A unified description of regular and chaotic motions in classical mechanics

CHChllCAL PHYSICS LETTERS

Volume 87, number 2

A UNIFIED

DESCRIPTION

OF REGULAR

AND CHAOTIC

19 Mrrch 1982

MO-I-IONS

IN CLASSICAL

LlECHANICS

*

RD. LEVINE The Fur-_ Habcr Research Cerrrer for Moleartar Dyrumrcs. T/K ticbrew CJnbcrsrry o/Jentwlcw

Jcnrwlcn~ 91904. tvact

and C E. WULFMAN Dcpamrerrr

ofPi~_vncs. Ururersir_v of rhe Pacfrc. Srochrorr. Calrjbrtrra 921

I

USA

Recrwed 11 August 198 I ; m fiid form 28 December 198 1

The datmctlon bct\\een rcgulsr ud chaotic mouon UI clx.wxl meclwucs depends upon trutmg time ds 3 pxtmctrr. A uniiied descnpuon oi both mottons cmergcs of rime IS rcgxded as 3 dyxuntcal vwabk bccausc the imtlal condrtlons arc then smgle-valued Elobd consizmts of the motion. Tune crolvu~g stxcs of IIICsystem m ordinxy ~IKIS~spxc ;~resutcs oi cqulhbrrum UI the eltendcd spxc

IS conslderable current interest m the dls-

There tincuon

between regular and chaotic motion

cal mechantcs WW,

Yet

[I-3].

such a lstmction

Jeclory

can be propagated

unique

fashion

from

nates and momenta

the current

point

m time,

m a

(1)

Q andP

slant. The dependence of P and Q upon

=

their

that of the tune-reversed

Q(~~),P(O).

argu-

systems

211 functIonally depend

the motion through

(3

It follows of Iz degrees

independent on time.

because

[4,5]

of freedom

constants

These

there

that classical

Chaotic

are global

is a unique

each q(r), p(t)

point,

possess

constants

classical moreover

of

trajecthey

are

Work supported by NSF Cnnt CHE 8014165 and by the US-Lsrael Birutiond SnenceFoun~tlon,Jerunlem,Israel.

0 009-2614/82/0000-0000/S

two uncoupled

ltarmomc

02.75 0 1982 North-Holland

separately

p,(O),

(3)

nme-indepcn-

smgle valued.

IS associated with tlms-depcndent

of the motion just as regular motions descnptlon

where the tmledepcndenr

how the dlstmction

Quahtauvely,

[ l-31,

rnd

the two and compu-

can be recovered.

the dIstinctIon

to miual conditions.

trajectories

con-

descrlptron

bcrween

types, made famdrar by many an.dytuxl studies

are. of

play a key role? The purpose of

thus paper IS to offer such a umficd also to show

rapidly

2,

The timcdependrnt

themselves are however

both types of motion,

tatlonal

oscillJ-

I = I,

t11e rime between

leads to a multkdued of the motion.

motion

fJmdiar

+ w7,(f)sln(w) 10 eliminate

Should there not then be a umtied

tivity l

andp?(O)

dent constant

tune-mde-

The most

p,@)=p,O)cos(qO Trying

lo elinunatc

of the morlon.

stants of the mollon

of the motion

if one attempts

For each oscdlator

constants

trqectory.

-0

for p(t).

hamiltonian

tory

is that of

constants

and slmdarly

which

example

~~(0)

con-

E\plhtly q(r)

conslants

IS constant.

t, and for any

are therefore

true.

pendent tars.

of the coordmatcs

at time f. For any time

is precisely

the lime between P and Q so as to obldin

P(O)=mJo).d4~0.

along the traJectory,

ments

longer

necessardy

JS the

these sfa1cments are no

Ira-

values of the coordl-

are the vtiues

and are at least as differenuablc

flow itself. As IS known,

of

A classIcal

10 the mibal ones:

and p(t)

and momenta

dl least one pomt

backwards

4W=P(q(O*P(f)7~). Here q(r)

from

is not obvious.

smgle valued

m classi-

For a

is based on the sensi-

chaotic

motion.

with very similar mitral conditions

(I e. exponentially)

two

will

diverge. Hence a given re-

Volume 87. number 2

CHEhlICAL

19 March 1982

PHYSICS LETTERS

gion m p.q space wdl rapldly deform, as III the Gibbs

man view fh3t f runs from

mivmg

mmcterize - to +-)_

metaphor.

ever be turned

Any parlicul3r backvxds

tr3jectory

in 3 unique

other words, the disrmctlon [l-3] and chaorlc

motion

is based

lied dcscriptlon elevatmg

provided

In

between regular

on comparmg

p, q or “phaw”

or 3re3s m ordmary

CIUI how-

f3shion.

space

nelghbormg

tmJ2ctories

vanable

by [6].

