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Symplectic Integrator and Beam Dynamics Simulations Lingyun Yang Hamiltonian Systems Symplecticness Differential 2-form Numerical Methods Numerical Integrators Applications and Examples Symplectic Integrator Implicit Symplectic Integrators Composition Method Generating Functions Lie Formalism Applications on Accelerator Beam Dynamics 1.1 (70) Chapter 1 Symplectic Integrator and Beam Dynamics Simulations Accelerator Physics Group, Journal Club November 9, 2010 Lingyun Yang NSLS-II Brookhaven National Laboratory
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Page 1: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.1 (70)

Chapter 1Symplectic Integrator and BeamDynamics SimulationsAccelerator Physics Group, Journal ClubNovember 9, 2010

Lingyun YangNSLS-II

Brookhaven National Laboratory

Page 2: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.2 (70)

Big Picture for Numerical Accelerator Physics

For single particle dynamics:

• Particle tracking is everything for a complex ring.(integrator)

• Dynamics and physics quantities are in the “map”:betatron oscillation, spin, ...

• Map can be extracted from the tracking. (TPSA)• Map analysis gives every quantity we want: twiss,

emittance, (normal form)

Here we focus on how to calculate the physics, instead of howto use it.

Page 3: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.3 (70)

1 Hamiltonian SystemsSymplecticnessDifferential 2-form

2 Numerical MethodsNumerical IntegratorsApplications and Examples

3 Symplectic IntegratorImplicit Symplectic IntegratorsComposition MethodGenerating FunctionsLie Formalism

4 Applications on Accelerator Beam Dynamics

Page 4: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.3 (70)

1 Hamiltonian SystemsSymplecticnessDifferential 2-form

2 Numerical MethodsNumerical IntegratorsApplications and Examples

3 Symplectic IntegratorImplicit Symplectic IntegratorsComposition MethodGenerating FunctionsLie Formalism

4 Applications on Accelerator Beam Dynamics

Page 5: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.3 (70)

1 Hamiltonian SystemsSymplecticnessDifferential 2-form

2 Numerical MethodsNumerical IntegratorsApplications and Examples

3 Symplectic IntegratorImplicit Symplectic IntegratorsComposition MethodGenerating FunctionsLie Formalism

4 Applications on Accelerator Beam Dynamics

Page 6: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.3 (70)

1 Hamiltonian SystemsSymplecticnessDifferential 2-form

2 Numerical MethodsNumerical IntegratorsApplications and Examples

3 Symplectic IntegratorImplicit Symplectic IntegratorsComposition MethodGenerating FunctionsLie Formalism

4 Applications on Accelerator Beam Dynamics

Page 7: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.4 (70)

Introduction

Over the last century (19th) attention has shifted fromthe computation of individual orbits towards thequalitative properties of families of orbits. Forexample, the question of whether a given orbit isstable can only be answered by studying thedevelopment of all orbits whose initial conditions arein some sense “close to” those of the orbit beingstudied.–M. V. Berry

The objective is to answer some questions (what-why-how)and to give some real life examples

1 What is symplecticness, in algebra, in geometry ?2 Why we have to use symplectic integrator ?3 How to prove the symplecticness of an integrator ?4 Popular symplectic integrators.5 Symplectic integrators in beam dynamics. Particle tracking

in different magnets.

Page 8: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.5 (70)

Hamilton Equations I

Hamilton equations

dqi

dt=∂H∂pi

,dpi

dt= −∂H

∂qi, (1)

or in a compact form

dzdt

= J∇zH(z, t) (2)

where z ≡ (q, p)T , q, p ∈ Rd, z ∈ R2d, and

J ≡(

0 Id

−Id 0

)(3)

The solution is a transformation mapping (flow map):

(q, p) = Φt0→t,H(q0, p0)

or simply φt,H if we set t0 = 0 as the starting time.

Page 9: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.6 (70)

Hamilton Equations II

The flow map φt,H given by a Hamiltonian system hassemi-group property, i.e. closed under composition operator:

φt+s,H = φt,H φs,H

When t = 0, the flow map φ0,H is the identity map.

φ−t,H φt,H = I

When using difference method to simulate the evolution of adynamical system, the time is discretized at t0, t1, . . . , tn, thesemi-group property is not exactly hold in general, but only upto certain order of step size.

Page 10: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.7 (70)

Hamiltonian flow maps are symplectic

A transformation (flow map) z∗ = ∂ψ∂z z is symplectic if

[ψz(z)]TJψz(z) = J

Theorem (Poincare 1899)

The flow map φt,H(z) of a Hamiltonian system is symplectic

Proof.

