iterative determination of invariant tori for time ... - Semantic Scholar

SLAC-PUB-5796 1992 April T/A

ITERATIVE TORI

DETERMINATION

OF INVARIANT

FOR TIME-PERIODIC

WITH

TWO DEGREES

-

HAMILTONIAN OF FREEDOM*+

W. E. GABELLA

of Physics,

Department

$

University of Colorado, Boulder, Colorado 84309

and Stanford Linear Accelerator Center, Stanford University, Stanford, California

94309

R. D. RUTH AND R. L. WARNOCK

Stanford Linear Accelerator Center, Stanford University, Stanford,

California 94309

ABSTRACT We describe a nonperturbative Jacobi equation solutions

of a nonlinear

which yield accurate

numerical Hamiltonian

approximations

suited to the case in which the perturbation has a periodic important in time.

and not necessarily

in accelerator

system.

We find the time-periodic

to invariant

The method integrable

smooth dependence on the time.

equation

is approximated

of solutions

by its finite-dimensional

.(

. .

oscillations

of a particle

method.

in the presence of step

by a shooting

algorithm.

method is tested in soluble models, and finally applied to a non-integrable the transverse

system

This case is

with respect to angle variables, then solved by Newton’s

we enforce time-periodicity

is

is a periodic step function

To avoid Fourier analysis in time, which is not appropriate functions,

tori.

to the underlying

theory, where the perturbation

The Hamilton-Jacobi

Fourier projection

technique for solving the Hamilton-

The

example,

beam in a storage ring, in two degrees of

* Work supported by the Department of Energy, contracts DE-AC03-76SF00515 and DE-FG02-86ER40302. t Submitted to Physical Review A. $ Currently visiting Fermilab MS345, P.O.Box 500, Batavia, IL, 60510, from University of California, Los Angeles. .* Submitted

to Physical Review A,

freedom. “2 l/2

In view of the time dependence of the Hamiltonian, in which phenomena like Arnol’d

degrees of freedom”,

this is a case with Diffusion

can occur.

systems abound in applications

of classical

I. INTRODUCTION Examples of nonlinear and semi-classical

Hamiltonian

mechanics:

or electromagnetic

the problem of N bodies interacting

forces, the beam-beam

interaction

in storage rings with collid-

ing beams, and the control of magnetic field configurations devices, to name but a few. tures and not completely the0rem.l

Interesting

ied intensely,

Most of these problems

integrable

in the technical

examples of completely

but they are essentially

via gravitational

in plasma containment

are not soluble by quadra-

sense of the Liouville-Arnol’d

integrable

systems have been stud-

different from generic problems of nonlinear

mechanics.2 We are concerned with systems that may be viewed as perturbed systems.

If the perturbation

is sufficiently

nian satisfies a non-degeneracy

condition,

theorem may apply.3 The unperturbed that are invariant the trajectory The KAM independent ficiently

theorem

certain of these tori,

is imposed.

torus.

those that have rationally distorted

These are called KAM

of large measure, but they are interleaved -..

surfaces in phase space

and these tori foliate the space. That is,

frequencies, survive in a slightly

weak perturbation

(KAM)

any point in phase space lies on an invariant

asserts that

perturbed

Hamilto-

the Kolmogorov-Arnol’d-Moser

system has toroidal

under time evolution,

passing through

small, and the unperturbed

integrable

form when a suf-

tori; they form a set

by regions in which tori need not exist,

the so-called resonant regions of phase space corresponding .* 2

to rationally

dependent

unperturbed a KAM

frequencies.

Resonances exist in an arbitrarily

small neighborhood

torus, so that the tori no longer foliate phase space.

For systems of sufficiently

small phase space dimension,

ant tori has a direct implication

for stability

of the motion.

the existence of invariThe effective dimension

of phase space is D = 2d + 7, where d is the number of mechanical dom, and r = 0 for an autonomous periodically

on the time;

an invariant outside.

(we exclude non-periodic

torus divides

the space into two disjoint

of stability,

regions, an inside and an

if the inside is a bounded domain

In this event, an orbit

initially

can follow a “stochastic

close to an invariant

plane into disjoint torus (though

by Arnol’d

with D = 5, and similar phenomena are referred to broadly For the study of stability to invariant

of nonintegrable

is equivalent

not on one)

invariant

invariant

deviate

in an example

as Arnol’d

DiffusionP

systems, it is useful to compute

tori, even in cases with D > 4. In contrast

tori, a family of approximate

ues of an approximately

regions.

web” associated with resonances, and eventually

greatly from the torus. Such an effect was demonstrated

approximations

at hand.

tori have too few dimensions to separate regions of phase

space, just as a point does not divide a two-dimensional

tori, corresponding

to exact

to various val-

action, may foliate a region of phase space. Such

to a canonical transformation

such that the action is an approximate

-..

If D 5 4

time dependence).

a desirable region of phase space for the problem

If D > 4, the KAM

variation

depends

It is clear that an orbit beginning inside the torus must stay there forever.

representing

a family

degrees of free-

system, while r = 1 if the Hamiltonian

This amounts to a useful statement

invariant

of

invariant.

to new action-angle By studying

the relatively

in time of the new action, one can set bounds on the motion

.. 3

variables, weak

for a long

but finite time.6 This argument

is in the spirit of the Nekhoroshev

Theorem7,

and

torus is useful in giving information

on

proceeds in the same way for any D. In a less formal way, a nearly invariant the dominant variables,

resonances. When the torus is represented as a Fourier series in angle

the magnitude

the strength

of the Fourier coefficient

of excitation

the Fourier coefficients the transformation been identified

of a resonance in that mode.

with

respect to action,

mode measures

Also, the derivatives

which determine

as a sensitive indicator

transform,

D = 3. Generally

good approximate

is associated with the approach to strongly

as some measure of nonlinearity Perturbative

methods

speaking,

a difficulty

to compute

approximate

invariant

been to adapt and improve

classical perturbative

so as to carry the perturbation

methods

Even with

in interesting

the onset of strong instability. a larger region of validity, perturbation .*

techniques,

the power of computer

may be ineffective

algorithms

series to relatively

has been to invent non-perturbative

of computers.

in finding unstable

is. increased.

Since the advent of the computer,

tendency

have

to see what happens to good invariants

ployed over many decades.

lation,

of

of the onset of large-scale chaos, at least

for a class of models5 with invariants

of

the Jacobian

of angle variables in an associated canonical

regions, and for that reason it is informative

-..

in a particular

tori have been emone tendency for machine

high orders.

calcu-

Another

again taking advantage

implementations,

perturbative

parts of phase space, particularly

Appropriate

has

non-perturbative

methods

near

may have

and even present some advantages in cases for which

theory is adequate. 4

This paper describes a particular iterative

solution

nonperturbative

of the Hamilton-Jacobi

handle a Hamiltonian

with periodic

equation.

cal perturbation

theory, the technique

the underlying variables

Following

representation

The generator

solution

as a finite

Fourier

from the method first used in iterative

equation:

we avoid Fourier analysis in time.

and poorly

convergent.

enforce periodicity conditions

and D = 5.

of this trans-

equation,

provides an

series in angle coordinates.

solution of the Hamilton-Jacobi

Since the perturbations

periodicity

of the generator in time by a “shooting until periodicity

of present

of Fourier analysis, we method” in which initial

is achieved.

the procedure with examples from accelerator theory with D = 3

The examples

deal with

oscillations

of the beam, the so-called betatron

tion arises from fields of sextupole dependence

The generator

of time, a Fourier analysis would be inefficient

Lacking the automatic

are varied systematically

We illustrate

direction

functions

variables of

torus.

