F. J. Lin and J. E. Marsden- Symplectic reduction and topology for applications in classical molecular dynamics

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Symplectic reduction and topology for applications in classical moleculardynamicsF. J. Lin and J. E. MarsdenDepartment of Mathematics, University of California, Berkeley, California 94720(Received 16 October 1991; accepted or publication 21 November 1991)This paper aims to introduce readerswith backgrounds n classical molecular dynamics tosome deas n geometric mechanics hat may be useful. This is done through somesimple but specific examples: (i) the separation of the rotational and internal energies n anarbitrarily floppy N-body system and (ii) the reduction of the phase spaceaccompanying the change rom the laboratory coordinate system to the center of masscoordinate system relevant to molecular collision dynamics. For the case of two-bodymolecular systems constrained to a plane, symplectic reduction is employed todemonstrate explicitly the separation of translational, rotational, and internal energiesand thecorresponding reductions of the phasespacedescribing the dynamics for Hamiltoniansystems with symmetry. Further, by examining the topology of the energy-momentummap, aunified treatment is presentedof the reduction results for the description of (i) theclassical dynamics of rotating and vibrating diatomic molecules, which correspond to boundtrajectories and (ii) the classical dynamics of atom-atom collisions, which correspondto scattering trajectories. This provides a framework for the treatment of the dynamics of largerN-body systems, ncluding the dynamics of larger rotating and vibrating polyatomicmolecular systemsand the dynamics of molecule-molecule collisions.

I. INTRODUCTIONBasic nterests n molecular physics nclude the studyof (i) the structure and dynamics of molecules, clusters,and complexes (such as hydrogen-bonded omplexesand

van der Waals molecules) and (ii) the dynamics of mo-lecular collisions, including the casesof atom-atom elas-tic scattering, atom-molecule inelastic scattering, andatom-molecule reactive scattering. While molecules nat-urally obey the laws of quantum mechanics, one fruitfulapproach has been to first examine the dynamics deter-mined by the laws of classical mechanics. This has beentrue both for the case of rotating, vibrating moleculesand for the caseof molecular collisions,2-8 ncluding ion-molecule reactions. In fact, quantum mechanicswas de-veloped by assuming a correspondence with classicalmechanics,t and this is one of the directions in which thegeometric approach o quantum mechanicshas continuedto develop.1-3This approach builds upon a backgroundin classical mechanics.14*2P5Beginning with principles and examples hat are fa-miliar to classical molecular dynamicists , we will intro-duce some deas from geometric mechanics. Specifically,we begin with Hamiltons equations n curvilinear coor-dinates and will show that they can be expressed eomet-rically in coordinate-free orm in phasespace.Other ideasto be discussedgeometrically, include the separation ofenergies (cf. Ref. 16) accompanying the appropriatechoice of a coordinate system, the description of dynam-ics in a moving coordinate system (cf. Ref. 17) in terms

of dynamics on a reduced phase space,* and the decou-pling of the dynamics corresponding to the internal androtational motions.These deas will be motivated by looking specificallyat (i) the dynamics of atomic clusters, which are alsorelevant to the dynamics of van der Waals molecules,iand (ii) the dynamics underlying atom-atom differentialcross sections.3Finally, we explicitly treat the two-body problem in acentral force field potential. After geometrically derivingthe separation of energiesand the reduced phase spaces,we eventually look at the topology of the reduced phasespaces or the case of a molecular interaction potential.The reduced phase spacesof different topological typescorrespond o different physical situations, including therotating, vibrating diatomic molecule and atom-atomelastic scattering.This work provides a framework for the geometrictreatment of the dynamics of N-body systemspreviouslytreated by other approaches,such as atom-diatom vander Waals complexesand atom-linear molecule van derWaals complexes (see, e.g., Ref. 22), or the dynamics ofa diatomic molecule in a time-dependentpotential, suchas an electromagnetic field (see, e.g., Ref. 23). Anotherpossible uture area of application is in geometric compu-tational quantum mechanical molecular dynamicsstudies.24While the rotating, vibrating molecule has recentlybeenexaminedgeometrically,25f26he previous results didnot discuss the optimal decoupling of the internal and

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rotational dynamics. Interest has been expressed n thegeometric approach to molecular collision dynamics2and in the topology of the scattering dynamics.3The to-pology of the Kepler problem has been examinedpreviously.28-30owever, the earlier studies have not ap-plied reduction techniques o the molecular problem, andthe topological relationship betweenbound and scatteringtrajectories has not been discussed.

