Upload
trandien
View
232
Download
3
Embed Size (px)
Citation preview
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
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.
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
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
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
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
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.
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.
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.
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.
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
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.
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)
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.
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.
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)
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)
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.
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 ?
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
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
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.
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.
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]
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
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
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.
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.
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.
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.
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
1.28 (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)
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
1.29 (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)
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
1.30 (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 ?
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
1.31 (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]
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
1.32 (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
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
1.33 (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:
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
1.34 (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]
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
1.35 (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
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
1.36 (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 ).
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
1.37 (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.
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
1.38 (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
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
1.39 (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.
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
1.40 (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.
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
1.41 (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)”.
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
1.42 (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.
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
1.43 (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.
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
1.44 (70)
Keppler Problems III
Figure: RK4, step size h = 0.0015
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
1.45 (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.
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
1.46 (70)
Keppler Problems V
Figure: RK4, step size h2 = 10h = 0.015
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
1.47 (70)
Keppler Problems VI
Figure: Euler-A and Euler-B, step size h3 = 100h = 0.15
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
1.48 (70)
Keppler Problems VII
Figure: RK4, step size h3 = 100h = 0.15
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
1.49 (70)
Keppler Problems VIII
Figure: Euler-A and Euler-B, step size h4 = 120h = 0.18
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
1.50 (70)
Keppler Problems IX
Figure: RK4, step size h4 = 120h = 0.18
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
1.51 (70)
Keppler Problems X
Figure: SV, step size h4 = 120h = 0.18
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
1.52 (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
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
1.53 (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 +
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
1.54 (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
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
1.55 (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).
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
1.56 (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.
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
1.57 (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.
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
1.58 (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.
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
1.59 (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.
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
1.60 (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)
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
1.61 (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)
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
1.62 (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)
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
1.63 (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(
1ρ
√(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)
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
1.64 (70)
Transverse Beam Dynamics V
• For straight magnet, ρ→∞. K = T(p) + V(q). Yoshidascheme works well.
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
1.65 (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].
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
1.66 (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).
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
1.67 (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).
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
1.68 (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
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
1.69 (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)
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
1.70 (70)
Differential k-form I
(w1 ∧ · · · ∧ wk)(ξ1, · · · , ξk) =
∣∣∣∣∣∣ω1(ξ1) · · · ω1(ξk)· · · · · · · · ·
ωk(ξ1) · · · ωk(ξk)
∣∣∣∣∣∣ (76)