By consid2rmg

P as coordmates.

will remam

the (pxtial)

close,

two

cf. lig. I.

ofP snd

urn2 drnvatives

Q It wdl however be shown

to discuss the

possible

is that

primary rno1iva1ion for the unified I[ en3bles

rium sraristxal

mechanics,

rims+xolving

st3t2s

toman lables

b3sed

oiequlhb-

to the descnptlon

and, III particular,

function function

Lhroughout

of

to collision

W=H(p,q)-E,

(4)

tions thereof. The Poisson brackets m (4) are defined as usual 2xcept, of coursz, that we now have two addltional dynamical variables. No12 in particular that dt/dr

= I and dE/dr

space with

IV =

= 0. W2 shall ref2r to p,q, E, c

H - E as psth space.

Timz-depsndent space

constants

are analogous

of the motion

to ordinary

m path

(time-mdrpendent)

A point in path space IS characterized by th2 values of p. q. E, 1. As the tr3JJectory

evolves

T (and

hence r)

p. q. E. t Sp3C2 IS visIted

IWIC~ by the same

or in more malhematical

language, a traJ2ctory is

homeomorphic 3 time-Independent

hamil-

H(p.q) whtch 1sassumed 10 b2 an of IIS arguments.

ar2 now of the form

continuously increase, cf. fig. I. Hence no point III

on timc-mdeprn-

[ 5.71.

We consider an3lytic

description

the machinery

of the motion,

dent constants phenomena

us fo apply

equations

(r runs from

constants of the motion m phase space.

sensitwity to mitml conchtions. II2

dz/dT = (1%‘.z} ,

Let T pa-

3 trajectory

where z IS any on2 of the dynamical vanables or func-

The um-

As wdl br argued, 311motions arc then regular ones. In pJrricular, by using Q and

Hamdton‘s

- to + mfinity.

along

“distances”

III this prtper IS obtainrd

time to a role of 3 dynamlcal

the progress

The dynarmcal var-

are now q and r and IheIr conjugates p and -E.

whrre E is the total energy and we take the n2wto-

p3th space

to the real line (i.e. every

c3n be rectified).

In parttcular,

tmjjsctory, tr3lectory

m

in thr p, q.

E, I space,P and Q are parameterlzed by 7 Ehmmanon of the progress paramerer, which, m p. q space, is the source of all dlfticulnes is here achieved ~13 the replacemenr

of 7 by r (dr/dr

= I), lcadmg to the well-

oi the motion Bven by eq. (1). Trajectories III tradItional phase space are obtained as projections from trajectories in path space. The distmction between regular and chaottc bounded motton appears only In the tradltional p, q space. W2 turn behaved constants

now to a proof Consider

of this asserlion.

the cas2 of common inrsrest where the

motion m phase space is confined to a bounded region. The action of th2 evolution oprraror along a Bven trajectory can thrn be of two types: (1) the regular

motion where the trajectory is confined to an

n-torus. In this case the evolurion operator IS a coupl-

(b)

ed rorarlon operator of the compact group SO(Z) X

SO(?) X .__X SO(Z) ()I-times). Explicitly, witlun p, q space

l-g

1. Evolution in path spxc. (a) Two kjecforlcs

mp.q.E,r

spscc (at Ihc s3me value of E), which, upon projection to p,q or phase space, arc seen to cxponcnt~~By dwerge. (By cons1ruciIon. Ihe dlstmctlon between regularand chaotic motions IS recovered upon proJection from the eutendcd.or path spxc 10 the conventional phase space ) (b) The same two trqxtones in P,Q,E,i space. 106

where L is the Liouville operator, Li = a/M, is the rotation operator and Bi is the angle variable. Note that while eXp (7O,Li) = exp [(T + Znn)w, L,] SO that the motion m 8, recurs, the entire trajectory is period-

ic only ii the Wi are rationally

dependent.