II. HAMILTONS EQUATIONS IN CURVILINEARCOORDINATES FOR THE STUDY OF CLASSICALMOLECULAR DYNAMICSThe Hamiltonian H in curvilinear coordinates for asystem with N degreesof freedom isHz: ,$ i 8PIpi + VP (2.1)t-1 j-l

written in terms of the fundamental metric tensor gii(Ref. 17) with inverse gi, i.e.,8=g; , i, j= 1,2 ...,N, (2.2)

momenta pk (k = 1, 2,..., N), and potential energy V.Then Hamiltons equations take the explicit form. aH

qk=& 9

k= 1,2 ...,N,(2.3) . aH 1=ap,=,PI

aH av pePr= - dr= -ar+mr3,

dHpk= ---J&y

g$pj+ V) , k= 1,2,...,N.(2.4)

In the examples,we will treat caseswhere gi is diag-onal, i.e.,g=6i#, i, j= 1,2 ...,N, (2.5)

and the potential energy V is a function of the coordinatesqk (k = 1, 2,..., N) alone. In these cases, Hamilton sequationssimplify to&=tikpk, k= 1,2,...,N (2.6)

N adk= - gk-; i;l a4k (g)p;, k= 1,2 ...,N. (2.7)

When a change of variables to an appropriate curvi-linear coordinate system such that k;k = 0 for somek canbe carried out, then one can reduce the dimensionality ofthe phase spacedescribing the Hamiltonian system. Thiscan be seen n the following simple example.Polar coordinatesx=r cos 8, (2.8)y=rsin 8, (2.9)

are a special case of curvilinear coordinates with funda-mental metric tensor gii satisfying

1282 F. J. Lin and J. E. Marsden: Symplectic reduction and topology

(2.10)

and with corresponding nverse1 0g=

i i1 .

O7(2.11)

Thus, expressed n polar coordinates, the Hamiltonian Hfor a system with a potential V describing a central forcefield takes the formfW,Qwd =&p: + T& * ;P; + V(r). (2.12)

Then Hamiltons equations n this coordinate system are(2.13)

(2.14). aH 18=-=-p*ape mr

aHk;e= -Y&=0.

(2.15)

(2.16)Hence the angular momentum pe is a constant and weobtain on the reduced phase space (r,p,) the reducedHamiltonian Hp,( r,p,), which we now express n the form

Hpo(r,p,) = ( 1/2m )pf + V,,(r),with effective potential VP,(r)

(2.17)

VPOr)=V(r) + (1/2mr2)p2 e (2.18)

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parametrically dependent on pe. This is the effective po-tential familiar to molecular dynamicists (see, for exam-ple, Refs. 6, 3, 7, or 8). It is comprised of the originalpotential for the potential for the central force plus apotential energy term describing a centrifugal barrier.Hamiltons equations on the reduced phase space are

JHP, av,,Pr= - dr=

-- ar a

(2.19)

(2.20)The reduced Hamiltonian has been employed withthe appropriate mass m (i) in the form given above bymolecular collision dynamicists to describe the classical

dynamics of atom-atom collisions3*6*7nd (ii) in the cor-responding quantum mechanical form by molecular spec-troscopists to treat the rotating, vibrating diatomicmolecule.31-33n these cases,examples of molecular in-teraction potentials that have been employed include theLennard-Jones34P35otential and the Morse36 potential,respectively. Both of these potentials have the essentialqualitative features of (i) attaining a very large positivevalue as the internuclear separation r approaches0, (ii)attaining a finite negative value (the wel l depth) at theequilibrium internuclear separation, and (iii) vanishingas the internuclear separation tends to 00, allowing thedescription of molecular dissociation.The Lennard-Jones potential V,(r) takes the formV&9 =k[ (o/r)12 - (cr/r)6], (2.21)

where E is the well depth and the potential vanishes atr = U. The corresponding effective potential then takesthe formVu,p,(r) = VU(~) +pv2mr 2. (2.22)

This can be expressed n dimensionless orm (called re-duced units in the chemical physical literature). Follow-ing Hirschfelder, Curtiss, and Bird,6 we define the follow-ing variables in dimensionlessunits:r *r/o, (2.23)P = V/E, (2.24)

p8 =pe/( 2mea 2, 2 (2.25)co= P/E (2.26)

Then, in dimensionlessunits, the effective potential cor-responding to the Lennard-Jones potential is given by

