▪ One can find approx. corrections by using a canonical perturbation method for ε << 1.

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( )4 2 2 41

4( , ) 8 24 3

VV r z z z r r

d= − +

( )22 2

2 20

2

12 ( , )

2 2 4

z rp eA eVp p

H z r eV r zm m r c d

+

= + − + − +

First order perturbation

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0 1( ) ( )+ ( , )H J H J H J

Start with an integrable system 0 ( )H H I=

0

0

( ) 0

( ) ( )

I H I

H I II

= − =

= =

0 1( , ) ( ) ( , )H I H I H I = +

The goal is to find new and such that ( , ) ( )J H I K J →

In the first order, we approximate J = I

How does one look for integrable systems for n>1?

▪ No general method of finding.

▪ One well-understood method (n=2): look for the second integral, quadratic in momenta➢ Turns out, all such potentials are separable in one of

these 4 coordinate systems (cartesian, polar, elliptic, parabolic)

➢ See Landau, Lifschitz “Mechanics” for reference

➢ First comprehensive study by Gaston Darboux (1901)

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Jean-Gaston Darboux

1842-1917

Darboux method

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▪ We are looking for integrable potentials such that2 2

( , )2

x yp pH U x y

+= +

),(22 yxDCppBpApI yyxx +++=

,

,2

,

2

22

axC

axyB

cayA

=

−=

+=

and the second integral:

[ , ] 0I H =

Darboux equation (1901)

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▪ Let a ≠ 0 and c ≠ 0, then we will take a = 1

▪ General solution

ξ : [1, ∞], η : [-1, 1], f and g arbitrary functions

( ) ( ) 033222 =−++−+− yxxyyyxx xUyUUcxyUUxy

22

)()(),(

−

+=

gfyxU

( ) ( )

( ) ( )c

ycxycx

c

ycxycx

2

22222

2222

+−−++=

+−+++=

Elliptical coordinates

The second integral

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▪ The 2nd integral

▪ Example:

( ) ( )

22

222222 )()(

2,,,

−

+++−=

gfcpcypxpppyxI xxyyx

( )22

2

1),( yxyxU +=

( )1

2)( 22

2

1 −= c

f ( )222

1 12

)( −=c

g

( ) ( ) 22222,,, xcpcypxpppyxI xxyyx ++−=

What’s the connection to traps?

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▪ Can one add an additional potential

such that the system becomes integrable and nonlinear?➢ The electrostatic potential must have

( ) ( )2 2 2

2 2 2 20

2 2

12

2 2 2 8 4

c cz r

p p m r eVH p p z r

m mr m d

= + + − + + −

( , )U r z

( , ) 0U r z =

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▪ Prolate spheroidal coordinates (u,v,ϕ)

▪ Oblate spheroidal coordinates (ξ,η,ϕ)

▪ Separable potentials:

( )( )

( )( )

2 2

2 2

1 1 cos

1 1 sin

x c u v

y c u v

z cuv

= − −

= − −

=

( )( )

( )( )

2 2

2 2

1 1 cos

1 1 sin

x a

y a

z a

= + −

= + −

=

( )( ) ( )

( )( ) ( )1 2 1 2

2 2 2 2, ; ,

g u g v f fU u v V

u v

+ += =

− +

Penning trap again

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Is separable in both prolate and oblate spherical coordinates.

Is a constant

Is separable if or

( ) ( )2 2 2

2 20

2 2, 2

2 2 8 4

c ceff

p p m r eVV r z z r

mr m d

= − + + −

1 2 3

1

2

3

2 22

0 26

c dmcV

e c

=

( )2

2 20

2 22 2 2

c

eff

p p VV z r

emr em d

= − + +

Now add non-linear potentials

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▪ In addition (the Laplace equation must be satisfied for electrostatic fields in vacuum):

▪ In prolate coordinates the solution has been know since Jacobi: the potential of Two Fixed Coulomb Centers (2FCC):➢ https://www.astro.auth.gr/~varvogli/two-fixed-centers.pdf

( )( ) ( )

( )( ) ( )1 2 1 2

2 2 2 2, ; ,

g u g v f fU u v V

u v

+ += =

− +

( , ) 0U r z = ( , ) 0V r z =

( ) ( )

32

2 22 2

( , , )U x y zz c r z c r

= +

− + + +

2 Fixed Coulomb Centers

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https://www.astro.auth.gr/~varvogli/two-fixed-centers.pdf

Oblate spheroidal coordinates

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▪ The solution is known since 1959: the Vinti potential

( )

( ) ( )

( ) ( )

1/22

2 22 2 2

1

2 22 2

4

, , ,

a a r z a r z

V x y za a r z a r z

− + + − − + =

+ + − +

Applicable to traps:

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The non-linear trap recipe

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1. Chose voltage (or magnetic field):For protons in 1-Tesla mag. field and d = 1 cm:

2. Add the non-linear potential, either 2FCC or Vinti

2 22

0 26

c dmcV

e c

=

0 1.6 kVV

BB

Veff+ 2FCC Veff+ Vinti

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Henon-Heiles paper (1964)

▪ First general paper on appearance of chaos in a Hamiltonian system.

▪ There exists two conserved quantities➢ Need 3 for integrability

▪ For energies E > 0.125 trajectories become chaotic

▪ Same nature as Poincare’s “homoclinictangle”

Michel Henon (1988):( )2 2 2 31 1

( , , )2 3

U r z r z r z z = + + −

Henon-Heiles system

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( )2 2 2 2 31 1

2 3x yH p p Ax By Dx y Cy= + + + + −

Given the Henon-Heiles system with adjustable

coefficients, are there any combinations that are

integrable? Yes, the following 4 are known to

be integrable:/ 0, any ,

/ 1, / 1

1/ , any ,

6

1 1/ , A /

16 16

D C A B

D C A B

D C A B

D C B

=

= − =

= −

= − =

Summary

▪ We have discussed a number of examples of integrable physical systems (linear and non-linear).

▪ Integrable systems form the “building blocks” about which perturbation theories can be developed.

▪ The non-integrable dynamic system actually constitute the majority of all dynamical systems. To find (or discover) integrable systems requires some luck.

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