Departing

interest are discontinuous

of canoni-

to new action-angle

of the Hamilton-Jacobi

of a nearly invariant

is represented

for the possibility

the pattern

system, and finds a transformation

an approximate

is designed to

makes use of the action-angle

so that the new action is nearly constant.

formation, explicit

integrable

The method

time dependence, allowing

that the time dependence may not be smooth.

based upon

technique:‘g

of the oscillation

of particles

oscillations.

transverse

Nonlinearity

magnets, which are introduced

frequencies

on the longitudinal

to the

of the moto counteract

momentum

of the

beam. In these examples the method has a large domain of convergence, including regions of strong nonlinearity. this is checked by following ..T

It produces close approximations single orbits, originating 5

to invariant

on the tori, through

tori;

accurate

numerical

integration

A comparison of our approach;

of the equations of motion.

to related work requires some awareness of technical for that reason we defer comparisons

features

to the final section of the

paper. In Section II, we derive the Fourier projection in a form suitable for the shooting algorithm. a fixed-point fixed-point

are briefly

on two soluble examples. example,

the problem

strong nonlinearities. We give conclusions Section VI. Appendix for sufficiently

discussed.

numerical

Calculations

oscillations

is applied

in a context

based on the contraction

In this section we derive the projected

mapping

We write the Hamiltonian describe the time-periodic the entire, nonlinear nian be independent .*

Hamiltonian

equation

a perturbed

for charged

of angles is the Hamilton-Jacobi

how-

integrable system.

to the action-angle

The equation requiring

6

OF

variables of the integrable

canonical transformation

Hamiltonian.

theorem.

The discussion is applicable,

representing

in the action-angle

algorithm

EQUATION

Hamilton-Jacobi

in a transverse magnetic field.

ever, to any time-periodic

with

of related work in

I contains a proof of convergence of the shooting

THE HAMILTON-JACOBI

motion

to a nontrivial

of tori for both D = 3 and D = 5 are discussed.

II. FINITE FOURIER PROJECTION

particle

is tested

in a model of an accelerator

and try to place the method

weak nonlinearities,

as

methods used to solve the

In Section IV, the technique

In Section V, the method

of betatron

equation

The shooting method is formulated

problem in Section III. Iterative, problem

of the Hamilton-Jacobi

part and

variables of

that the new Hamilto-

equation, a partial differential

equation

for the generating

function

of the transformation.

We project

finite Fourier basis in the angle variable and find a set of equations conditions

for the Fourier amplitudes

For a single charged particle of a cyclic (“circular”) transverse motion (linear)

Hamiltonian

as

in the static,

in the action-angle

transverse

magnetic describing

field the

variables of the unperturbed

= qs)*I+v(aq,s)

variable s is the independent

resents the particle’s

azimuthal

along a cjosed reference orbit d-component

moving

10

H(@,I,s)

The time-like

and boundary

of the generator.

accelerator or storage ring, the Hamiltonian

can be written

it onto a

(1)

.

variable of Hamilton’s

location in the accelerator, of circumference

C.

equations; it rep-

measured by arc length

Boldface

characters

vectors, where d = 1 or d = 2 in all of our numerical

indicate

examples.

For

d = 2 each component of any vector corresponds to one of the directions transverse to the reference orbit.

The unperturbed

Hamiltonian,

where /?i is th e C ourant-Snyder

of pi(s) = l/n;(s),

the linear aspects of the applied magnetic

perturbation

V are periodic in s with period C.

space coordinates

(1) is derived from an initial

(xi,pi),

. I, is defined in terms

beta function

izes entirely

The Hamiltonian

a(s)

which character-

fie1ds.r’ Both /3i and the

formulation

in terms of phase-

where xi is the transverse displacement

of the particle

from the reference orbit,

and pi = dxi/ds.

unperturbed

variables (Qi, Ii) by a canonical transformation,

angle-action

.* 7

These coordinates

are related to the

Xi

=

JZ@j

COS Qi

Pi = -j/WE@ = -dpi(s)/ds.

where 2ai(s)

,

(Sinai

+ cYi(S)COSQi)

I n t erms of (xi,pi),

,

i =

1,2

,

the unperturbed

Hamiltonian

takes the form

(3) where the functions

Ki(s),

periodic with period C, describe linear focusing forces

from quadrupole magnets. The definition

of 1Ci and the determination

of pi from

I(; is explained in Ref. 10. When V = 0 the motion is integrable, since it follows immediately

from Hamil-

ton’s equations that Ii is constant, and that @ i(s) = @ i(O) + J” da/Pi(a).

V = 0 it is possible to make a further canonical transformation

With

to variables that

represent simple harmonic motion, but it is not convenient to do so in the following work. Synchrotron Synchrotron

oscillations

oscillations

and synchrotron radiation

(oscillations

of the particle are ignored.

in energy associated with the electric r.f. ac-

celerating field) occur on a time scale much longer than the betatron oscillation

time. They can have an important

to a more complicated radiation

(transverse)

effect on long-term stability,

but lead

problem than the one we wish to study here. Synchrotron

tends to damp the betatron

oscillations

in electron accelerators, and in

fact improves stability.

. .

.*

The form of V for a string of normal sextupole magnets distributed 8

about the

circumference

of the ring is

V(x1,

x2,4

=

5

fn(s)$x:

-

3x14)

(4)

*

n=l

The function it is unity.

is zero everywhere

The sextupole

(e/cpo) d2B2/i3xf derivative

fn(s)

strength

except inside the n-th magnet

S,, has units of mV3, and is defined as S =

in cgs units, where po is the reference momentum,

of the vertical

where

and the second

magnetic field B2 is evaluated at x1 = x2 = 0.

We now consider the full nonlinear problem and seek a canonical transformation to new action-angle

variables,12

(@,I) -+ PII, J> , such that the new action J is invariant. mation

The generating

function

of the transfor-

is denoted as

F2(Q,J,s)

= +J+G(@,J,s)

where the first term represents the identity formation

(5)

,

(6)

map. The equations defining the trans-

are

Q=@+GJ(Q,J,s) I=

J+G*(@,J,s)

Hl(J,~,s)=H(Q,J+G~p,s)+G, 9

,

(7)

,

(8) .

(9)

The subscripts

represent partial

Hamilton-Jacobi a function

differentiation,

for instance G+ = {dG/dQi}.

equation expresses the requirement

that the new Hamiltonian

If G satisfies this partial of Hamilton’s

differential

equation, then J is invariant,

G of Eqn. (10) that is periodic

a (d.+ 1)-dimensional

invariant

(Q(s),I(s)),

torus, through

tori.

The value of the constant

Since only approximate

computations

of

Eqn. (8). Since J will be constant

solutions

which means that I(@, s) is periodic

vector J serves to distinguish

accuracy, as verified over a finite interval Because of the periodicity

periodicity

our

only to a certain

of s.

in the angles, it is natural

Fourier basis in @. Accordingly,

different

of (10) can be achieved numerically,

will lead to tori and actions J that are invariant

thus guaranteeing

representation

all points of the orbit lie on the surface I = I(@,s; J)

specified by Eqn. (8). Th e surface is toroidal, in @ and-s.

as a direct result

in @ with period 27r, and

in s with period C. Such a solution provides an explicit

along an orbit

(10)

.

equations in the new variables.

We seek a solution periodic

be

of J and s alone:

WJ,s)=W-(J+G+)+V(@,J+G+,S)+G, .-

The

to study Eqn. (10) in a

we expand the generator in a finite Fourier series, in a:

G(@, J,S) = g(0, J,s) +

c

s(m,

J,4eim’*

7

(11)

mEMUW 2%

s(m, J+> =

J

-im’oG(@,

0

10

J, s)

.

(12)

In Eqn.(ll),

the mode index vector m = (ml, ma,. . . , md) runs over a finite

set

denoted as M U M, which does not include m = (0, 0, . . . , 0). The set of indices for the independent,

complex Fourier modes is M. The set

i’@ is simply related to M and represents Fourier modes that are not independent of those with indices in M. vectors (mr,ma)

In the case d = 2, the set M is the set of all integer

such that ml E [0, Ml], and 7732E [-M2, M2] when ml > 0, while

m;! E [l, M2] when ml = 0. The set i@ is just the negative of M: &f = {(ml,

-m2)

b-m,

E M}.

Since G is real, the Fourier coefficients with indices in &l are

related to those with indices in M, g(-ml, For ease of notation,

Written

using indices in the union of

the corresponding

amplitudes

the set M, the summation

using just

c mEM 2 Re [g(m, .7, s)eim”] A projection

-m2) = g*(ml, m2).