Otherwise

it is quasiperiodic. of an rrregular tory cannot

(2) The second

or “chaottc” be confined

posstblhty

This dtsttnctlon

to a torus and the action

which

apply

IS that

Here the tralec-

motion.

the evolutton operator on rhe trajectory resented as in (5) above.

cmnot

of be rep-

I” phase

cJn vary rv1111tm12. If. how2v2r.

Jnd the entropy

time is regarded

IS possible in p, q space

I” p. q. E, f space. Mule

does not occur

19 Blarch 1981

CtlChIlCAL PtIYSICS LETTCRS

Volume 87. number 2

of motton

vartablc,

as a dyxtmtcal

the cqttauon

IS [5]

.

dlV (up) = 0

17)

where u differs

irom u by h.tvmg one additIonal

ponent

drldt

namely

COIII-

= I. In the cxtcndcd.(subp.q)

space the evolution operator may or may not xt on the trajectory as a coupled rotation, it WIII certamly

E. I. space Ltouville’s theorem holds Jnd rltc most gcncral solurlon oi the equatton for p IS on-e ~gdm J iunclton

not do so on any of the trajectoncs

of the tlmc-dcpendcnt

tn path space.

TIIIS IS due to the term alar which tn p. 9, E, I spice is part of the Liouvtlle

operslor.This

term IS a translJ-

non operaror

or. as we hJvc said before.

tn path spxe

does not recur

to plus mfimty. the concrpt

A slmdar

iortime

picture

of dynamlcal

obtains

that the intersrcuon

Hence.

therr

of tlmc steps. By takrng

dependrnt

constant

rem dp/dr

= 0). Hsnce

mlwng

not hold in p,q, E. f space. Indeed descrlptlon

uon tnp.q constants

Such a functton

the Liouvtlle evolving

rquatlon.

Consider

iamdiar

is the most general

states of equthbnum by the property

rium after spac2.

the proJectton

For systems

of many

function

of freedom

splat +

111equlhb-

it IS custom-

have been aver-

of motion

of the denstty

space is then of the form

dtv (up) = 0 . zero and hence

[j]

(6)

whereu 1sthe \elocLLy hector. necessarily

to p. q

sub p_ 9

in a projectsd

The equauon

In a subspace

Ltouvtlle’s

still

theorem

&v u is SOL need not

or “following”

stJnds Jr2 not

nanc2

[ 101 but Jrc seldo~n consldcred workers

mer~cally

states

[I

euauunrd

iound.

pu~stlon.ll

in otlter

Gelds.

in the Pomxrf

reJdcr

tnnc-mdep2ndcnt

One can suggest

J similar

by our previous ts Invited

sur-

tlic t>rCscoiiLonstrtlc-

E. t space but only rcguldr mouon

will bc

considcrJUons

to vcrtfy

this by a corn--

CUmplc of lhc scnsitivny

can however

Consldcr

by 29. (1). Thrse

tune-dependent

CklSSlCJl

constant

(IV. q = 0, cf (S)!,

mcch.mic3l

To dtlf2rcntlJtr

typr’s ofdgnJmicJl

to bring 1112propertics

given

CoiiStJIIts oi the

Jr2 ttme-dcpcndcnt

of 11s hJmiltonrJn.

twe2n dlffer2nt

xmstdnts

JgJin the Irr const3n1s

for any conservattvc

trrespecttvc

to iiiiti~l con&ions

be dtscussed using tune-dependen.

tlis motion.

motion

the behwrour

JS prcordJtncd

The sc2pticJl

oi tli~s

nlJgllCtlC WSO-

.I.1 11in IIt2 present area hdvc nu-

of Lrcll-bchJvrd

0i tlic motion.

system. bc-

11is ncs-

behavior

of the hamdtomut

2x-

To do so. nmc th.tt if 1 IS J of the motion

then so IS a//&

= 0 or

[d//dr

whcrc

artar = (I. II)

@I

and the Poisson

bracket

in (5) IS WIIII respuct

only. (To prove ths property wtlh rzspect iunctIon

in

co-

taking advJntJg2

in IIUCkJr

as 3 medns of d2monstrattng

bon tn p. 4.

0i

E\pcriments

UnCOIIllnOn

oi slates

wc Jr2 c\amtning

tn P. Q space .my

words.

pl~c~tly into the dtscussion.