F. J. Lin and J. E. Marsden: Symplectic reduction and topology 1283

PLpe(r *)=4[ (r *> -I2 - (r *) -6] +pz2/r *2.(2.27)

Above the valuep$ = 1.569, he effective potential has noinflection point and hence s purely repulsive for all val-ues of r * (Ref. 6, p. 554; Ref. 7, p. 54).The Morse potential V,(r) takes the form

V,(r) =&-2a(r-rO) - 2&-a(-rO), (2.28)where D is the equilibrium dissociation energy, r. is theequilibrium internuclear separation, and a is a positiveconstant determining the shapeof the potential well. Thecorresponding effective potential is

VM,p,( r) =De - 2a(r- 0) - 2De - dr- b) + p28/2mr 2.(2.29)

We now define the following variables in dimensionlessunits:r *=r/r 07 (2.30)V* = V/D, (2.31)a* =aro, (2.32)p$=pe/(2mDri)2, (2.33)V&= VP/D. (2.34)

Then the dimensionlesseffective potential correspondingto the Morse potential isV*M,pg(r*)=e-2a*r*-I) -2e-*(*-) +pQ2/r*2e

(2.35)Note that the dimensionless effective potential corre-sponding to the Morse potential contains an adjustableparameter a* that determines the width of the potentialwell and that did not appear n the dimensionlesseffectivepotential corresponding to the Lennard-Jones potential.We choose a* = 5 so that the potential wells ofPM,ppe(r) and PU,pO(r ) have approximately the samewidth when plotted as a function of r *. Numerically, wefind that the valuepe -- 1.413 s the critical value for theMorse potential with a* = 5, i.e., for values of p8> 1.413, the effective potential is purely repulsive for allvalues of r *.These ideas also extend to larger N-body problems,such as the dynamics of atomic clusters (discussed n thenext section), weakly bound van der Waals molecules,21atom-atom collisions (discussed n Sets. IV and VI), andpolyatomic reactions (discussed briefly in Sec. VII; seeRef. 8). As motivation for the more general and more



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mathematical treatment contained in Sets. V and VI, thenext two sections (III and IV) each present a physicalexample ollowed immediately by its corresponding geo-metric treatment.Ill. SEPARATION OF THE ENERGY OF ROTATIONAND THE INTERNAL VIBRATIONAL ENERGYA. Instantaneous rigid body momentum

Jellinek and Li2 showed hat the total kinetic energyof any nonrigid N-body system can be expressedas thesum of its overall rotational energy, which we will callsimply the rotational energy, and its internal vibrationalenergy.We ecall their result in Eq. (3.8) below.Let {ri : i = l,...,N) be the coordinatesof the particlesand {pi : i= l,...,N) be the momenta of the particles. Aforce-free rotation of a true rigid body is characterizedbythe time evolution of its angular velocity wrb( ). LetI*(t) be the instantaneous nertia tensor of the rigid bodywith respect to the lab-oriented system of coordinates(assumed for simplicity to be an inertial frame). Therigid body total angular momentum Lrb in the lab frameis

L*=Ib(t)*&(t) (3.1)and is a constant of the motion. For a nonrigid system,one defines he instantaneousangular velocity o;*(t) asfollows: The total angular momentum L of the nonrigidsystem s the product of the instantaneous nertia tensorI(t) and the instantaneous angular velocity o;(t) :

L=I(t)*oyqt). (3.2)Define the rigid body momentum P;~

p;rb=mi(wLhXri), i=l,...,N (3.3)and the difference Api between t and the momentum pi

Api=pi - pi*, i= l,..., N. (3.4)Then the total vibrational angular momentum vanishes,i.e.,

,?l riXAPi=O (3.5)because he total angular momentum of the system isgiven by

NL= E riXpi, = ; riXp;rb+ C riXApi (3.6)i=l i=l i=l

and the instantaneous igid body angular momentum sat-isfies


(3.7)Then the total kinetic energy of the nonrigid system s;* J&j, i!!$+ i* g+ ;, p:y?