The Jacobian of the flow map F(t) ≡ ∂

∂zφt,H(z) has F(0) = I2d.

K ≡ F(t)TJF(t) is a constant, i.e.dKdt

= 0.

Page 11: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.8 (70)

Symplecticness is the preservation of area

ηTJξ is the oriented area of the parallelogram determined by ηand ξFor a parallelogram P having a fixed vertex at (q, p), and twovectors as sides, η and ξ. The parallelogram P∗ aftertransformation ψ has sides ψzη and ψzξ. Vertex (q, p) are nowψ(q, p). Now P and P∗ have same area if and only if

ηTψzTJψzξ = ηTJξ

Clearly, this holds if and only if ψzTJψz = J

Figure: Symplectic map preserves the area

Page 12: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.9 (70)

Preservation of area: differential forms I

• We can use differential forms as an alternative languageto express the preservation of area.

• The meaning and properties of differential forms is outsideof this discussion of numerical simulation, (Arnold 1989,chapter 7).

• The algebraic manipulation is easy, and good enough toprove the preservation of area.

• The wedge product of ω1(ξ1, ξ2) and ω2(ξ1, ξ2) is definedas

(ω1 ∧ ω2)(ξ1, ξ2) =

∣∣∣∣ω11(ξ1, ξ2) ω12(ξ1, ξ2)ω21(ξ1, ξ2) ω22(ξ1, ξ2)

∣∣∣∣=ω1(ξ1, ξ2)ω2(ξ1, ξ2)− ω2(ξ1, ξ2)ω1(ξ1ξ2)

(4)

It is the oriented area of a parallelogram determined by ω1and ω2 in (ω1, ω2) plane.

Page 13: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.10 (70)

Preservation of area: differential forms IIz∗ = (q∗, p∗) is given by the map ψz(z), and we have

dp∗ =∂p∗

∂pdp +

∂p∗

∂qdq, dq∗ =

∂q∗

∂pdp +

∂q∗

∂qdq (5)

The wedge product is

dp∗ ∧ dq∗ =∂p∗

∂p∂q∗

∂pdp ∧ dp +

∂p∗

∂p∂q∗

∂qdp ∧ dq

+∂p∗

∂q∂q∗

∂pdq ∧ dp +

∂p∗

∂q∂q∗

∂qdq ∧ dq

(6)

the wedge product is skew symmetric,

dp ∧ dp = dq ∧ dq = 0, dp ∧ dq = −dq ∧ dp (7)

Then for one degree of freedom (d = 1)

dp∗ ∧ dq∗ = (∂p∗

∂p∂q∗

∂q− ∂p∗

∂q∂q∗

∂p)dp ∧ dq =

∣∣∣∣∣∣∣∂p∗

∂p∂p∗

∂q∂q∗

∂p∂q∗

∂q

∣∣∣∣∣∣∣ dp ∧ dq

(8)

Page 14: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.11 (70)

Preservation of area: differential forms IIIThe invariant of dp ∧ dq is equivalent to detψz(q, p) = 1, i.e. thepreservation of area. (for 2× 2 matrix, det M = 1 is equivalentto MTJM = J).In General case, d > 1, the transformation ψ is symplectic if anonly if

dp∗1 ∧ dq∗1 + · · ·+ dp∗d ∧ dq∗d = dp1 ∧ dq1 + · · ·+ dpd ∧ dqd (9)

ordp∗ ∧ dq∗ = dp ∧ dq (10)

• We can use this to check if a numerical algorithm issymplectic or not.

• Suppose ψz(z) maps from one iteration zn(tn) to the nextzn+1(tn+1), i.e. an integrator, we can check whether it issymplectic by proving

dpn+1 ∧ dqn+1?= dpn ∧ dqn (11)

• The concatenated map of two symplectic map issymplectic. This is obvious from the invariant of dpi ∧ dqi.

Page 15: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.12 (70)

Remarks on symplecticness and differential two-forms

• Differential 2-form is an alternative language to describethe preservation of area.

• For one-degree-of-freedom systems, symplecticnessimplies preservation of area (differential 2-form).

• For higher dimensions, the conservation of volume followsfrom Liouville’s theorem (differential 2n-form).

• Symplecticness is stronger than conservation of volume.• In the following sections we can see differential 2-form is

easier to use in proving the symplecticness of a numericalintegrator.

Page 16: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.13 (70)

Numerical Integrator I

A systems of differential equations

dxdt

= f(t, x) (12)

where x = (x1, x2, . . . , xn)T and f(t, x) = (f1(t, x), . . . , fn(t, x))T .Using t as independent variable and evaluate x at discretizedtime point t0, t1, · · · , tn, · · · . We define hi ≡ ∆ti = ti − ti−1 orsimply h as the current step size in each iteration.