Th2

from p. q, E. I down degr2es

In olher

aspect

2ssJry

in p, 4 space are now tisthat they r2mam

the evolution

in the reduced

of a ttms-

p, q, E. t space.

space in which some of the coordtnates ag2d over [5,9].

of

m p. 9 space corresponds

In the enlarged

ary to consldrr

solution

Hence the dcscnption

ttnguished

of

tn p, 9, E, I

ch.trJctcr

in eii2ct

in a “rotattng.’

system.

Tlic notion

func-

now a functton the mollon

state of the system

to equihbnum

the dcnslty

only of tmie-mdepcndent

thr ttme-drpendentconstantsoi spxe.

analo-

of the mouon.

system

tn cquilrbmtrrz tn

of systems

of the motion.

does

w2 now a19112 th.tt

At equihbnum

space is a function

theo-

metaphor

m path space is exactly

gous to the drscnption

ordmtte

2ncc or Jbsence

(by Liouvdlc’s

the Ctbbs

IS to note thx

fJc2 ofs2ction

p m phassc space IS tlself a tome-

of the motton

phase space.

on2 c.m

tn psth spacr.

function

t spx2

Previous

of the two ~21s IS smpty.

is no mning

Th2 density

the untticd

of

entropy

of ttme steps to be large enough,

ensure

ordinary

m trrms

an Initial set tn p, 4, E. I space and its

[S]. Constdrr

p.q.E.

constants

view of the slationary

die ttnie evolulion

a tral2ctory

runs from mmus

(or Kolmogorov)

Image aft2r a fintte number thr number

Anoth2r

to I, noting

drifer2ntiatc

rap.,!

{IL’. fl = 0

that IV IS not an e\phctt

of time.)

A set of Irr ttmz-drpcndent and one whtch

does provtdc

const.mts dynamical

of the motton. tns&t

IS thcrc-

fore given by i(O),jr(O) wltere 1112dot denotes IIIC (partial) Lime dcnvarlre. This new SeL Ls functionally dcpcndent on the prevtous

srt.q(O),p(O),

but IS not, in general. 107

Volume

lmearly

on it. If it IS (as would be the case

dcprndznr

for a harmonic stablhty rei.

CHEMICAL

87. number 2

poteniial),

snalysis (ref.

then one can carry

out a

[ 131. and also sectlon V.B of

[ 131). If not. 11IP possible to add higher-order

tinwz derivarwes, that (1r.H)

looking

for a set of constants

of members of thz set. CWII

such a set, the trajectory

equations

of motion

equations

wlrh consram coefficlenrs

tradnlonal

frsuch

can be expressed as a hnear combinarion

can be written

analysis ofstabdlly

as differential and hence the

apphcs.

The Frrtz Habcr Research Center 1s supported hlmer~a Cesellschaft

13r die Forschung,

by the

mbH. Munich.

FRG.

References ] I ] G. Casm and J. Ford, cds. Srochsslic bchwlor in clawcal and quamum mcchsmcal hanuhonnn systems (Sprmger. Bcrhn. 1979).

I08

PHYSICS

LETTERS

19 hkch

1981

[2] S. lorna. ed., Topics In nonlinear mechanla (AIP, New York. 1978). [3] V. Arnold, hl a th rmatial methods in class~cd mcchznics (Sprmger, Berlm, 1978). [4] J. Dothan, Phys. Rev DZ (1980) 1943. R.L. Anderson, T. Shibuya and C_E. Wuliman. Rev. Mcu. Fa. 23 (1973) 257. [S 1Y. Alhassld and R.D. Lennc. Phys. Rev. CX (1979) 1775;

Al8 (1978) 89. [6] J.L Synge, in: Encyclopedia

of phyucs, Vol. Ill/l, ed

S. Flugge (Sprmger. Berhn. 1960). [7] R D. Levme, Advan. Chcm. Phys. 47 (1981) X9. [8] Ya G. Smti, Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961)

899; 30 (1966) 15 [Am. Math. Sot Trawl 39, No. 1 (1964) 83;68,No. Z (1967) 341. [9] R. Zwaru~, Phys Rev. II!4 (1961) 983. [ 101 A. Abragam.The prinnplcs of nuclear magncrism (Clxcndon Press. Ouiord, 1961). [I I] hl. Henon. and C. Hrdes. Asrron. J 69 (1964) 73; GH N’alkcr,and J Ford, Phys. Rev. 188 (1969) 416. [ 121 C E. Wulfm~n md R.D. Levme, Chem Phys Letters 84 [l3]

(1981) 13. P. Bruner,Xdvan

Chem Phys 47 (1981)

701