Birkhoff normal forms for betatronic motion and stability ... · Birkhoff normal forms for...

G. Turchetti

Dipartimento di Fisica e Astronomia UNIBO

collaboration with F.Panichi University of Szczecin (PL)

Celestial and beam dynamics models

Birkhoff normal forms

Dyamic aperture

Lyapunov and noise induced errors

Conclusions

Birkhoff normal forms for betatronic motionand stability indicators

CELESTIAL MECHANICS

MODELS

Fronteers in Physics

Ultimate structure of matter:

the Higgs boson and beyond

Planetary systems in our

galaxy, earth like worlds,

life beyond the earth

Birkhoff normal forms for betatronic motion

Large Hadron Collider

Kepler Space Telescope


Models in celestial mechanics

A planet in the solar system and of a proton in a ring have dynamical

analogies and comparable stability times.

The 3 body problem: sun, Jupiter with m1>m2 on circular orbits and a

satellite with m3 0. Equilibrium at vertices of an equilateral triangle.

1 2

3

x

y

O

x1=-m

x2=1-m

x3= ½ -m

y3=√3 /2

m =m2/(m1+m2)

L4

restricted planar3 body problemLagrange equilibrium

Scaling coordinates and time (r12=1, T=2p) the Hamiltonian in corotatingsystem where V is the gravitational potential vx=px+y vy=py-x .

H= ½ (px2+py

2) + ypx-xpy + V(x,y)

Normal form near L4 where X=Y=0 and Jx= ½ (X2+Px2) Jy = ½ (Y2+Py

2)

H= w1 Jx + w2 Jy + H3 (Jx, Jy) + . . . .+ RN w1 w2 <0

H has a saddle at L4 no Lyapunov stability !!


Error plots. Poincaré section y=yc vy>0 H=Hc+10-5 Hc ~1.5


x x

vx

vx

x

vx

x

vx

x

vx

Reversibility error (REM) Lyapunov error (LE)

x

x-xc y-yc

vx, vy

Distance from L4 Projections and section

tmax= 200 T

The Hénon-Heiles model.

Motion of a star in an elliptical galaxy

H= T+ V =½ (px2+py

2) + ½ (x2+y2) -x3/3 +xy2

V=0 has a minimum at x=y=0 and three saddle points at the vertices of an equilateral triangle where V=1/6

stability H<1/6 boundary H=1/6

Birkhoff normal forms and betatronic motion

Isolines of H : innerst curve H=1/6 stability boundary

px=py=0 y=py=0 x=y=0

y

x

px

x px

py

Error plots Poincaré section y=0 py>0 H=E , boundary H(x,px,0,0)=E

Plots in x,px plane for y0, py0 fixed, boundary H(x,px,y0,py0)=1/6 tmax=20T


y0=py0=0 y0=py0=0.2 y0=py0=0 y0=py0=0.2

E=1/6 E=0.1 E=1/6 E=0.1

xxxx

xxxx

pxpx

px

px px

pxpxpx

Reversibility error (REM) P. section y=0 Orbits P section y=0

REM x px plane H value x, px plane

Tools for dynamic analysis

Normal forms. Hamiltonian invariant under a symmetry group

up to a remainder. Basically an analytic tool.

Frequency map error. The FFT of a quasi periodic signal n=2m

gives tunes n=(n1, n2) . Error e(n) = || n(n) - n(n/2) ||

Lyapunov error . Induced by an initial displacement

Reversibility error. Induced by noise or round-off


BEAM DYNAMICS

MODELS

Beam dynamics models: betatronic motion

Unlikely the Kepler problem the circular motion of a charge under a uniform

magnetic field B is not stable (drift along B).