• Explicit Euler (Forward Euler)

xn+1 = xn + hf(xn) (13)

• Implicit Euler (backward Euler)

xn+1 = xn + hf(xn+1) (14)

• Implicit Midpoint(symplectic, order 2)

xn+1 = xn + hf(xn + xn+1

2) (15)

Page 17: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.14 (70)

Numerical Integrator II

• Explicit 4th order Runge-Kutta

k1 =f(tn, xn)

k2 =f(tn +h2, xn +

h2

k1)

k3 =f(tn +h2, xn +

h2

k2)

k4 =f(tn + h, xn + hk3)

xn+1 =xn +h6

(k1 + 2k2 + 2k3 + k4)

(16)

Page 18: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.15 (70)

Euler Method, a simple example, non-Hamiltonian

xn+1 = xn + hf(xn) (17)

It is a first-order integrator, therefore x(tn) will approach the truesolution linearly as h→ 0.

Page 19: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.16 (70)

Simple Harmonic Oscillator I

For simple harmonic oscillator

H = p2 + q2 (18)

with initial condition q0 = 0, p0 = 1, the solution is q = sin t,p = cos t. The period T = 2π.The solutions are circles in phase space, they are concentricfor differential initial conditions.

What is wrong in the following numerical solutions ?

Page 20: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.17 (70)

Simple Harmonic Oscillator II

Figure: Forward Euler method: xn+1 = xn +hf (xn), step size h = 2π/12

Page 21: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.18 (70)

Simple Harmonic Oscillator III

Figure: Backward (implicit) Euler method: xn+1 = xn + hf (xn+1), stepsize h = 2π/12

Page 22: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.19 (70)

Simple Harmonic Oscillator IV

Figure: RK4 method, step size h = 2π/5.

Page 23: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.20 (70)

Simple Harmonic Oscillator V

Figure: RK4 Gauss method (symplectic), step size h = 2π/5.integrate with longer time.

Page 24: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.21 (70)

Pendulum

Figure: Area preservation of the flow of Hamiltonian system [8]

Page 25: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.22 (70)

Lennard-Jones oscillator I

H = T + V, V ≡ φL.J.(r) = ε[(rr)−12 − (

rr)−6] (19)

Figure: Euler method, h = 0.0002, 180k steps

Page 26: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.23 (70)

Lennard-Jones oscillator II

Figure: RK4, h = 0.0002, 180k steps

Page 27: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.24 (70)

Lennard-Jones oscillator III

Figure: Stormer-Verlet, second order, h = 0.0002, 180k steps.

Page 28: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.25 (70)

Lennard-Jones oscillator IV

Figure: Stormer-Verlet, second order, step size h2 = 10h = 0.002, 18ksteps.

Page 29: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.26 (70)

Some remarks on symplectic integrator

• Symplectic Integrator A.K.A Geometric Integrator A.K.ACanonical Integrator.

• classical theories of numerical integration give informationabout how well different methods approximate thetrajectories for fixed times as step sizes tend to zero.Dynamical systems theory asks questions aboutasymptotic, i.e. infinite time, behavior.

• Geometric integrators are methods that exactly conservequalitative properties associated to the solutions of thedynamical system under study.

• The difference between symplectic integrators and othermethods become most evident when performing long timeintegrations (or large step size).

• Symplectic integrators do not usually preserve energyeither, but the fluctuations in H from its original valueremain small.

Page 30: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

Applications and Examples

Symplectic IntegratorImplicit SymplecticIntegrators

Composition Method

Generating Functions

Lie Formalism

Applications onAccelerator BeamDynamics

1.27 (70)

In general, “one size fits all” doesn’t happen a lot. Whensolving numerical Hamiltonian problems, we will treat ourproblems differently.

1 Composition method, if H is separable as solvable pieces,e.g. H(q, p) = T(p) + V(q).

2 Implicit method, if H can not be solved by parts.

We start from the general case where H can not be solved byparts.

Page 31: Symplectic Integrator and Beam Dynamics Simulationsphysics.indiana.edu/~shylee/ap/wshop10/geometric-integrator-beamer.… · Symplectic Integrator and Beam Dynamics Simulations ...

Symplectic Integratorand Beam Dynamics

Simulations

Lingyun Yang

Hamiltonian SystemsSymplecticness

Differential 2-form

Numerical MethodsNumerical Integrators

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1.28 (70)

Euler-A and Euler-B are symplectic integrator I

For Hamilton equationsdqdt

= ∇pHdpdt

= −∇qH(20)

Using differential two-forms we can prove Euler-A and Euler-Bmethod described in the following are symplectic. They are firstorder, but we will not prove here (will not prove any “ordercondition” in this talk).

