Birkhoff normal forms for betatronic motion and stability ... · Birkhoff normal forms for...
Transcript of 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
Birkhoff normal forms for betatronic motion
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 !!
Birkhoff normal forms for betatronic motion
Error plots. Poincaré section y=yc vy>0 H=Hc+10-5 Hc ~1.5
Birkhoff normal forms for betatronic motion
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
Birkhoff normal forms and betatronic motion
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
Birkhoff normal forms and betatronic motion
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
Birkhoff normal forms and betatronic motion
:
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
Birkhoff normal forms and betatronic motion
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)
Birkhoff normal forms and betatronic motion
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)
Birkhoff normal forms and betatronic motion
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
Birkhoff normal forms and betatronic motion
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
Birkhoff normal forms and betatronic motion
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
Birkhoff normal forms and betatronic motion
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)
Birkhoff normal forms and betatronic motion
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
Birkhoff normal forms and betatronic motion
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
Birkhoff normal forms and betatronic motion
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)
Birkhoff normal forms and betatronic motion
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
Birkhoff normal forms and betatronic motion
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
Birkhoff normal forms and betatronic motion
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
Birkhoff normal forms and betatronic motion
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
Birkhoff normal forms and betatronic motion
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 >
Birkhoff normal forms and betatronic motion
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
Birkhoff normal forms and betatronic motion
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
Birkhoff normal forms and betatronic motion
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.
.
Birkhoff normal forms and betatronic motion
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)
Birkhoff normal forms and betatronic motion
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
Birkhoff normal forms and betatronic motion
Power law growth of e(n) for Hénon map
Birkhoff normal forms and betatronic motion
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
Birkhoff normal forms and betatronic motion
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
Birkhoff normal forms and betatronic motion
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
Birkhoff normal forms and betatronic motion
LE FE
RE RE round off
Comparison of frequency map error and REM
Birkhoff normal forms and betatronic motion
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
Birkhoff normal forms and betatronic motion
y0=py0=0
y0=0.15
py0=0
x x
x x
px
px
px
px
Frequency error REM
Frequency error REM
CONCLUSIONS
Birkhoff normal forms and betatronic motion
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
Birkhoff normal forms and betatronic motion
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)
Birkhoff normal forms and betatronic motion