H= ½(px2+py

2) + ½ + V(x,y,s) s=v0t px=dx/ds

Multipoles contribution

V= - 1/2 K1 (s) (x2 -y2) - 1/6 K2(s) (x

3 - 3xy2) + …

Explicit map for thin multipoles Km with m ≥ 3


:

x2

R2

x

z

y

sOO

B

Linear lattice M(x)= L x conjugated to a rotation

L= W R(w) W -1 W = = Wx-1

Courant-Sneider coordinates x’, p’x Change of section from sk-1 to sk

Lk = Ak Lk-1 Ak-1 Lk=L(sk)

Exact recurrence for bk=b(sk),

ak=a(sk) and phase advance


b1/2 0

-a b-1/2 b-1/2

x’

p’x

x

pxnormal form

The 2D Hénon map

One turn map for linear lattice with a thin sextupole in scaled coordinates

X = ½ bx3/2 k2 W-1 x for a flat beam

= R(w)


Xn

Pn + Xn2

Xn+1

Pn+1

n=0.21 orbits normalized REM Lyapunov error N=200

The 4D Hénon map

One turn map for a linear lattice with a thin sextupole in scaled coordinates

= b=

= R(w)


X n+1

Px n+1

Y n+1

Py n+1

nx=(3- √ 5 )/2 ny= √2 -1 b=1 projection of orbit with x0=y0=0.2 px0=py0=0

R(wx) 0

0 R(wy)

X n

Px n + Xn2-bYn

2

Y n

Py n -2b Xn Yn

bx

by

BIRKHOFF NORMAL FORMS


The one turn (superperiod) map M for thin multipoles is a polynominal whichcan be truncated to order N. In Courant-Sneider coordinates

MN(x)= R(w) (x + P2(x) + .. +PN(x) ) M(x)=MN(x) + O(|x|N+1)

A nonlinear symplectic tranformation x= F (X) changes M(x) into

a new map U(X) which is invariant under the group generated by R(w)

M(x)= F o U o F -1 (x) U(RX)=RU(X)

U(X) = R exp(DH) X H(RX)=H(X)

F(X) = exp ( DG ) X

H is the interpolating Hamiltonian, F a symplectic coordinates change


If w is non resonant (n=w/2p irrational) R(w) generates a continuous group

of rotations J= ½(Px2+X2) is the invariant.

If w=wR is resonant or quasi resonant w=wR+e (nR=m/q e<<wR)

R(wR) generates a discrete group of rotations and

H(J,q) = hk (J) cos(k q q + ak)

is invariant under the group. The level lines of H exhibit a chain of q islands


k

½

X Xx x

Px Pxpxpx

Quasi resonant normal forms for v=0.255(q=4) in x,px and X,PX planes

Orbits for v=0.21 in x,px plane and quasi resonanantnormal form (q=5) in the X,PX plane

Nekhoroshev stability estimates and analyticity

The series defining the normal forms are divergent due to singularities

associated to resonances. Conjugation with normal form up to a remainder

M = FN o (UN+EN) o FN-1 UN = R exp(D )

For a 2D where r=(X2+Px2)1/2 =(2J)1/2 estimate

|EN-1| <A (r/rN)N r<rN = 1/(CN)

Minimum achieved for N=N*=(e C r)-1 and r/rN*= e-1 . In a disc of radius r

|EN*| <A exp(-N* )=exp(-r*/r) r*=(eC)-1

Orbits starting in a disc r/2 remain in disc of radius r for n|EN* | <r/2

n < ½ Ar exp (r*/r)


H N

Singularities of normalizing transformations

The Birkhoff series diverge due to an accumulation at the origin of the

complex r=2J plane of singularities associated to the resonances.

If w=2p p/q +e for e 0 the conjugation function F behaves as

a geometric series. F has a pole at r=rq where W= dH/dJ is resonant.

W=w+ r W2 = 2p p/q rq =- e/W2

In the generic case varying r the frequency crosses infinitely many

resonances. The leading ones correspond

to the continued fraction expansion pj/qj

of the tune n and are located approximately

at rj=-ej/W2 where ej = w - 2p pj / qj

A rigorous analysis confirms this picture.

If rj>0 we have a true resonance (chain

of islands). The resonance is virtual if rj<0


Im r

Re r

analiticity region

DYNAMIC APERTURE

Work point. From H= wxJx+wyJy+ ½(h11 Jx2+2h12 JxJy+h22Jy

2) to any

resonance kxWx+kyWy= 2p m corresponds a line axJx+ayJy=b in action space


Tune plot

|kx|+ky|≤5

Resonancelines actionspace

nx=1/5

ny=1/4

Tune plot

magnif.