(3.8)The last term is the Coriolis term (cf. Refs. 14 and 37)

N p;rb.Api Nci=l mi = igl (~~wXri)*bi9

rb=WL riX Api,

=o, (3.9)which vanishes since the total vibrational angular mo-mentum vanishes.Thus the total kinetic energy separatesinto two terms. The vanishing of the Coriolis term iscrucial for the separation of energies n the general N-body case.The identity (3.8) is used as one ingredient in thereduced energy-momentum method discussed below.While equation (3.8) separates he energy, it does notseparate he dynamics. The block diagonalization resultsof Simo, Lewis, and Marsden do separate the dynam-ics near a relative equilibrium and will be sketchedbelow.B. Application of the reduced energy-momentummethod

Let Q be the configuration manifold and P be theassociatedcanonical phasespace, .e., the cotangent bun-dle P = T*Q. The phase space s endowed with the ca-nonical symplectic two-form a. Let H : P+R be aHamiltonian function of the form kinetic plus potentialenergy . We assume hat the Hamiltonian system pos-sesses ymmetry induced by a Lie group G, which acts onP by canonical transformations. Let 3 be the Lie algebraof G. Let Y * denote the dual of the Lie algebra 9, andlet J : P-+ 3 * be the corresponding conservedquantity.The Hamiltonian H : P-R can be expressedasIH(z)=&-p,(z)~~-, +$J(z)+--(q)J(z) + V(q)(3.10)HerepJ(z) is the momentum associatedwith the instan-taneous angular velocity c(z) of the rotating body, i.e.,the Legendre ransformation FL of the instantaneousve-locity field

P.,(Z) =WlXz)&) IgTfQ, (3.11)J. Math. Phys., Vol. 33, No. 4, April 1992

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Vl =vcm + Wl, (4.5)v2=v,, + w2.

Let the relative velocity v be defined o be(4.6)

v=v2 -v*. (4.7)Deiine r;, r;, rim, vi, vi, vim, s;, s;, w;, w;, and v to bethe corresponding ec tors after the collision. By conse r-vation of total linear momen tum, we havevim = v,, andhence i, = re m+ v,,c. For an elasticcollision, the mag -nitudes of the initial and final velocities are equal, i.e.,I vl I = I vl I. By look ing at the velocitiesvi and vi afterthe collision in the laboratory system, t is not obviouswhethera final velocity s due o a large mpact parameterdeflection (forward scattering n the center of masssys-tem) or due to a head-oncollision (rebound collision in

the centerof masssystem). However , his can be readilydeterminedby looking at the Newton diagram4*5or thecollision, .e., by looking at the velocitiesw; and wb n thecenter of masssystemafter the collision.Furthermore,after making the transformation o thecenter of mass coordinate system, he differential crosssection ( 0)) i.e., the intensity of deflecti onsas a functionof the scatteringangle8, can hen be expressed lassicallyin terms of the impact parameter b and the deflectionfunction ,y for one pair of colliding particles as

withx=rr-26

in terms of the classical urning point ro, the intermolec-ular potential V(r), and the total ene rgyE. Hence,givenan experimentally determined differential cross section,one can gain nformation about he intermolecularpoten-tial. While mol ecular nteractions are quantum mechan-ical and the classicalmechanical reatment includes sin-gularities, e.g., at the glory angle and at the ra inbowangle, he classicaldifferential crosssection provides m-portant information about the intermolecular potential(see, or exampl e,Ref. 8 for further discussion).In this example, t is the velocity phase pace versusthe momen tum phasespace ) hat is important. One firstemploys he well-known fact that the two-body centralforce problemcan be reduced o a one-body roblem. 4 Inorder to determine he relationshipbetween i) the angledefinedby the differencebetween he final an d initial di-rectionsof the relative position vector between he parti-clesand (ii) the scatteringangleof one of the particles nthe l aboratory coordinate system,one must perform the

1286 F. J. Lin and J. E. Marsden:ymplectic reduction and topology

above ransformation between he laboratory and centerof masssystems.Specifically,one must look at the veloc-ities (instead of the momenta) in the center of masssys-tem. This is due to the fact that, in the center of ma sscoordinate system, he total linear moment um is zero,and the two particles move, of course, with equal andopposite momenta. In the next subsection,we will givethe geometric relationship between the initial velocityphasespaceand the reducedvelocity phasespace.Note that, while equations 4.8) and (4.9) determin-ing the differential cross ectionare derivedby employingconservationof energyand conservationof angular mo-mentum, they are not Hamiltons equati ons on the re-ducedphase pace cf. Eqs. (2.19) and (2.20)]. However ,the e nergy and angular momentum dependence f thetopology of the dynami cson the internal phase pacewillbe discussedurther below (cf. Sec.VI).