• Euler-A

qn+1 = qn + ∆t∇pH(qn+1, pn)

pn+1 = pn −∆t∇qH(qn+1, pn)(21)

• Euler-B

qn+1 = qn + ∆t∇pH(qn, pn+1)

pn+1 = pn −∆t∇qH(qn, pn+1)(22)

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1.29 (70)

Euler-A and Euler-B are symplectic integrator II

We prove Euler-B is symplectic by proving

dqn+1 ∧ dpn+1 = dqn ∧ dpn (23)

From Equation. (22) we have

dqn+1 = dqn + ∆t[Hpqdqn + Hppdpn+1] (24)

dpn+1 = dpn −∆t[Hqqdqn + Hqpdpn+1] (25)

where Hqp, Hpp and Hqq are Jacobian matrix

Hqq =∂2H∂qi∂qj

, Hqp =∂2H∂qi∂pj

, Hpp =∂2H∂pi∂pj

, Hpq = HTqp

(26)Using the skew symmetry and bilinear property

da∧db = −db∧da, da∧(αdb+βdc) = αda∧db+βda∧dc (27)

We havedqn ∧ Hqqdqn = 0, dpn ∧ Hppdpn = 0 (28)

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1.30 (70)

Euler-A and Euler-B are symplectic integrator III

dqn+1 ∧ dpn+1 = dqn ∧ dpn+1 + ∆tHpqdqn ∧ dpn+1 (29)

whiledqn ∧ dpn+1 = dqn ∧ dpn −∆tHT

qpdqn ∧ dpn+1 (30)

We usedda ∧ Adb = (ATda) ∧ db (31)

Sodqn+1 ∧ dpn+1 = dqn ∧ dpn (32)

Euler-B method is symplectic.Similarly we can prove Euler-A is also symplectic.

Now, we have a first order, implicit, symplectic integrator.(disappointed at “first order” and “implicit” ?)

How about higher order ?

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1.31 (70)

Runge-Kutta Methods and Symplecticness IThe most popular RK4 method can be written as

k1 =f (tn, xn)

k2 =f (tn + h/2, xn + hk1/2)

k3 =f (tn + h/2, xn + hk2/2)

k4 =f (tn + h, xn + hk3)

xn+1 =xn +16

(k1 + 2k2 + 2k3 + k4)

(33)

012

12

12 0 1

21 0 0 1

16

26

26

16

Figure: Explicit RK4 [8, Hairer 2006]

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1.32 (70)

Runge-Kutta Methods and Symplecticness II

Let bi, aij, (i, j = 1, . . . , s) be real numbers and let ci =∑s

j=1 aij.An s-stage Runge-Kutta method is given by [8]

ki = f(t0 + cih, x0 + hs∑

j=1

aijkj), i = 1, . . . , s

x1 = x0 + hs∑

i=1

biki

(34)

The explicit Runge-Kutta methods have aij = 0 whenever i ≤ j.The coefficients are usually displayed as

c1 a11 · · · a1s...

......

cs as1 · · · ass

b1 bs

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1.33 (70)

Symplectic Runge-Kutta methods I

Conditions for symplectic Runge-Kutta methods

Assume the coefficients of the method (34) satisfy the relations

biaij + bjaji − bibj = 0, i, j = 1, . . . , s (35)

Then the method is symplectic.

The proof using differential forms can be found in ref. [4]. Sameas the proof for Euler-A and Euler-B method, we need to provethe differential 2-form is conserved.

Now for arbitrary form of H, we have a 4th order symplecticintegrator, but it is still implicit as shown in the following lemma.

Lemma

Symplectic Runge-Kutta methods are necessarily implicit, i.e.aij 6= 0 for some i, j ∈ 1, 2, . . . , s, j ≥ i

A class of RK methods usually called Gauss methods meetthis condition:

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1.34 (70)

Symplectic Runge-Kutta methods II

1 s = 11/2 1/2

12 s = 2

12 −

√3

614

14 −

√3

6

12 +

√3

614 +

√3

614

12

12

3 s = 312 −

√15

10536

29 −

√15

155

36 −√

1530

12

536 +

√15

2429

536 −

√15

24

12 +

√15

10536 +

√15

3029 +

√15

155

36

518

49

518

4 s = 4 can be found in ref. [1]

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1.35 (70)