Resonancelines actionspace

nx=0.21ny=0.19

Short term dynamic aperture of 2D Hénon map

Boundary of stability domain of H for unstable resonances.

Resonance 0: as w 0 interpolating Hamiltonian H = ½ w (P2+X2) - X3/3

Scaling X=wx, P= w p and H=w3 h boundary h=1/6

h - = - (1-x)2 (1+2x) =0

Resonance 1/3: as e=w-2p/3 0 interpolating Hamiltonian

H= e J- 4 J3/2 cos(3q)


n 0 n - 1/3 0

Dynamic apertureof Henon map from resonant normal forms

6

1 p2

2 6

1

At the critical saddle points of h = 4/27 After a scaling with e

Short term dynamic aperture of 4D Hénon map

For w1=w2=w 0 after scaling X= w x, …, Py=w py and H= w3 h

where h is Hénon-Heiles hamiltoian

h - = + (1+2x ) ( 3y2-(1-x)2) = 0

stability region h<1/6 boundary h=1/6


1

6

px2+py

2

2

1

6

Dynamic aperture and REM error plots for nx =ny =0.01 Iterations n=5000

Plots in x,px y, py and x, y planes. Coordinates scaled by w=2 p n x


py=0 y=0 y=0.2 y=0.5

py=0 px=0 px=0.2 px=0.5

px=0 x=0 x=0.2 x=0.5

x px plane

purple curveanalytic boundary

x y plane

Iterations 5000

y py plane

coordinates scaled by w

LYAPUNOV AND NOISE

ERRORS


Errors due do a small displacement or small noise allow stability assessments

Lyapunov error LE

Small initial displacement: let iterates of x0 and x0 + e h ||h ||=1 be xn and xn + e hn +O(e2) Normalized Lyapunov error

eL(n,h) = ||hn || hn= DM(xn-1) hn-1

or

eL(n,h) = || An h || An=DMn(x0)

DM denotes the tangent map. To have a result independent from the vector hwe sum over the errors for any orthonormal basis obtaining

eL(n) = ( Tr(AnT An) )

1/2


Forward error FE

Noise of vanishingly small amplitude: xn a random vector <xn xmT> = I d nm

xe,n = M(xe,n-1 ) + e xn = x n + e Xn +O(e2)

Global stochastic perturbation

Xn = Lim

non homogeneous recurrence

Xn = DM(xn-1) Xn-1 + x n X0= 0

The normalized forward error is defined by

eF(n)= <Xn Xn >1/2 = Tr( Bk(n) BTk(n) ) Bk(n)= DM k(xn- k)

Notice e2F (n) is the trace of covariance matrix < Xn Xn

T >


1/2

k=0

.n-1

e 0

xe,n –x0

e

Reversibility error RE

with respect to the initial condition after n iterations forward and backwards with noise. Let x e, -m, n be the error after n iterations with M and m with M-1

xe,-m,n = M-1(xe,-m+1,n ) + e x –m = x n-m + e X -m, n +O(e2)

Global stochastic process

XnR = lim = Xn + DM-k (xk) x –(n-k)

Error is defined by

eR(n)= <XnR Xn

R>1/2 = Tr ( Ck(n)CkT(n) ) + Tr ( AI k(n) AIk

T(n) )

where AIk=DM-k(xk) and Ck= AInBk


k= 0

n-1

1/2n-1 n-1

k= 0k= 0

.

x e,-n,n – xn

ee 0

Linear maps

Errors asymptotics elementary for linear maps DM(x) = L x

eL(n)= Tr( (Ln)T Ln) eF(n)= eL2 (k)

elliptic fixed point

e(n) ~ 1 for LE e(n) ~ n 1/2 for FE, RE

parabolic f. p.

e(n) ~ n for LE e(n) ~ n 3/2 for FE, RE

hyperbolic f. p.

e(n) ~ e ln for LE, FE, RE


k=0

n-11/2 1/2

Power law growth and oscillations

Rotation:

L=R(w) eL(n)=√2 eF(n) =(2n) 1/2

L = F R(w) F –1 eL(n)= (A - (A-2) cos(2nw) )1/2A ≥ 2 eL (n) oscillates

F

Integrable map in normal form M(x)= R(W(||x||2/2) x

eL2(n) = 2 + ( W’ ||x||2 )2 n2

eF(n) ~ n3/2 eR(n)= eF(2n)

Integrable map not in normal form M= F R(W) F -1 power law error growth with oscillations.