Fourier series are written

the two sets M U A?, even though pendent.

m2) :

in Eqn. (11) becomes

; this formula is used in numerical

of the Hamilton-Jacobi

equation

are not inde-

computations.

onto the above Fourier

basis

gives

&g(m,J,s)+im~Sl(s)g(m,J,s)+~(m,J,s;g)=0

7

Here v(m, J, s; g) is the Fourier transform of the Fourier coefficients

of the perturbation E M}

{g(m, J,s),m

through

me

Me

(13)

and is a functional

the action transformation

in Eqn.(8):

+-n, J, w) = Ga(tD, J,s)

=

J

2* d@ -e-‘m’*V(@,

J + G+(@, J, s), s)

o Wd c

img(m,

mEMU@

11

J,s)eim”

.

,

(14) (15)

Recall that the set M U &l does not contain the mode m = 0, so that Eqn. (13) does not involve m = 0, and is independent be solved for g(m, J,s) completely

with m # 0 subject to the s-periodicity

determines

Projection

of the choice of HI (J, s). Hence, it can condition.

This

Ga when J is specified.

of the Hamilton-Jacobi

equation onto the m = 0 mode gives

.Hl(J,

s) = a(s) . J + &g(O, J, s) + v(0, J, s; g)

Since ~(0, J, s; g) is determined G*, we determine

HI by an arbitrary,

It is convenient definitions

s-periodic

(16)

of Eqn. (13) gives

choice of g(0, J,s).

to choose g(0, J, s) to be zero. Different

choices give different

of the new angle !V, but the differences are innocuous, in that the change

of Xl? during number,

by G+ alone, and the solution

.

a period of s is always the same.

The perturbed

i’ gives the change in !I! in one turn normalized

27rv’. From Hamilton’s

equation for the evolution

c

1 Y’ = g

tune, or winding

by 27r: Q(C)

- q(O) =

of e’, the tune is

c

J

dJH$,s)ds

= Y+ &

, JaJ@,J,w)ds (17)

0

0

where Y is the tune of the unperturbed

motion,

c

2wu =

J

Ct(a)da

.

(18)

0

Owing to periodicity

.

in s, the term &g in HI does not contribute

(17) , and the tune is invariant .-

to the integral

to changes in the choice of g(0, J, s). 12

The linear term in Eqn. (13) can be eliminated exp (im . X(s)),

by using the integrating

factor

where

(19)

S

X(s) =

J

St(a)da

0

is the linear phase advance, 9(s) - a(O). Eqn. (13) becomes -

&h(m,

J, s) = -eim’X(s)

, m EM ,

~(m,J,wdh))

(20)

where h(m, J, s) = eim’x(s)g(m, The.periodicity

of g in s implies the boundary

J, s)

condition

h(m, J, C) = ezaim”h(m,

J, 0)

.

(21)

on the new variable h:

.

(22)

Notice that h has the nice property .of being constant over any interval of s in which the perturbation

vanishes, i.e., any region in which magnetic

fields are linear or

of a solution g(m, J, s) of (13) implies periodicity

of its derivatives

zero. Periodicity

with respect to s, at least at generic points where V and St are sufficiently Suppose, for instance, that a(s) hood of SO, and that

V is continuous

asg(m, J, s) is continuous tiating

as a function

near so, and periodic

(13), and assuming more smoothness

conclusions

‘..__

and V(@, J, s ) are continuous

about higher derivatives.

of J.

13

in s in a neighbor-

Then (13) shows that

in s with period C. By differen-

of V and a, one can make similar

In the accelerator

be lacking only at sharp edges of magnets. .*

smooth.

problem,

smoothness will

The differential summarized

relations

and boundary

conditions

for the h coefficients

J, s; g(h))

,

are

below:

&h(m, v(m,

J,s) = -e im.x(s)v(m,

J,s;g(h))

= T$$emirn”v(fft,

m E M

J + G+,s)

,

,

(23) (24)

0

.Ga=

imh(m,

c

J, s)e

im.(O-X(s))

7

(25)

mEMUM

h(m, J, C) = e2*im*vh(m,

J, 0)

.

P-3

In the next section, we describe a method to find solutions of Eqn. (23) consistent with Eqn. (26).

III. SOLUTION BY THE SHOOTING METHOD In this section, we discuss the-solution plitudes

h(m, J, s). We formulate

a nonlinear

for the Fourier am-

the problem as one of finding the fixed point of

map. We close this section with a discussion of the numerical

for evaluating

methods

the map and for finding its fixed point.

Eqns. (23)-(25)

define th e evolution

the final values h(m, J, s = C). initial

of Eqns. (23)-(26)

condition

of the initial

The integration

conditions

h(m, J,s = 0) to

of Eqn. (23) from an arbitrary

h(0) t o a final value h(C) defines the map 17: h(C) - h(0) = U(h(0))

.

(27)

Here and elsewhere we suppress reference to m and J, and write h(s) for the vector

. .

with .*

components

h(m, J,s).

The boundary 14

condition

(26) is not satisfied for an

arbitrary

h(0). I n t erms of the evolution map U, the boundary

condition

demands

that h(0) satisfy the equation

h(m,

J, 0) = e2*imtv_ 1u(m, J, h(O)) *

(28)

In other words, we seek a fixed point h(0) of the map A ,

h(O)= A(W))

(29)

,

where

(30)

A(m, J, h(O))= e2rimtu_ 1u(m7 J?h(o)) An essential property

of this formulation

is that A is proportional

to the per-

V. This makes it feasible to solve the fixed point problem by

turbation

strength

iteration,

when V is sufficiently

from zero by an appropriate

small, and the divisor e2sim’Y - 1 is bounded away

choice of the mode set M and the unperturbed

Y. The change of variable from g to h was needed to create an operator tional -to V; the corresponding

step in quantum

tune

propor-

mechanics is to use the interaction

picture. To solve Eqn. (28), we use standard of nonlinear

maps.

13,14 W

techniques from the study of fixed points

e d escribe these methods in detail below.

15

A. Simple Iteration The most obvious method to find the fixed point of Eqn. (28) is referred to The iteration proceeds in the following manner, h’+l(O) =

here as simple iteration.

A(h’(0)) , with the superscript i labeling different iterates. A proof of convergence to a unique fixed point is described in Appendix .-

Our initial

I.

guess, ho(O), will be the approximate

lowest order in perturbation

solution of Eqns. (23)-(26) to

theory. This is obtained by putting

G+ equal to zero

in the right hand side of Eqn. (24). Then Eqn. (24) can be evaluated explicitly using Eqns. (2) and (4) . The resulting initial

iterate is

C

h’(m,

’J

eim’X(s) o(m, J, s) ds

J, 0) = 2~~~~v, .

,

(31)

0

with +-G&s)

= $5

.f72(s)S,(S(mL

(3,0))i(J1P1(S))3’2

n=l + S(m

-

m E M.

with

(qm

-

(ho))

-

(&MA(s))~‘~

W))

The S(m - (j,k))

m = (j, Ic) and 0 otherwise,

+ qm

-

-

(JlPl(4)“2J2B2(4)

(1, -2)))~(J,p,(s))‘/2ma(s)}

are Kronecker

and the summation

deltas with

(32)

the value 1 when

is carried out over N sextupole

magnets of strength Sn; see Eqn. (4). Notice, at s = 0 the h and g coefficients are equal. The notorious

problem of small divisors near resonances can be seen in the

leading factor in Eqn. (30). Th e d enominator

(...

is an integer. The iterative .*

vanishes when m + u = p, where p

procedure cannot succeed unless the mode set M and 16

the unperturbed

tune u are chosen so that the denominator

smaller the minimum

value of the denominator,

cannot provide arbitrary

by expanding .-

-

accuracy.

the smaller the perturbation

If one attempts

the set of Fourier modes, the minimum

there are vectors u, with rational see, this deterioration

components

The

V

It follows that for a fixed V this

must be to secure convergence of the iteration. method

is nonvanishing.

to increase accuracy

divisor tends to zero, since

close to any u whatever.