Symplectic Runge-Kutta methods III

12 − ω2 ω1 ω′

1 − ω3 + ω′4 ω′

1 − ω3 − ω′4 ω1 − ω5

12 − ω′

2 ω1 − ω′3 + ω4 ω′

1 ω′1 − ω′

5 ω1 − ω′3 − ω4

12 + ω′

2 ω1 + ω′3 + ω4 ω′

1 + ω′5 ω′

1 ω1 + ω′3 − ω4

12 + ω2 ω1 + ω5 ω′

1 + ω3 + ω′4 ω′

1 + ω3 − ω′4 ω1

2ω1 2ω′1 2ω′

1 2ω1

ω1 = 18 −

√30

144 , ω′1 = 18 +

√30

144 , ω2 = 12

√15+2

√30

35 ,

ω′2 = 12

√15−2

√30

35 , ω3 = ω2( 16 +

√30

24 ), ω′3 = ω′2( 16 −

√30

24 ),

ω4 = ω2( 121 + 5

√30

168 , ω′4 = ω′2( 121 −

5√

30168 , ω5 = ω2 − 2ω3,

ω′5 = ω′2 − 2ω′3.5 s = 5 can be found in ref. [1]

12 − ω2 ω1 ω′1 − ω3 + ω′4

32225 − ω5 ω′1 − ω3 − ω′4 ω1 − ω6

12 − ω

′2 ω1 − ω′3 + ω4 ω′1

32225 − ω

′5 ω′1 − ω

′6 ω1 − ω′3 − ω4

12 ω1 + ω7 ω′1 + ω′7

32225 ω′1 − ω

′7 ω1 − ω7

12 + ω′2 ω1 + ω′3 + ω4 ω′1 + ω′6

32225 + ω′5 ω′1 ω1 + ω′3 − ω4

12 + ω2 ω1 + ω6 ω′1 + ω3 + ω′4

32225 + ω5 ω′1 + ω3 − ω′4 ω1

2ω1 2ω′164225 2ω′1 2ω1

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1.36 (70)

Symplectic Runge-Kutta methods IV

ω1 = 322−13√

703600 , ω′1 = 322+13

√70

3600 , ω2 = 12

√35+2

√70

63 ,

ω′2 = 12

√35−2

√70

63 , ω3 = ω2( 452+59√

703240 ), ω′3 = ω′2( 452−59

√70

3240 ),

ω4 = ω2( 64+11√

701080 ), ω′4 = ω′2( 64−11

√70

1080 ), ω5 = 8ω2( 23−√

70405 ),

ω′5 = 8ω′2( 23+√

70405 ), ω6 = ω2 − 2ω3 − ω5, ω′6 = ω′2 − 2ω′3 − ω′5,

ω7 = ω′2( 308−23√

70960 ), ω′7 = ω′2( 308+23

√70

960 ).

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1.37 (70)

Remarks on Integrating a General Hamiltonian System

What we know so far:• Explicit Euler method is not symplectic.• No explicit RK method is symplectic.• Implicit Midpoint method is symplectic. (did not prove)• Euler-A and Euler-B are first order symplectic.• RK method with Gauss scheme are symplectic

These are for general form of f(x). i.e. no special form ofHamiltonian are assumed.

• What about H(q, p) = T(p) + V(q) orH(q, p) = H1(q, p) + H2(q, p) where H1 and H2 are solvable?

• We always hear about drift-kick, what does it really meanfor symplectic integrator.

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1.38 (70)

Partitioned Euler Method IIf H = T(p) + V(q), then

• Euler-A

qn+1 = qn + ∆t∇pH(qn+1, pn)

pn+1 = pn −∆t∇qH(qn+1, pn)(36)

becomes

qn+1 = qn + ∆t∇pT(pn)

pn+1 = pn −∆t∇qV(qn+1)(37)

It is explicit now, and a “drift-kick” scheme.• Euler-B

qn+1 = qn + ∆t∇pH(qn, pn+1)

pn+1 = pn −∆t∇qH(qn, pn+1)(38)

becomes

qn+1 = qn + ∆t∇pT(pn+1)

pn+1 = pn −∆t∇qV(qn)(39)

This is a “kick-drift” scheme

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1.39 (70)

Stormer-Verlet Scheme

pn+1/2 = pn − ∆t2∇qV(qn) (40)

qn+1 = qn + ∆tpn+1/2 (41)

pn+1 = pn+1/2 − ∆t2∇qV(qn+1) (42)

We will see that Stormer-Verlet method, which is popular inmolecule dynamics simulations, is a second order symplecticintegrator.

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1.40 (70)

Composition and Symmetry I

H = T + V⇒ φt,T+V?= φt,T φt,V

If the above is true, for H = T(p) + V(q), the over all order willdepend on the integrator used in T(p) and V(q). We maychoose a higher order explicit integrator for it.