B. Definition of tangent bundle reduction via thesymplectic reduction theorem and theLegendre transformationWe first summarize he results of application of thesymplectic reduction theorem. For the N-body prob-lem, let the configuration manifold Q be R3Nand let thephasespaceP be the cotangentbundle T*Q = T*R3Nwith the canonical symplectic structure. Let the Liegroup G = R3 act by translation on Q = R3N or eac hcomponent of the configuration manifold, inducing an

action by cotangent ifts on T*Q, i.e., for xeR3X(ql,..., &PI ,..., PN) = (ql + x,...,qN + %h-**,PN) -(4.10)

The momentum map isJ(q, ,..., QN,Pl,...,PN) PI + * + PN=PO, (4.11)

corresponding o the total linear momentum. The re-duced phasespaceJ - * po)/GP,, where Gp, s the isot-ropy subgroupofpo in G, is symplectically diffeomorphicto T*R3(N-) with the canonical symplectic structure.Coordinates n the reducedphasespaceare often cho sento be Jacobi coordinates.Furthermore, he dynamicscan then be described nthe reduced phasespace.The Hamiltonian flow on theoriginal phasespace nducesa Hamiltonian flow on thereducedphasespace.The reduced Hamiltonian HP0 onthe reducedphasespace s

where H is the G-invariant Hamiltonian o n the originalphasespace, po : J - ( po) -+ Pp, is the canonicalpro-jection and ip, : J - ( po) --+P is the inclusion.



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In order to define a tangent bundle reduction consis-tent with the above cotangent bundle reduction, we em-ploy the Legendre transformation FL : TQ- T*Q, whichrelates he Lagrangian formulation on TQ and the Hamil-tonian formulation on T*Q.12 The tangent bundle reduc-tion is defined so that the Legendre transformation com-mutes with reduction, i.e.,

reductionTQ, - TQ2FL n,reductionT*QI - T*Q2

(4.13)

where T*Q2 = (T*Q,>,. In other words, one hasTQF~--WTQI)I,. (4.14)

Specifically, we haveTR3(N- )EFL - ( [FL( TR3N)]Po). (4.15)

V. SEPARATION OF ENERGIES AND REDUCEDPHASE SPACES BY SYMPLECTIC REDUCTION: THEEXAMPLE OF TWO PARTICLES IN A CENTRALFORCE FIELDWe illustrate the separation of translational, rota-tional, and internal motions in the classical dynamics oftwo particles restricted to motion in a plane.The configuration space Q of the system is R2x R2. Coordinates on Q are pairs (rl,r2) of vectors inR2 describing the positions of the masses ml and m2,respectively, n an inertial frame. Further, we will assumethat r2 - r+R2\{O} so that the two particles do not oc-cupy the same position in configuration space. The La-grangian L on the tangent bundle TQ is given by

L(r9ki&) =fmlllhl12 + ~m21ji2112 V(llr2 - rlll>,(5.1)where we assume that the potential energy V dependsonly on the distance l/r2 - rill between massesml andm2. We apply the Legendre transformation to obtain theHamiltonian H on the cotangent bundle T*QHhr2,rwd = 2m,+z+ V(llr2-rrll). (5.2)The Euclidean group E(2) = R2 @ SO(2) acts on thissystem on T*Q by translation by and rotation about thecenter of mass of the system. In the case of identicalparticles, there is a further discrete symmetry Z2 corre-sponding to the interchange of the two particles.

F. J. Lin and J. E. Marsden: Symplecti c reduction and topology 1287

A. Separation of the translational energy from thesum of the rotational and internal energiesThe center of mass roeR2 of the system s defined bythe equation(ml + m2h=mlrl+ m2r2. (5.3)

The total linear momentum po~R2 of the system s a con-served quantity and satisfies he relationshipPo=Pl+ Pz- (5.4)

Hence, we define the relative coordinate r and the relativemomentum p:rrr2 - rl (5.5)P=P2-P1, (5.6)

and we obtain the following linear equations relating rl,r2, ~1, and PZ o rs, r, PO,and P:rl=r0 + [ml/Cm1 + m2)lr, (5.7)r2=ro - [m2/(ml + m2) r (5.8)PI =$o - 5P (5.9)

P2iPO + 5p. (5.10)We can transform from Cartesian coordinates (r1,r2) tothese coordinates (r,ro) to recoordinatize the originalphase space (rl,r2,pl,p2) as the phase space (r,ro,p,pdand then restrict to the reduced phase space (r,p) sincep. is a constant and r. is a function of time satisfying1r0(f) =rdO) + ml+m2 Pot*

This is an example of an equation describing a movingcoordinate system or, equivalently, an equation describ-ing a time-dependent holonomic constraint (cf. Ref. 17).This is the physical motivation for the first cotangentbundle reduction to follow now.An elementa of the translation group R2 acts on Q by