As we shall

of convergence is clearly observed in computations.

B. Newton Iteration To expand the domain of convergence, we turn to Newton’s method. divisors have the same impact obvious.

Nevertheless,

The small

in this method, even though their role is a bit less

for a given M and u, the Newton method succeeds for much

larger V than can be handled in simple iteration. The desired fixed point is the solution to the complex-valued

F(h(0))

In Newton’s

= A@(O)) - h(O) = 0

method we make a first-order

vector equation,

.

(33)

Taylor expansion of F around a given

iterate to define the next iterate as a solution of linear equations. The derivative ory:

F is not an analytic

real equations, defined. -....

of F is not well-defined

and L(O) = .*

of h(0).

function

of F with

then the derivative

We use a compact (Reh(O),Im

notation

h(0))T.

in the sense of complex function Eqn. (33) should be written

as two

respect to Re h and Imh

is well-

for the real equations,

&’ = (Re F, Im F)*,

Then Eqn. (33) takes the form #(i(O)) 17

the-

= 0.

To state equations

in components,

we write i(m)

for i(m,

J,O).

Then Newton’s

method is defined by the equations

P(m,ki)

B( m, n, Xi) . [L’+‘(n)

+ C

- i’(n)]

= 0

,

(34

NM

.

.-

with B(m,n,?1)

=

aReF(m,a)/dReh(n)

aReF(m,i)/dImh(n)

aImF(m,?l)/dReh(n)

aImF(m,k)/dImh(n)

This system of equations method

(Gaussian

numerically

elimination).

by divided

by a small amount, matrix

is solved for the new iterate The Jacobian

ki+‘,

given &‘, by a direct

fi of the map is approximated

differences.

Each component

and fl computed

at perturbed

of i is perturbed and unperturbed

dReh(n) dF(m,X) dImh(n)

F( m, h + (E(n), O)T6h) - F(m, Sh =

points.

The

F( m, h + (0, ie(n))T6h)

h) 9

- F(m,

,

6h

=

in the direction

corresponding

to mode n.

In practice,

The larger domain of convergence of Newton’s putational

expense that

due to the Jacobian

becomes significant

evaluation.

With

(36)

h)

where-6h is a small real number and e(n) = {S(p - n), p E M}

method

for d 2 2.

is the unit vector

we take 6h = (10m6 -

is achieved at a comThe expense is mainly

d = 2 the total number of independent

modes in the set M is 2M1 M2 + Ml + M2. have Ml .*

separately

elements of the Jacobian are then found as

bF(m,k)

(...

* (35) >

= M2 = 15, hence 480 independent 18

In a typical

calculation

we might

modes and 960 map evaluations

to

approximate

For a general d-dimensional

the Jacobian.

of independent

components

more than 2d-1Ml M2 . . . Md. The calcula-

is a little

tion of the Jacobian quickly

mode vector the number

becomes untenable as the mode set or the number of

degrees of freedom is increased. In contrast,

the simple iteration

requires only one

of the map A(h) at each iteration.

evaluation

We employ two methods to reduce the time for the Jacobian calculation.

.-

Broyden’s update method is used to approximate many modes within

the Jacobian.

First,

Second, we discard

the mode set M.

C. Newton-Broyden Iteration For the Newton-Broyden only once and is afterward

iteration,

at the i-th iteration

iteration

is

=

pi

+

[j?(ii+l)

then the update of the Jacobian

_

jyii) (ii+1

The h are treated

as column

The costly computation

vectors,

-

fji

_

ii)T

(ii+1 . (ii+1

_

hi)] _

,...

conditions .*

_ ii)T (37)

&,i)

and the iT as corresponding

row vectors. iteration

P(L’+‘).

The domain of convergence of the Newton-Broyden

In practice

. (Li+l

at the (; + 1)-th

(36) is used only to find fro; each subsequent

requires only one new map evaluation,

computation

fully

updated using the Broyden algorithm. 15-17 If bi is the

Jacobian

* i+l P

the Jacobian of the map is calculated

iteration

is still large while

times are much more reasonable than those of a full Newton iteration. it converges for strong effective nonlinearities, close to the dynamic

in particular,

aperture in accelerator problems. 19

for initial

A more daring and still more economical If A were zero in Eqn.

(33), the Jacobian

(37), we were surprised

to find that the Broyden

of convergence substantially to the calculation .-

-

approximation

somewhat

would be

-1.

can be attempted. Putting

fro =

updates still provided

larger than that of simple iteration.

-1 in

a region

In comparison

with fro from divided differences, the region of convergence was

smaller, and more iterations

were required for adequate convergence.

D. Mode Selection To achieve a certain maximum

accuracy,

mode numbers M;.

are usually

many amplitudes

one has to choose a minimum

value for the

On the other hand, for a given choice of M; there of modes with

We apply a simple technique to identify

Im;] < Mi that are quite negligible.

and eliminate

the negligible

modes within

the set M. We carry out one simple iteration A above. We then compute m the maximum small fraction computations. amplitudes

using the full mode set M, as in Section

the evolution

over s of 1h( m, J,s)l.

in s of the resulting When the latter

h(O), and for each

is less than some fixed

of max, maxs Ih(n, J, s)I, we throw away mode m in all subsequent In our examples with-d

are retained.

= 2, no more than 130 to 200 complex

Since the time to approximate

down with the square of the number of amplitudes,

20

the initial

Jacobian goes

a great deal of time is saved.

E. Numerical Integration The nonlinear map U of Eqn. (28) must be evaluated by numerical integration of Eqn. (23). W e use a fourth order Runge-Kutta changes only over the support

of the nonlinear

example the numerical integration .-

algorithm.

Recall that h(m, J, s)

V.

perturbation

Thus, in our

need be performed only over the extent of the

sextupole magnets. The final value h(m, J, C) o bt ained by the integration in Eqn. (27) to calculate the map U.

After the iteration

is used

converges to the fixed

point h(m, J,O), th e coefficients can be evolved in s using the same fourth order Runge-Kutta

algorithm.

It has been observed that the number of Runge-Kutta nonlinear linearity

element must be increased to maintain is increased by going to large amplitudes

modes must increase. The number of integration

(d + 1)-dimensional sent variation

integration

steps per

accuracy as the effective nonJ. Similarly,

the number of @

steps in the s-direction

along the

torus is analogous to the number of Fourier modes to repre-

along the @ direction.

As the nonlinearity

increases, the invariant

torus becomes more distorted in both the s and the @ directions,

so that both the

number of steps and the number of modes must go up. Working

at large amplitudes

that the iteration integration intermediate

can reach a fixed point only for a relatively

steps. To perform the calculation

point is then used as the initial

with a larger number of integration .*

small number of

at a large number, the result from an

number of steps must be used as the starting

found at the intermediate

‘(.__

and using the above stated guess for ho, we find

point.

The fixed point

iterate for a calculation

steps. This has the added advantage of cal21

culating

the initial

saving computing

Jacobian with a smaller number of integration

steps and thus

time.

IV. TESTS OF THE METHOD We test the technique introduced .-

that can be solved analytically. harmonic merical

oscillators, solution

and numerical is perturbed

in the previous section on two model problems

For the case of two autonomous,

we compare the exact invariant

of the Hamilton-Jacobi

equation.

linearly

surface with

coupled

that from nu-

We also compare the analytic

tune shifts-for a linear example in which uncoupled betatron by a quadrupole

motion

term.

A. Linearly coupled harmonic oscillators For two linearly

coupled harmonic

oscillators,

the Hamiltonian

can be written

as

NOW the Pi are constants,

but they are defined in analogy to the pi(s) of Eqn. (1).

The x and pz are column vectors of coordinates respectively. mation

Since Eqn. (38) is independent

producing

tion diagonalizing

an uncoupled the matrix

For the numerical . . . .

F&s2 .*

Hamiltonian.

(21, ~2)~ and momenta

(~1 ,PZ)~,

of s there exists a canonical transforThis is equivalent

to a transforma-

KC.

solution of the Hamilton-Jacobi

serves as the perturbation.