Theorem

If H = H1 + H2 + · · ·+ Hn is any splitting into twice differentiableterms, then the composition method

Ψ∆t = φ∆t,H1 φ∆t,H2 · · · φ∆t,Hn (43)

is (at least) a first order symplectic integrator.

This can be proved from the Taylor expansion ofφ∆t,H1(φ∆t,H2(z)).

Even Hi are exactly solvable, the overall effect may be only firstorder.

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1.41 (70)

Composition and Symmetry II

Here comes the symmetry.

• The adjoint method of Ψ∆t is defined by Ψ∗∆t = [Ψ−∆t]−1

(∆t↔ −∆t, zn ↔ zn+1).• Euler-A is the adjoint method of Euler-B.• A method is symmetric if Ψ∗∆t = Ψ−∆t.• We can prove that Ψ∆t = Ψ∗∆t/2 Ψ∆t/2 is symmetric.

Theorem

The order of a symmetric method is necessarily even (nothingto do with the symplecticness).

Compose Euler-A and Euler-B together, we have a symmetricsecond order symplectic integrator.“drift(half)-kick(half)-kick(half)-drift(half)”.

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1.42 (70)

Keppler Problems I

H = T(p) + V(q) =p2

1 + p22

2+

1√q2

1 + q22

(44)

The solution in polar coordinate is r =a(1− e2)

1± e cos θ, the

eccentricity of the ellipse is e =√

a2−b2

a2 , a,b are semimajor andsemiminor axis.

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1.43 (70)

Keppler Problems II

Figure: Euler-A and Euler-B, step size h = 0.0015

The angular momentum are exactly conserved, because it isthe differential 2-form.

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1.44 (70)

Keppler Problems III

Figure: RK4, step size h = 0.0015

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1.45 (70)

Keppler Problems IV

Figure: Euler-A and Euler-B, step size h2 = 10h = 0.015

With larger step size, quantitatively the results changed, butqualitatively the result did not change.

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1.46 (70)

Keppler Problems V

Figure: RK4, step size h2 = 10h = 0.015

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1.47 (70)

Keppler Problems VI

Figure: Euler-A and Euler-B, step size h3 = 100h = 0.15

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1.48 (70)

Keppler Problems VII

Figure: RK4, step size h3 = 100h = 0.15

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1.49 (70)

Keppler Problems VIII

Figure: Euler-A and Euler-B, step size h4 = 120h = 0.18

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1.50 (70)

Keppler Problems IX

Figure: RK4, step size h4 = 120h = 0.18

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1.51 (70)

Keppler Problems X

Figure: SV, step size h4 = 120h = 0.18

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1.52 (70)

Generating Function I

R. Ruth and K. Feng developed symplectic integratorindependently using generating functions.

The generating function methods

1 use generating function to get the iteration scheme, and

2 the new Hamiltonian are approximated to certain order ofstep size.

These two points guarantee the symplecticness and the order.

• Ruth’s method (1983) [2]

General Third Order

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1.53 (70)

Generating Function II

H = g(p) + V(x, t), f = −∇xV

p1 =p0 + c1hf(x0, t0) x1 =x0 + d1hdgdp

(p1) (45a)

p2 =p1 + c2hf(x1, t0 + d1h) x2 =x1 + d2hdgdp

(p2) (45b)

p =p2 + c1hf(x0, t0) x1 =x0 + d3hdgdp

(p) (45c)

where c1 = 7/24, c2 = 3/4, c3 = −1/24, d1 = 2/3, d2 = −2/3,d3 = 1.

• Feng and Qin (Implicit)[3]

pi =p0i − hHqi(

p + p0

2,

q + q0

2)− h3

4!(Hpjpkqi Hqj Hqk +

2Hpjpk Hqjqi Hqk − 2Hpjqkqi Hpj Hqk − 2Hpjqk Hpjqi Hqk− (46a)2Hpjqk Hpj Hqkqi + 2Hqjqk Hpjqi Hpk + Hqjqkqi Hpj Hpk )

qi =q0i + hHpi(

p + p0

2,

q + q0

2) +

h3

4!(Hpjpkpi Hqj Hqk +

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1.54 (70)

Generating Function III

2Hpjpk Hqjpi Hqk − 2Hpjqkpi Hpj Hqk − 2Hpjqk Hpjpi Hqk− (46b)2Hpjqk Hpj Hqkpi + Hqjqkpi Hpj Hpk + 2Hqjqk Hpjpi Hpk )

where subscripts of H are partial derivatives, e.g.Hpjpkqi = ∂3H

∂pj∂pk∂qi. Same indices are summed up according

to Einstein rules.

