Hamiltonian Dynamics
or
Analytical Mechanics:
An Introduction with Applications to Accelerator Physics
Lecture 8 Accelerator Physics III: Review, Brief Outline of Some Related Topics
Dr Chris Warsop,
ISIS Facility, Rutherford Appleton Laboratory, Oxon.
Visiting Lecturer, JAI, Oxford
Course Outline
1
2
3
4
5
6
7
8
Introduction and Lagrangian Formulation
Hamiltonian Formulation
Canonical Transformations
Hamilton-Jacobi Theory
Perturbation Theory
Accelerator Physics I:
Aspects of Longitudinal Dynamics
Accelerator Physics II:
Aspects of Transverse Dynamics
Accelerator Physics III: Review and Brief Outline of Some Related Topics
8 Accelerator Physics III: Review, Brief Outline of Some Related Topics
Contents of this Lecture
8.1 Review of Material
8.2 Some Related Topics
8.3 Suggested Reading
8.4 Acknowledgements
8.1 Review of Material
1
2
3
4
5
6
7
8
Introduction and Lagrangian Formulation
Hamiltonian Formulation
Canonical Transformations
Hamilton-Jacobi Theory
Perturbation Theory
Accelerator Physics I:
Aspects of Longitudinal Dynamics
Accelerator Physics II:
Aspects of Transverse Dynamics
Accelerator Physics III: Review and Brief Outline of Some Related Topics
1 Introduction and Lagrangian Formulation
• Mechanical System with Constraints
• Generalised Coordinates (q1, ..qn) → Configuration Space
• Optimal description in the required number of degree’s of freedom
• Principle of Virtual Work
• D’Alembert’s Principle
• Lagrange’s Equations: n second order equations of second order ( q, qɺ )
L ( q, qɺ, t ) the Lagrangian
L = T −V
d  ∂L

dt  ∂qɺ j
 ∂L
 − ∂q = 0

j
 1.32 


 1.33 


2 Hamiltonian Formulation
• Hamilton’s Principle → Basic Principle for Mechanics
q
t2
t2
δΙ = δ ∫ L ( q1,..., qn , qɺi ,..., qɺn , t )dt = 0
 2.2 
 
t1
t1
t
• Hamilton’s Principle → Lagrange’s Equations → Starting Point for Key Theory
• Legendre Transformation:
(q, qɺ, t ) → (q, p, t )
• Move to Hamiltonian Formulation: 2n first order equations in 2n variables
The Hamiltonian
qɺi =
∂H
∂pi
H (q, p, t )
 2.29a 


−pɺi =
∂H
∂q i
 2.29b 


• Phase Space (q,p) → surface of H for conservative systems
n of each
( q, p )
3 Canonical Transformations
• Canonical Transformations:
Transform from (q,p) → (Q,P) whilst preserving canonical properties
• F1, F2, F3, F4 Generating Functions and Associated Transformations
• Symplectic Form → Condition for Canonical Transformations
ηɺ = J
∂H
∂η
[ 3.26 ]
=J
MJM
[ 3.34 ]
• Conservation of Phase Space Volume: Liouville’s Theorem
 ∂2 H
∂2 H 
∇ ⋅ v = ∑
−
=0
q
p
p
q
∂
∂
∂
∂
i i
i i 
i 
• Poisson Brackets
[ u, v ]q, p =
∂u ∂v ∂u ∂v
−
∂qi ∂pi ∂pi ∂qi
[ 3.35 ]
4 Hamilton-Jacobi Theory
• Hamilton-Jacobi Equation → Generating Function to Constant (q,p) solution

∂F2
∂F2  ∂F2

H q1,..., qn ,
,...,
, t  +
=0

∂q1
∂qn 
∂t
J = ∫ pdq
• Action Angle Variables
∂H ( J )
Jɺ =
=0
∂w
• Transformation Equations
q=
J
sin 2π w
π mw
H = H (J )
[ 4.20]
wɺ =
 4.3 
 
∂H ( J )
∂J
=ν ( J )
[ 4.21]
[ 4.24]
( w, J ) → ( q, p )
[ 4.30 − a ]
p=
mwJ
π
cos 2π w
[ 4.30 − b]
5 Perturbation Theory
• Approximate Solutions for Difficult Problems
H ( q, p, t ) = H 0 ( q, p, t ) + ∆H ( q, p, t )
 5.1 
 
• Time Dependent Theory ~ Variation of Constants
αɺ i = −
∂∆H ( α, β, t )
 5.3 − a 


∂βi
βɺi =
∂∆H ( α, β, t )
∂αi
• Found secular changes in frequency of perturbed system
βɺ = −
ν ′ = ν + βɺ
J
32π 2 ml 2
• Time Independent Theory
En ( J ) = Vn (ϑ0 , J )
∂S n (ϑ0 , J )
∂ϑ0
=−
1
Vn (ϑ0 , J ) − Vn (ϑ0 , J ) 

w0 
 5.3 − b 


6 Accelerator Physics I: Aspects of Longitudinal Dynamics
• Hamiltonian for Single Harmonic RF System
hw 02 η
e
U (φ )
2
2
π
2β Es
• If Conservative get much info from H(φ,W) surface
H ( φ,W ) =
Surface of H ( φ,W )
W2 −
Contours of H ( φ,W )
with velocites ( φɺ,Wɺ )
• Adiabatic Invariants
 2 β Es
J = ∫ 
2
 hw0η
2
1
2
e