Written 22

equation, the coupling term

in the action-angle

variables

of the

unperturbed

Hamiltonian,

H(@,I)

the full Hamiltonian

=~o~I+21’~~cos~1cos~2

where 520 = (l/pl,l/&) Let (u,pU) .-

are related

variables

appears uncoupled.

by th e orthogonal

(x,pZ)

transformation

Kc; we let Kd = STKcS, where Kd = diag(l//?f,,

I(, through

@iu for the uncoupled the above definition

(39)

pi = -dmsinQi.

be variables in which the Hamiltonian

new beta parameters the matrix

,

and xi = dmcos

’

(42)

with

The parameters . . . .

are chosen so that the right hand side of Eqn. (42) is positive.

@iu are defined to be positive. .s 23

The

We take the following to compare introduce

with

steps to generate points on an invariant

the numerical

the action-angle

solution

of the Hamilton-Jacobi

surface I(@) equation.

variables (IU, au) of th e uncoupled motion.

To generate

points on the torus, we hold I, fixed and allow ep, to vary on a uniform grid. Working

through

We

40 by 40

the transformations

-

PJU)

I-b (u, Pu) b-b (X,Pz)

,

I-+ (@,I)

(43)

we finally obtain points I(@) with invariant

action I, to compare with the numerical

solution

The action-angle

of the Hamilton-Jacobi

equation.

transformation

is

(44) As a measure of the difference between.the analytic

solution

Hamilton-Jacobi

IA, we compute the normalized

solution

IHJ and the

sum of deviations

(45)

The summation TABLE

is over the 40 by 40 grid in the QU space.

I gives the values of 6; found for several coupling strengths r with I, =

(10B5 m, 10e5 m),

,& = .50336 m,

1 m. The mode spectrum complex

.(...

modes.

the method .*

p2 = 1.29322 m, and a ring circumference

was truncated

of

to Ml = M2 = 7, giving 112 independent

A subset of significant

modes within

described in the previous section. 24

this set was selected by

The analytic

and Hamilton-Jacobi

TABLE I. Comparison of surfaces from the Hamilton-Jacobi equation with analytic surfaces for the linear coupling model with different coupling strengths I?. The parameters 6i, defined as in Eqn.(&), measure the discrepancy. C = l.Om,

r

Ml = M2 = 7

I, = 10B5m,

(mB2)

61

modes

62

selected

invariant

0.04

1.406.10-4

2.881.10-4

5

0.10

5.799.10-4

6.606.10-4

7

0.50

5.430.10-5

6.186.10-5

42

0.65

3.782.10-4

11.480.10-4

38

solutions

agree very well. Notice that at I’ = 0.50 mm2 more modes are

kept and a significant

increase in accuracy over the other cases is achieved.

FIGS. 1 and 2 show the two components of the invariant 11, and 1%) at any s (time-independent displayed

as functions

with 0 at the origin.

1 + Gai(‘,I,)/‘itt,, surfaces.

problem)

with

i = 1,2.

Notice that the departure

The constants

we are plotting

I, serve to distinguish

This nonlinearity

is f61%,

Ii/Ii,

scale

=

different In

and in FIG. 2 showing the

displayed by the coupled variables 1; is of

course not an essential feature of this basically genuine nonlinearity

by 27r, and on a vertical

function,

by

They are

from a plane surface is quite pronounced.

FIG. 1 showing the 11 surface, the distortion

12 surface, it is f55%.

and for I’ = 0.65 m-‘.

of the angles 9, normalized In terms of the generating

surface, normalized

linear problem.

In problems with

one must be careful to separate any effects of linear coupling,

which can be confused with the real effects of interest in plots like those of Figures 1 and 2.

25

B. Quadrupole perturbation For a quadrupole perturbation,

it is interesting to compare the tune shift found

by matrix

methods with that from the numerical solution of the Hamilton-Jacobi

equation.

The perturbation

due to a quadrupole is V = K(s)(z2

- y2)/2, where

K(s) is constant within the magnet and zero elsewhere. In action-angle variables, it -

takes the form V = K(s)(p 1I 1 cos2 @r - ,&I2 cos2 Qpz). The strength I( is measured in units of rnm2. From Eqns. (16) and (17) we see that the tune shift in the Hamilton-Jacobi formalism

is

(46) Given G+, the integral is calculated with a Simpson’s rule for the s integration with

a fast Fourier transform

for the @ integration.

The derivative

and

is estimated

with a simple divided difference. According to lowest order perturbation

theory, the tune shift for a weak quadru-

pole magnet with strength I< is

(47) This can be found using Eqn. (46) and setting G+ to zero. An exact analytic result for the tune shift can be found using matrix methods. Consider two magnetic lattices, one where the first element is a drift ,...

responds to free particle .^

motion)

(i.e., it cor-

and another where the drift is replaced by the 26

quadrupole

perturbation.

of the two lattices the matrices.

can be compared,

If the full-turn

is T, then the full-turn quadrupole

The 4 x 4 matrix

representations

of the full-turn

and the tunes extracted

maps

using the traces of

map for the lattice that begins with the drift element

map for the same ring, but with the drift replaced by the is T’ = T.T~~*TQ, where TD is the matrix

perturbation,

for the drift and TQ is the matrix

representation

for the quadrupole

representation perturbation:

TD =

cosfid TQ =

i

drift,

cosad

-flsinfid

These matrices

0

0

0

0

matrix

0

0

0

cash A/?? d flsinh

d

fl

d

--& sinh fi

act on the phase space vector (2r,~l,x2,~2)~.

If there is no coupling

obtained

0

d

cash fi

I (48)

and the quadrupole,

block diagonal;

d

-& sin a

is d and the quadrupole

strength

between x1 and x2 motions,

the formula

T(‘) is represented in a standard notation T(‘) =

COS

27TUi+ Cri sin 27rVi

-(I again with 2oi = -dpi/ds.

cos 27rui = cos fi

+ Cyf) Sin27rUifPi Th e analytic

as

is I

’

(49)

forms for the tune shifts are found to be

d cos 27ry + d (’ lpp”

sin 2rul

> .*

27

cm 27ru1 + ( fi

The lattice

functions

ai(

d al - A/??& -

pi(s) are evaluated at the beginning

of the drift.

We studied several cases with a single quadrupole of strength varying from low5 used for T is that given in the next section in TABLE

to 0.3 mS2. The lattice with

the sextupoles

is either the drift

removed,

i.e., replaced by drifts.

or the quadrupole

perturbation

strength,

are compared

and the (x) equation. wy2/

marks

C(wHJ)2]1/2,

is 9.97 + 10S6 for Aul/I a t s-dependence

s = 0 and are plotted

map as defined in Section III.

of the Hamilton-Jacobi

In TABLE curve with

again using an explicit

conditions

turns,

thus defining ITR(s

PJ(qys

= T&C)),we

each

integration

by comparing

of Hamilton’s

symplectic

equations

surface, and each is tracked for 1000

the corresponding

on the tra-

values on the computed

surface,

define an error parameter

61= max gi!!,” IIyJ(@lTR(S= nc)) - pcs = nC>I {@.101 -g&ylI,“qq”(s = nc,) - Jll * The summation

is over the number of turns the trajectory

in this case), and the maximum uniformly

distributed.

Notice that 61 is normalized

(51)

was followed (1000 turns

is taken over a set of 16 initial

in the departure from linear motion.

angles which are

so that it measures the error

Instead, one could normalize

by replacing the

denominator

in (51) by J1, so as to measure the error in 11 itself; this would give

considerably

smaller values.

Several other parameters variant

are given in TABLE

III.

The distortion

surface from a plane gives a measure of the strength

similar quantity ._.

the

integrator. 2o Sixteen

To compare the values of action

a trajectory. with

61 is estimated

fourth-order

are chosen on the invariant

= &),

with the nonlinear

Several items characterizing

III, the accuracy of the solution

initial

of @1/27r. The

equation are given in Table III.

the result of accurate numerical

(“tracking”),

jectory,

as a function

can be found by evolving the Fourier coefficients

time evolution solution

These curves are sections of the 2-torus

of the in-

of the nonlinearity;

is called “smear” in accelerator physics. 21 We define the distortion

to be the average of the maximum .*

excursion 32

above the mean and the maximum

a

excursion below the mean, divided by the mean, Jr. A characteristic of the trajectory nonlinear

displacement

is the value of x1 at ipr = 0,s = 0: x10 = d-.