Solve the wrong problem in a right way – K. Feng

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1.55 (70)

Lie Formalism and Yoshida Scheme I

Yoshida developed a method, for H = T(p) + V(q), a 2n-thorder symplectic integrator can be constructed from 2(n-1)-thorder symplectic integrators.

• the solution of linear differential equation z′ = Az is z = etA

• etAetB 6= et(A+B) in general

• Using BCH formula, etAetB = et(A+B)+t2/2[A,B]+··· ≡ etD

• For H = T(p) + V(q), T(p) and V(q) are solvable (as inEuler-A and Euler-B methods)

• The symmetry can make the integrator be even order.• Discretize the time step ehH = ec1hTed1hV · · · ecnhTednhV ≡ ehK

• Use BCH form to approximate K to H up to certain order ofstep size h, and solves for ci and di.

• This can be generalized to nonlinear differential equationz′ = f (z).

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1.56 (70)

Lie Formalism and Yoshida Scheme IIA 2n + 2-order integrator can be constructed by three 2n-orderintegrators [5]:

S2n+2(τ) = S2n(z1τ)S2n(z0τ)S2n(z1τ) (47)

where (there is a typo in Yoshida’s paper)

z0 = − 21/(2n+1)

2− 21/(2n+1), z1 =

12− 21/(2n+1)

(48)

They are solved from the order condition z0 + 2z1 = 1,z2n+1

0 + 2z2n+11 = 0.

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1.57 (70)

Lie Formalism and Yoshida Scheme III

Figure: 2nd order symplectic integrator of Yoshida’s scheme.Composed by Euler-A and Euler-B method.

Figure: 4th order symplectic integrator of Yoshida’s scheme. It iscomposed by three second order integrators symmetrically.

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1.58 (70)

Lie Formalism and Yoshida Scheme IV

Figure: 6th order symplectic integrator, composed by three 4th ordersymplectic integrators. The step 4 and 5 can be combined, so canstep 8 and 9.

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1.59 (70)

Comments on Generating Function and Lie Formalism

• Yes, using the generating function, we have canonicaltransformation, it is symplectic. But what about the order ?

• Yes, Lie algebra can give us canonical transformation, e:f :zis symplectic, but e:f : =

∑∞n=0

1n! :f :n, it may be not

symplectic if the trucation are not careful.

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1.60 (70)

Transverse Beam Dynamics I

H = −(1 +xρ

)√

p2 − (px − eAx)2 − (pz − eAz)2 − eAs (49)

Where s is the independent variable.

Bx = − 1hs

∂As

∂z, Bz =

1hs

∂As

∂x, hs = 1 + x/ρ (50)

If px and pz are small,

H ≈ −p(1 +xρ

) +1 + x/ρ

2p[(px − eAx)

2 − (pz − eAz)2]− eAs (51)

Divided by p0, the norminal momentum, δ = (p− p0)/p0,choose Ax = Az = 0, As 6= 0, now px is px/p0, py is py/p0.

K = −(1 +xρ

)√

(1 + δ)2 − p2x − p2

z −eAs

p0(52)

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1.61 (70)

Transverse Beam Dynamics II

• For a sector dipole B0ρ = e/p0 for on momentum particle,Bx = 0, Bzρ = e/p0, and eAs

p0= − e

p0Bz(x + x2

2ρ ) = − xρ −

x2

2ρ2

K = −(1 +xρ

)√

(1 + δ)2 − p2x − p2

z +xρ

+x2

2ρ2 (53)

For driftK1 = −

√(1 + δ)2 − p2

x − p2z (54)

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1.62 (70)

Transverse Beam Dynamics III

px(s) = px (55)dxds

= px/√

(1 + δ)2 − p2x − p2

z (56)

pz(s) = pz (57)dzds

= pz/√

(1 + δ)2 − p2x − p2

z (58)

δ(s) = δ (59)

? =∂K1

∂δ= − 1 + δ√

(1 + δ)2 − p2x − p2

z

(60)

(61)

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1.63 (70)

Transverse Beam Dynamics IV

For dipoles, it is not in the form of H = T(p) + V(q) that weare familiar. Etienne gives a split of K [9, Forest 2006]:

K = −(1 +xρ

)√

(1 + δ)2 − p2x − p2

z + b1(x +x2

2ρ)︸ ︷︷ ︸

H1

−b1(x +x2

2ρ)− eAs

p0︸ ︷︷ ︸H2

(62)Each part are analytically solvable for arbitrary bn 6= 0.