.


Averaging on oscillations

Local error growth rate given by

De(n)= nR De(n)= nN

D na= a D eln = l n nR.

To damp oscillations of De(n) double average was proposed

Y(n) = 2 <<D e >>(n)

Y(n) is the mean exponential growrh factor of nearby orbits (MEGNO), Cincotta et al (2001)


d log e(n)

d log n

log e(n+1) –log e(n)

log (n+1) – log n

In the next slides we compare the errors for the 2D Hénon map

LE normalized Lyapunov error eL(n) . The error eL(n,h) is

avoided since h introduces a bias.

FE and RE The exact formulae involve the tangent map

REM reversibility error (method) due to round off.

The computation of REM requires just n iterates of the map followed

by n iterates of the inverse map. One can avoid the tangent map

in computing LE choosing e=10-14 and two orthonormal vectors h


Power law growth of e(n) for Hénon map


2

13

LEFEREREM

YLE YFE

YRE Y REM

orbits n= √ 2 -1 errors 1: x0=0.05 p0=0 2: x0=0.6 p0=0

averages 1: x0=0.05 p0=0 << 2: x0=0.6 p0=0 3: x0=0.75 p0=0

Integrable map. Near a stable resonance n=m/q +e/2p with q>4 the

resonant normal form gives a good approximation up to the dynamic aperture.

The errors growth follow a power law eL(n) ~ n |W’(J)| J. Near the separatrix

W(J) ~ 1/ log (Js-J) and W’ ~ (Js-J)-1 still a power law

Tq(x) Tchebychef


log

10

eL

Y

Orbits of H near 1/5 resonance Lyapunov error eL (n) MEGNO average YL (n)

xn= exp(n Dt DH) x0 with Dt=0.2 n=1000

For the Hénon map the boundary of a chain of islands is a chaotic layer, where the errors growth is exponential. Plots for e(n), Y(n) when n=0.21


eL eF

eReREM

YLE YFE

YRE Y REM

Obits magnification errors log10 e(n)

averages Y(n) errors zoom averages Y(n) zoom

Normalized REM error color plots for n= √2 -1 N=100


LE FE

RE RE round off

Comparison of frequency map error and REM


n= 2 11 n=2 14

n= 2 14 n=2 14

Frequency map error REM

Frequency error and REM: 4D Hénon map tunes n1=0.21 n2=0.25

x, px plane


y0=py0=0

y0=0.15

py0=0

x x

x x

px

px

px

px

Frequency error REM

Frequency error REM

CONCLUSIONS


Beam and celestial dynamics share the same tools

The Birkhoff normal forms describe the non linear betatron motion

Resonance induced singularities make the series asymptotic

Applications: tune diagrams, dynamic aperture, slow extraction

The Lyapunov and noise induced errors provide stability portraits

The round off reversibility error gives comparable results

The error analysis applies in absence of a first integrals H


REFERENCES

[1] Bazzani, P Mazzanti, G Servizi, G Turchetti. Normal forms for Hamiltonian maps and nonlinear effects in a particle accelerator. Il Nuovo Cimento B 102,

51-80 (1988).

[2] A Bazzani, M Giovannozzi, G Servizi, E Todesco, G Turchetti. Resonant normal forms, interpolating Hamiltonians and stability analysis of area preserving maps. Physica D: Nonlinear Phenomena, 64, 66, (1993).

[3] A. Bazzani, E. Todesco, G. Turchetti, G. Servizi A normal form approach to the theory of nonlinear betatronic motion CERN Yellow Reports 94-02 (1994)

http://cds.cern.ch/record/262179/files/CERN-94-02.pdf

[4] F. Panichi, L. Ciotti, G. Turchetti Fidelity and reversiblity in the restricted 3 body problem Communications in Nonlinear Science and Numerical Simulation 35 , 53-68 (2015)


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