The

tune shift as defined in the previous section, and the CPU time to find

the fixed point on the IBM 3090 Model 200 E, are also shown.

TABLE III. Parameters for numerical solutions of the Hamilton-Jacobi equation with d = 1. The corresponding approximate invariant curves are given in FIG. 6.

.-

case

tune

cpu time

shift

IBM 3090

Au

(4

constant

mode

tracking

initial

action

set

comparison

offset

A(m)

Ml

2.10-7

15

1.51 * 1o-5

2.1

f2.8

-1.068.

1O-4

2.10-s

31

1.04 * 10-4

7.5

f9.0

-1.073

* 10-3

19

2. 1O-5

63

3.67 - 1O-3

21.4

f31.6

-1.220

* 10-2

28

61

As the amplitude

xlo(mm)

%

5

Jr increases, a larger mode set (all modes in the set are

being selected) is required to maintain approximately

distortion

linear with

amplitude,

even moderate

accuracy.

The tune shift is

as is expected for sextupole

magnets.

The

initial- offset x10 for case (c) comes close to the value 22.5 mm given in the ALS Conceptual dynamic

Design Report lg for the maximum

aperture

study, without

._.

of the ideal lattice

x1 coordinate

(i.e., the configuration

errors).

..

33

of the short-term of magnets that we

B. Two-Dimensional Example We present several results for the full two-dimensional discuss the tune shifts found from a family of approximate they are almost linear in the constant actions.

sextupole problem.

We

solutions and show that

We also present two solutions in

detail: one for small J and one for large J. In FIGS. 7 and 8, the nonlinear solutions,

are plotted

tune shifts,

found from several numerical

in a contour plot as functions

of the constant

actions Jr

and J2. The (x) marks points where numerical solutions of the Hamilton-Jacobi equation were found.

The tune shifts were fitted to a global, cubic polynomial

in

the constant actions, and this was used to make the contour plots. The discrepancy between the fit and the data, measured by the root mean square of the deviation normalized

by the data, was 2.61 . low3 for Av~ and 2.27 . 10S3 for AZQ. From

the figures and the fit, it is clear that the tune shifts are nearly linear functions of the actions.

This is what is expected for sextupole nonlinearities

perturbation

theory.

The solutions

from low order

found for FIGS. 7 and 8 used the modest mode set of Ml

=

it42 = 7. Of the 112 modes only the 30 largest were selected to do the calculation. This kept the computation

time for the solution at the largest value of the action

J = (6 . 10S6m, 6 . 10S6m) to 271 seconds on the IBM short term tracking,

3090. Comparison

with

as described above, gives Sr 5 0.31 and S2 < 0.56, and these

large values only for the largest J. The majority

of the solutions in the figures give

Si 5 10m2. Th e t wo -d’lmensional 6i are defined in analogy to the one-dimensional case. In the two-dimensional .

definition,

the initial angles (@lo, @20) are distributed

.. 34

evenly in a 4 by 4 grid in the @ plane. In FIGS. 9-12, we give three-dimensional

A(@,J, O>/Jl and k@, J, 0)/J 2

plots of invariant

as functions

2. lo-’ m and J1 = 52 = 4. 10m6 m, respectively.

of (@p1/27r,@2/27r) for J1 = J2 = As expected, the distortion

a planar surface is greater for the case with larger constant .-

action space of FIG. 5, these two solutions TABLE

action.

from

In the initial

correspond to points A and B.

IV gives some relevant parameters

for the two-dimensional

shown in FIGS. 9-12, as well as two other solutions, invariant

surfaces, showing

surfaces we do not show. The corresponding

solutions

indicated

as C and D, whose

initial

actions for C and D

are given in FIG. 5. All solutions have Mr = Ms. The number of modes actually used in the calculation

is shown, along with the total number of independent

in the set from which they were selected. The initial integration

number and final number of

steps per sextupole are given under NRI( (number of full Runge-Kutta

steps, each requiring equation).

modes

four evaluations

of the right-hand

side of the differential

The parameters giving the comparison to short term tracking

as 6;. The distortion as the.average

from a planar surface is reported for each case; it is defined

of the excursions above and below unity of the ratio Ii/J;.

we give the total

computation

program diagnostics,

are shown

time, including

Finally

both the surface calculation

and

on the SLAC IBM 3090 Model 200 E ; the time is dominated

by the surface calculation. We see from TABLE

IV that for larger actions, and greater distortions,

a larger

mode set must be used and more modes must be chosen to represent the surface accurately.

Larger actions also require more integration

to get numerical ..

convergence to a fixed point. 35

steps per nonlinear element

This leads to a big increase in the

TABLE IV. Parameters for representative numerical solutions in two-dimensions including those in FIGS. 9-12. These correspond to the initial conditions marked in FIG. 5. case -. .-

Jl,Jz

s+odes

/ NRK /

( 10S6m)

Mi I

I

distortion

tracking

I

61

cpu time

I1

62

A

0.2, 0.2

15

501480

2/10

2.66. 1O-4

4.37 - 10-4

-B

4,4

31

180/1984

7/16

3.88. 1O-2

1.62 . 1O-2 f33.3

-c C

1,3 1, 3

31

1251480

5/16

5.64. 1O-3

5.16 s 1O-3 f35.4

D

3, 1

31

180/1984

4/16

5.73. 1O-4

9.59 * 1o-4

computation

(%)

f6.2

f15.3

time.

We compute

the offsets in (ICI, 22) for the surfaces of cases A and B. For

@ = 0, we use xi0 = Jm; For J1 = J2 = 2. lo-’

the beta functions

at s = 0 are given in TABLE

II.

m, case A, we find ~~10= 2.2 mm and 520 = 1.3 mm.

For

J1 = J2 = 4. low6 m, case B, we find &o = 10.8 mm and ~20 = 6.1 mm. The approximate lation.

solutions can be sensitive to the mode set used for the calcu-

As the set is increased, the likelihood

and the whole technique

can break down.

of encountering For a suitable

as to spoil convergence, the Fourier amplitudes siderable scatter,

a trend of exponential

One expects this for a function

a resonance increases mode set, not so large

of G+ appear to follow, with con-

decrease with increasing

that is analytic

in a strip about the real axis in

the complex plane of @I or @2. That is, the Fourier amplitudes should be bounded in modulus by an exponentially Imp/, and the rate of decrease is conditioned theory, ..

G+ is indeed analytic

in strips.

of such a function

decreasing function

by the width

An argument 36

IrnrI and lrnzl .

of [ml I and

of the strip.

based on analyticity

In KAM is not

directly

relevant

in our approximation,

number of modes and is analytic

however, since our Ga has only a finite

in the entire complex plane of Qi.

In FIGS. 13 and 14, we give a logarithmic Fourier -. ._

amplitudes

of Ga,, namely

dimensional

solutions,

magnitudes

of the Fourier

roughly

follow

diminishes

experience

that

invariant

with

coefficients

increasing

of

versus lrnrl for two one-

6 and Table III.

do not decay exactly Moreover,

trend.

modulus

As expected, exponentially,

the but

the rate of decrease of Fourier

J1, reflecting

the general expectation

and

computation

of

surfaces at large actions.

Fourier amplitudes

amplitudes,

surfaces, the graphic representation

with

case. For a graphical

rithms

Jl,O)l/J1

more and more modes are needed for accurate

For two-dimensional

Im;h(m,

Imlh(mr,

cases b and c of FIG.

an exponential

coefficients

plot of the normalized

mode index is not as obvious as in the one-dimensional

display we have made a least-squares

for cases A and B given in TABLE

J,O)l/llJII

M Cexp(-almrl

- blmzl).

of the data and the fitted function

versus~ulmll

of the decrease of the

IV, to an exponential

of

function,

In FIGS. 15-18, we plot the loga-

(crosses and dashed line, respectively),

+ blm2l. A s in the one-dimensional

dent, albeit with considerable scatter.

fit of the moduli

case, an exponential

trend is evi-

Again, the decrease is slower at the relatively

large action of FIGS. 17 and 18. In this section, we have shown that the technique for solution of the HamiltonJacobi equation

works well in non-integrable

s-dependent

cases with d = 1, even

in regions of phase space close to domains of large scale instability.