V0 ( ( cos φ − cos φs ) + (φ − φs ) sin φs )   dφ
 Hb −
2π


7 Accelerator Physics I II: Aspects of Transverse Dynamics
• General Hamiltonian
H =
m 2c 4 + c 2 ( P − eA ) + eφ
2
• For transverse motion

x
e
x
H = − 1 +  + 12 px2 + 12 p y2 + 1 +  B0 x + 12 B′ x 2 − z 2 + 16 B′′ x3 − 3 xz 2 + ... higher order multipoles
p ρ 
 ρ
2
2
2
2
eAs1 eAx px 1  eAx  eAz pz 1  eAz  1  x   
eAx  
eAz  
+
−
+ 2
+ 2
 −
 + 2     px −
 +  pz −
 
p
p
p
p
ρ
p
p
p
 




 
 
+ (other terms from sqrt expansion)
(
(
)
• Perturbed Hamiltonian in Action Angle Coordinates
H = Qx J x + pn ( ϕ )J 2 cosn ( Ψ )
n
• Invariant Hamiltonians for Resonance
(
)
)
8.2 Some Related Topics …
• Envelope Equation for Kapchinskij-Vladimirskij (KV) beam - or RMS equivalent
H env =
1
2
(p
2
a
)
+ p + K x a + K z b − 2 K sc ln ( a + b ) +
2
b
2
2
Canonical coordinates: beam envelope, and it’s derivative
ε x2
2a
2
+
ε z2
2b 2
( a, a′, b, b′) = ( a, pa , b, pb )
Envelope oscillations (a,a’) - under space charge near resonance
6
6
4
4
4
2
2
2
0.6
0
1
0.8
1.2
1.4
1.6
0.6
0.8
1.2
1.4
0.5
-2
-2
0
-0.5
-4
-1
-4
-1
-0.5
0
-6
0.5
1
So we take generalised coordinates as second moments - envelopes and derivatives …
[for basic theory see S Y Lee Chapter 2]
1.6
8.2 Some Related Topics …
• Vlasov Equation: Space Charge Effects
from Liouville’s Theorem for distribution function
f ( x, p x , y , p y )
df ∂f
∂f
∂f
∂f
∂f
=
+ xɺ + yɺ
+ pɺ x
+ pɺ y
;
∂x
∂y
∂px
∂p y
dt ∂t
f = f 0 ( H 0 ) + f1 ( x, p )
Can use to determine behaviour of distribution under space charge, moments etc
Coherent modes, resonances - loss mechanisms.
[ M Reiser; e.g. Hofmann Phys Rev E Vol. 57 No. 4 p4713]
• Vlasov Equation: General Instability Theory
The Vlasov Equation, and associated perturbation formalism, is the basis of general
instability theory.
[A W Chao, Physics of Beam Instabilites, Chapter 6]
8.3 Books ~ 1
Analytical Mechanics
Traditional Approach
H Goldstein, Classical Mechanics, Second or Third Edition, Addison Wesley
C Lanczos, The Variational Principles of Mechanics, Fourth Edition, Dover
D Boccaletti, G Pucacco, Theory of Orbits, Vols 1 & 2, Springer
D T Greenwood, Classical Dynamics, Dover
M G Calkin, Lagrangian and Hamiltonian Mechanics, World Scientific
L N Hand, J D Finch, Analytical Mechanics, Cambridge University Press
I Percival and D Richards, Introduction to Dynamics, Cambridge University Press
W Greiner, Classical Mechanics, Springer
D Morin, Classical Mechanics, Cambridge (plus Chapter on Hamiltonian Dynamics on WWW)
More Modern and/or Advanced Approach
J V Jose, E J Saletan, Classical Dynamics, A Contemporary Approach, Cambridge
V I Arnold, Mathematical Methods of Classical Mechanics, Springer
A J Lichtenberg, M A Lieberman, Regular and Chaotic Dynamics
8.3 Books ~ 2
Accelerator Physics
E Wilson, An Introduction to Particle Accelerators, Oxford
S Y Lee, Accelerator Physics, World Scientific
H Wiedemann, Particle Accelerator Physics, 2nd edition or 3rd edition, Springer
M Reiser, Theory and Design of Charged Particle Beams, Wiley
D A Edwards, M J Syphers, Introduction to the Physics of High Energy Accelerators
J D Lawson, The Physics of Charged Particle Beams, Oxford
S I Tzenov, Contemporary Accelerator Physics, World Scientific
A A Kolomensky, A N Lebedev, Theory of Cyclic Accelerators, North-Holland
L Michelotti, Intermediate Classical Dynamics with Applications to Beam Physics, Wiley
E Forest, Beam Dynamics: A New Attitude and Framework, Hardwood Academic
Plus proceedings of CERN Accelerator Schools, CERN and SLAC Reports
G Guignard, A General Treatment of Resonances in Accelerators, CERN 78-11, 1978
A Schoch, Theory of linear and non-linear perturbations of betatron oscillations in AG
synchrotrons, CERN 57-21, 1958
R D Ruth, Single Particle Dynamics in Circular Accelerators, SLAC-PUB-4103, October 1986
E D Courant, R D Ruth, W T Weng, Stability in Dynamical Systems, SLAC-PUB-3415, August 1984
8.4 Acknowledgments
A number of books have been used in preparing this course, the full list is above, but the
following have been particularly valuable sources of material: Goldstein, Lanczos, Greenwood,
Wiedemann, Lee.