x(s) =ρ

b1(

√(1 + pt)2 − px(s)2 − p2

y −dpx(s)

ds− b1) (63)

px(s) = px cos(sρ

) + (√

(1 + pt)2 − p2x − p2

y − b1(ρ+ x)) sin(sρ

) (64)

y(s) = y +pysb1ρ

+py

b1(sin−1

(px√

(1 + pt)2 − p2y

)− sin−1(

px(s)√(1 + pt)2 − p2

y

))

(65)

py(s) = py, pt(s) = pt (66)

t(s) = t +(1 + pt)s

b1ρ+

1 + pt

b1(sin−1

(px√

(1 + pt)2 − p2y

)− sin−1(

px(s)√(1 + pt)2 − p2

y

))

(67)

(68)

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1.64 (70)

Transverse Beam Dynamics V

• For straight magnet, ρ→∞. K = T(p) + V(q). Yoshidascheme works well.

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1.65 (70)

Summary I

• Symplecticness is a fundamental geometric property forHamiltonian system that an integrator should preserve.

• Using differential 2-form, we can prove the symplecticnessof an integrator.

• Symmetry may increase the order of an integrator.• For most of the case, H = T(p) + V(q), composition

methods are convenient. We can have an integrator up toarbitrary order (not necessary the most efficientintegrator).

• Sector dipole, rectangular dipole and all the straightmagnets can be modeled by high order integrator usingthe exact form of H.

• What about matrix code, still has any attractiveadvantages ?

• Moving to integrator based tracking code ? [9, Forest2006].

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1.66 (70)

Reference I

J. C. Butcher, Implicit Runge-Kutta Processes,Mathematics of Computation 18, 50-64 (1964).

Ronald D. Ruth, Nuclear Science, IEEE Transactions On30, 2669-2671 (1983).

K. Feng, M. Z. Qin, The symplectic methods for thecomputation of Hamiltonian equations, In Y. L. Zhu and B.Y. Guo, editors, Numerical Methods for Partial DifferentialEquations, Lecture Notes in Mathematics 1297, pages1-37. Springer, Berlin, 1987.

J. M. Sanz-Serna, Runge-Kutta schemes for Hamiltoniansystems, BIT Numerical Mathematics 28, 877-883 (1988).

H. Yoshida, Construction of high order symplecticintegrators, Physics Letters A 150, 262–268 (1990).

J. M. Sanz-Serna and M. P. Calvo, Numerical HamiltonianProblems, 1st ed. (Chapman & Hall, 1994).

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1.67 (70)

Reference II

Benedict Leimkuhler and Sebastian Reich, SimulatingHamiltonian Dynamics (Cambridge University Press,2005).

Ernst Hairer, Christian Lubich, and Gerhard Wanner,Geometric Numerical Integration: Structure-PreservingAlgorithms for Ordinary Differential Equations, 2nd ed.(Springer, 2006).

E. Forest, Geometric integration for particle accelerators,Journal of Physics A: Mathematical and General 39,5321–5377 (2006).

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1.68 (70)

Two-form

A two-form on R2d is a skew-symmetric bilinear function Ω(ξ,η)with arguments ξ and η. The symplectic two-form is defined as

Ω(ξ,η) = ξTJη ξ,η ∈ R2d (69)

The geometric interpretation of the two-form Ω for d = 1 is theoriented are of the parallelogram spanned by the two vectors ξand η. ξTJη = ξ2η1 − ξ1η2, ξ = (ξ1, ξ2)T , η = (η1, η2)T

For d > 1, we define

Ω(ξ,η) =

d∑i=1

Ω0(ξ(i),η(i)) (70)

where Ω0 is standard two-form of a pair of vectors. Ω(ξ,η) isthe sum of the oriented area of the parallelograms spanned bythe pair of vectors ξ(i) and η(i)

Differential

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1.69 (70)

Examples

1 Free particle in R3

H = p2/2m, (71)

the flow map is

φt,H(q, p) =

(q + t

m pp

)(72)

2 The pendulum.

H = T + V =p2

2− cos(q). (73)

3 Kepler’s problem.

H = T + V =p2

1 + p22

2+

−1√q2

1 + q22

(74)

4 Modified Kepler’s problem.

H = T + V =p2

1 + p22

2+

−1√q2

1 + q22

− ε

2√

q21 + q2

2

(75)

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1.70 (70)

Differential k-form I

(w1 ∧ · · · ∧ wk)(ξ1, · · · , ξk) =

∣∣∣∣∣∣ω1(ξ1) · · · ω1(ξk)· · · · · · · · ·

ωk(ξ1) · · · ωk(ξk)

∣∣∣∣∣∣ (76)


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