-I.__

In similar cases

with d = 2, it was difficult

to approach such domains in reasonable computation

time. ..

accurate

Nevertheless,

rather

invariant 37

surfaces could be obtained

under

conditions

of substantial

nonlinearity.

VI. COMPARISON WITH OTHER WORK AND CONCLUSIONS -.

._

We have demonstrated

a possible nonperturbative

tion of the Hamilton-Jacobi

equation

odic function

of the independent

The primary

goal of such a solution

tori.

Consequently,

Hamilton-Jacobi

any method

equation

method for numerical

when the nonlinear

variable,

in particular

is to construct

that yields invariant

perturbation a periodic

solu-

is a peri-

step function.

approximations

to invariant

tori, whether

based on the

or not, should be evaluated

in competition

with

the

present method. We have found that the technique described is quite expensive in computation time when applied for d = 2 at large amplitudes accuracy is required.

of oscillation,

This means that it is not very promising

at least when high as it stands for a

fully realistic model of accelerators, which must have d = 3 to allow for synchrotron oscillations

as well as betatron

economical

approach, either by improving

if one is to study long-term mentioned

oscillations.

stability

It is therefore imperative

to find a more

the present method or by other means,

of higher dimensional

systems along the lines

in the Introduction.

A well-established

method in nonlinear mechanics is to work with the Poincare

return map, which takes a surface of section in phase space into itself. one effectively

eliminates

there are corresponding a convenient ..

one dimension

In this way

of phase space, and one can hope that

advantages in computational

cost. In accelerator

surface of section in the (2d + 1)-d imensional 38

physics

extended phase space

is defined by specifying map then propagates

a point

on the reference orbit,

Eqn.

An invariant

map”.

section of the (d + 1)-d imensional

surface of the return torus.

necessary information

on stability.

of the Hamilton-Jacobi

G+(Q, J,s = 0) as initial

provides all

equation

with respect

condition.

We review two approaches to determination

of invariants

of the return

In the first approach, which might be called the “many orbit picture”,

scribing

that an exact invariant the invariants

coefficients,

or non-perturbative

surface or invariant

by appropriate

one finds approximate

by

In any case, the full torus is easily found from

the section by a simple integration

equations

map

It is represented

to s = 0. This section of the full torus usually

(8)restricted

to s, taking

The return

the other 2d phase space variables once around the ring, to

s = C; it is called the “full-turn is a d-dimensional

say s = 0.

parameters,

function

one states

must satisfy.

De-

for instance Fourier or Taylor

solutions of the equations through

determination’of

map.

the parameters.

perturbative

In the second approach, the

“single orbit picture”,

one takes advantage of the fact that a single orbit is tran-

sitive on an invariant

surface; i.e., it comes arbitrarily

surface.

Knowing

points on the orbit, the surface.

it should be possible to fit a surface to a subset of

again by determination

Notice that the many orbit

the historical spectively;

the orbit,

viewpoints

and ordinary

coefficients pictures

differential

equation for consideration

describing

correspond

to

equations,

re-

of all orbits at

equations for single orbits.

An obvious necessity in applying sentation .m

of appropriate and single orbit

based on partial

that is, the Hamilton-Jacobi

once, and the Hamilton

close to any point on the

the return

map method

of the map that embodies the dynamics 39

is to have a repre-

of the system with

sufficient

For studies of real laboratory

accuracy.

systems such as complex

simple formulas for maps like those popular in the literature ics (standard

map, quadratic

adopt formulas -.

._

allowing

map, etc.)

are usually

accelerators,

of nonlinear

not adequate.

dynam-

One has to

greater complexity,

for example a power series in Carte-

sian phase-space variables with a substantial

number of terms:’ or a finite Fourier

series in angle variables with some flexible representation of the coefficients

(say through polynomials

or spline functions).23

tions are being studied for accelerator theory, to enforce the symplectic liable approach integration

24,25

is to define the return

through

approximations

condition.

22,23

along with necessary corrections

A more cautious

to maps defined by symplectic

based on the many orbit picture.

formulated

as a functional

difference equation.

paper 3 on the Twist

Theorem

As formulated

(KAM

theorem

For proof of the Twist

by a sequence of transformations,

be known as “super-convergent

perturbation

theory is awkward to implement

esting to consider either ordinary for numerical

by the return

numerical

solution

perturbation

of the functional

The requiremap can be

by Moser in his

for area-preserving

this is an equation for the canonical transformation

the map to a pure rotation.

perturbation

more re-

integrators.

a surface in phase space be left invariant

equation

and currently

map as the result of symplectic

ment that

of the.plane),

Such representa-

one period in s. In fact, formulas for maps are best derived as

Let us first consider methods

original

for the action dependence

maps

that conjugates

Theorem,

Moser solved the

using an algorithm

that has come to

theory”.

Since the super-convergent

in numerical

computation,

theory or nonperturbative

it is intermethods

equation.

There are at least two ways to formulate .. 40

the functional

equation,

correspond-

ing to different parametrized

ways of parametrizing

by the angle variable

the representation

@ of the underlying

of Eqn. (8), th en the functional

the Hamilton-Jacobi

surface. 26 If the surface is

the invariant

integrable

equation is quite analogous to

equation, in that the constant action J is an input parameter,

and G+ (now evaluated at s = 0 only) is the unknown function. the surface may be parametrized unknown

is a pair of functions

tion to new variables.

equations

J(Q, I), !&(@,I)

to date in nonperturbative

motion

with

A program

return

Nonperturbative

interesting, principle, that

solutions

was written

briefly

in Ref.

by a symplectic solutions

of the functional

we treated

27 , confirmed

a better control

of small divisors.

canonical,

integrator

for G+ here in

that considerable

rather than by an explicit Moser equation

would be

for d = 2. That equation

allows, in

In practice

it gives a transformation

however, since the transformation

form rather than in the implicit

form determined

is obtained

calculation

of the return map may be cast in the language of Birkhoff

leading to a recursive algorithm

that allows a practical

tive series to rather high order.28’2g In applications .m 41

in

by a generating function.

When the map is given as a power series, the perturbative variants

difference

to solve the equation

of the generalized

but have not yet been attempted

is not precisely

explicit

generalized

time could be saved in comparison to the present method, even if the

map is represented

formula.

gives Moser’s equation,

d = 2, for the same model that

Section 6. The results, reported computation

defining the canonical transforma-

dimension.

is quite limited.

for betatron

On the other hand,

by the new angle variable Q, in which case the

This latter formulation

to systems of arbitrary Experience

system, as in

generation to accelerators

of in-

normal forms, of the perturbathis approach

has been much aided by a new method

to compute

map,22’28 given

for the accelerator

method

a symplectic

uses automatic

integrator

differentiation

(“differential

tives of the map defined by the integrator, -.

._

A drawback in principle compute an invariant invariant

function

to generate deriva-

to machine precision. theory is that it aspires to

exists in an exact sense, and the formal series

diverges. The series has an asymptotic invariants

aspires to find only an isolated invariant be defined, it avoids this limitation

character at best. Indeed, in

improve in quality

as the order of the calculation

is increased.

initially,

surface, on which invariant

is one such that

a least-squares combination

K(z)

equation

of the function

= K(M(z)).

In Ref.

in the components

invariant

in a least-squares

function

can

func-

sense. If

K of the return

30, this equation

sense on a finite mesh in z-space, with

of monomials

functions

in principle.

z = (q, p) is a point in phase space, an invariant M

but finally

Since Moser’s method

A more direct way to find a globally defined and approximately tion is to solve the defining

The

defined globally on phase space, rather than a single

practice one finds that computed deteriorate

of the

model at hand.

algebra”)

for this type of perturbation

surface. No such function

for the function

the Taylor coefficients

map

was solved in

I< represented

as a linear

of z. For an example in 1 l/2

de-

grees of freedom, the Henon map, it was found that curves around the origin and a fifth-order Turning mining

next to the single orbit approach, we consider the problem

a surface so that it passes through

the return

. ..__

island chain could both be described by the same function.

map.

a subset of points on a single orbit of

As in the case of functional

depend on the choice of parametrization ..

equations,

of the torus. 42

of deter-

the fitting

algorithm

will

Suppose that the torus is

parametrized

as in Eqn. (8), so that the “curve parameter”

is the unperturbed

angle a. Then the problem can be stated as one of fitting

a function

sented as a finite Fourier series, to orbit points (I(s),+(s)),

s = 0

efficient

method

to perform

of Section V, at amplitudes

the method

produces an invariant

Ji close to those of Case D in Table IV,

torus with all modes such that lmil 5 Mi = 30

in 2 minutes on the IBM 3090, versus 2 hours for the calculation

62 M 2 x 10s6. The fitted orbit was calculated

integrator

by using an explicit

formula for the map, rather than the integrator. method seems uneconomical

is there any reason to consider it further

in comparison

for practical

There could very well be a reason, if the method of integrating equation

with respect to s could be made more efficient.

We gave little attention

used a method that was familiar It involves making transform, equation myriad

for every evaluation

._-

of the right-hand

(23); i.e., four such transform transforms

efficient,

and in fact

far from optimal.

and also an inverse Fourier

side of the ordinary

pairs for every Runge-Kutta

differential step. These

might be avoided by using values of G on a mesh, rather than

its Fourier coefficients, might .*

Fourier transform

interest in

is convergent and

from earlier work but probably

a d-dimensional

computations?

Our primary

to making the s-integration

to sur-

the Hamilton-Jacobi

this paper was to show that the shooting method for s-periodicity workable.

by the same

used in Section V. A further saving in time might be attained

Since the Hamilton-Jacobi face fitting,

of Table IV. More-

is much more accurate, since fewer modes were discarded; it

gives 61 M 2 x 10-7, symplectic

(mod C). An

such a fit was described in Ref. 31. When applied to

the problem

over, the calculation

I(@), repre-

as unknowns.

be done by using periodic

One has to enforce periodicity

B-splines to interpolate 43

in $, but that

the function

values.

A

point in favor of the Hamilton-

Jacobi shooting method is that it seems to have a

bigger region of convergence than the analogous algorithm

based on the functional

27

equation.

If the invariant constant

surface is parametrized

by the new angle !P, conjugate

to the

action J, then the surface to be fitted to orbit data at constant J is given

by two finite Fourier series,

q(V!,J)

= C

q,(J)eim’*

,

p(XP, J) = C

mES

Here (q, p) are globally coordinates

defined phase-space coordinates; variables of the underlying

along orbits should be given directly = 2run,

termined.

this on the right-hand

linear system; their values orbit

n = 0, 1, . . ., for some tune u that has to be de-

v and the Fourier coefficients

this is not a standard ematical

they could be Cartesian

side of Eqn. (52), and orbit points

n = O,l,. . .) on the left-h an d si‘d e, we get a set of equations to

F&L

determine

(52)

by the return map. On a non-resonant

we should have Q(&)

MnCL

.

mES

or angle-action

Substituting

pm(J)eim’*

theory.

problem

Nevertheless,

(qm, pm). Because Y is initially

unknown

in Fourier analysis, backed up by a sound mathauthors working

in molecular

dynamics,

32

plasma

t heoryf3 and celestial mechanics34 have dealt with a similar problem in an heuristic way, by an adroit use of windowing

functions

with discrete Fourier transforms.

It is not clear that the methods used to date are accurate and efficient enough for our purposes,

especially

for s-periodic

would seem to deserve further 34 are for autonomous our nonautonomous .*

perturbations

investigation.

systems with

in d = 2, but the problem

The examples given in Refs. 32, 33,

d < 2, thus considerably

case with d = 2. 44

less difficult

than

As is emphasized in Ref. 32, orbits close to resonances (within d = 1) may lie on invariant

surfaces that can be parametrized

always with the same choice of (q, p). By contrast, only for surfaces that “surround into surfaces I=constant, by our methods, dynamical

the origin”,

Poincark framework,

as in Eqn. (52),

the representation

(8) is useful

i.e., that can be deformed continuously

9 E [0, 27rld. T o t reat surfaces associated with resonances

one has to make a preliminary

information

islands, in case

change of coordinates,

is required to choose those coordinates. one may be seeking invariant

and some

Moreover,

in the

surfaces of higher powers of the

return map, rather than of the map itself. The foregoing impression

that the question of how best to compute

closed subject, origin.

brief review, which is far from comprehensive,

particularly

invariant

when the surfaces in question

should give the surfaces is not a

do not surround

the

The present study adds a new entry to the list of possible techniques for

studying

this long standing problem.

ACKNOWLEDGEMENTS One of the authors (W. E. G.) would like to thank John Cary of the University of Colorado for his support and the Accelerator at SLAC

for their hospitality.

Energy contracts

Theory and Special Projects group

This work was supported

DE-FG02-86ER40302

by the Department

(C o 1orado) and DE-AC03-76SF00515

(SLAC).

45

of

APPENDIX I. CONVERGENCE OF THE SHOOTING METHOD In this appendix we discuss the convergence properties of the shooting algorithm,

as realized by solving the fixed point problem (29) through simple iteration.

We show that under certain restrictions

the iteration

is governed by the contrac-

tion mapping theorem 35. It follows that there exists a unique fixed point of A in a certain metric space. The argument is valid only when the set of Fourier modes of the generating function

is finite, and includes no resonant mode.

Recall that the fixed point problem,

ho = A(ho), has the following

form ex-

pressed in terms of vector components labeled by the mode index m:

c ho(m) =

where the.function

e2rim.V

’

_

1

dsf(m,

h(s;

ho),

4

(53)

,

J 0

f is defined by.

f(m, &; ho),s) = -eim’x(s)

j?[

$1

e-im’*+,

J + c

inhcn,

s; ho)ein’(=+)),

s>

,

(54)

n

and h(m, s; ho) is the solution condition

of the following

differential

equation with

initial

ho:

dh(mi:’ ho) = f(m, h(s; ho), s) To indicate that a function

.

(55)

depends on an entire vector having components labeled

by the mode index, we suppress reference to the index.

.* 46

A. Model Problem For the discussion of contractive

properties, we consider an analogous problem

with a complex ho that has only a single component (mode). We thereby simplify the notation

without

losing any essential features. A later generalization

the full system with many modes will be immediate. ._

to cover

The equations for the model

fixed point problem take the form

ho = $7

dsf (h(s; ho),s)

,

(56)

0

g(s;ho) =f(&; ho),s).

(57)

with h(0; ho) = ho. The dependence of the solution of the differential the initial

condition

is displayed explicitly.

equation on

The small divisor is represented by D

and is analogous to the divisor, exp(2@m * Y) - 1, that appears in Eqn. (53). The contraction

mapping theorem, which will be applied at two different levels

to solve this problem, is as follows.

Suppose that an operator F maps a complete

metric space S into itself, and is contractive

on S. Then there exists a unique fixed

point z = F(z) in S. Moreover, x may be computed by iteration, where x(O) is any element of S. The contraction

d(F(xl),I+$) for all z1,x2

I +wx2)

E S and a fixed CYE (0,l).

-~._.

,

is that

(58)

H ere d(xl, x2) is the distance between

x1 and x2 in the metric of S, and the iteration sense d(x(P),x)

condition

x(P+l) = F(x(P)),

= O(aP).

.* 47

converges to the solution x in the

We suppose that function

the complex

function

f(h, s) is piecewise-continuous

of s on [O,C], and is bounded and Lipschitz-continuous

as a

as a function

of

y , as follows:

If(Y,S)lGo , MY174 ._

- f(Y2,S)I

IYl - y2I