QP,(rz,r2) =a*(r1,r2) = (rl + a,r2 + a), (5.12)its tangent map T@, acts on TQ by

T@a(rl,r2,i&) =(~a(rl,r2),~a(rl,r2) * (Wd)= (rl + a,r2 + a,h,i2), (5.13)

and hence the cotangent lift T*@* defined by



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= h,r2,p1,p2)TW+~ w2hp2)acts on T*Q by

(5.14)

T*R,hwwd = (rl + v-2 + amd. (5.15)The correspondingmomentum map J : T*Q + R2 isdefinedby

J(w2,p~,pd PI + PZ. (5.16)We start with the original phasespaceT*Q with thecanonical ymplectic structure dr, A dpl + dr2A dp2.Thechangeof variables to the center of mass frame corre-sponds o a symplectic reduction, i.e., the quotienting ofthe inverse mage (in the original ph ase space) of the

constant otal linear momentum p. = p1 + pzby transla-tions a = [PO/(ml + m2)] et. In other words, he reducedphasespaceat p. is the quotient spaceW*Q),,=J- (Po,&,~, (5.17)

where GPOs the isotropy subgroupof p. with respect othe coadjoint action of R2 by cotangent ifts T*@, onT*Q. Further, by the cotangent bundle reduction theo-rem, one has the symplectic diffeomorphism(T*Q)po=T*(Q/Gp,,). (5.18)

Specifically, we have T*(R2~R2)p0 z r*(R2\{O}>,wherewe haveassumed 2 - rleR2\{O}. Further, the re-ducedphase pacehas he canonicalsymplectic structuredr A dp. (The assumptionof the atoms not occupying hesame ocation in configuration space s implicitly under-stood in our notation for the original phasespace.) ThereducedHamiltonian HP0on T* ( R2 {O}) satisfiesHpo(r,p)= llPl122bv2mq + m2)) + ~,,(llrll) (5.19)

with amende dpotential VP, satisfyinglIPoIl~PoWll)=V(ll4l) + 2crn,+ m2> (5.20)

In the new coordinate system, Hamiltons equationsbecomeaH,o

=w (5.21)aH,o$= --

ar (5.22)

a40 1i=apo= m+m2 Pot


(5.23)

(5.24)and r. is a cyclic coordinate .Hence he dynamics can bedescribedby the first two equations, he reducedHamil-tons equations. n other words, the dynamics drop to thereducedphasespace r,p).6. Separation of the rotational and internal energies

For a central force potential, the angular momentu mPO

Pe=xPy - YPX (5.25)is a conserved uantity. We make he following changeofvariables rom (x,y>~R~\{o} to (r$)~(O,co ) X 5 :

x=rcos 8, y=rsin 8, (5.26)to transform from Cartesian coordinates (x,y) to polarcoordinates (r,(3) to recoordinatize the original phasespace from its description in Cartesian coordinates(x,y,px,p,,) to its description in polar coordinates( r,8,p,,pe). We then restrict to the reduced phasespace(r,p,) sincepe is a constantand the time dependence f 8is given by

e(t) =e(o) + at, (5.27)wherew is the angular velocity. Again, the expression orthe eliminated coordinate in this case,e(t)] is an equa-tion describing a moving coordinate system or, equiva-lently, an equation or a time-dependent olonomic con-straint.As in the previousdiscussion or the initial reduction,an elementR4 of the group SO(2) acts on the configu-ration space 0,~ ) X S by

QRp,e) =~,&e) = he + +), (5.28)its tangent map TCJ~+acts on T( (0, CO X S) by

T~Rq(~,e,kt)) = (~Rp,w~Rp,e)~ (kh )= w + +,4), (5.29)

and hence he cotangent ift T*aR4 definedbyT*~R,(~,e,P,Pe)(~~~(r,e),(r,B))

=(r,e,P,Pe)T~~~~~(r,e),~B) (5.30)



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acts on T*(( 0, CO x 5) byT*%,(r&P,Pe) = (r# + i&P&). (5.31)

The corresponding momentum map JO : T*( (0,~ )x S) -+ R is defined by

J&&wd =pe. (5.32)The change of variables to the rotating frame corre-sponds to another reduction, specifically, the quotientingof the inverse mage [in T*( (0, co> X S)] of the constanttotal angular momentumpe = Iw, where I is the momentof inertia, by rotations. The new reduced phase space sthe quotient spaceT*t(Qa ) XS),,= J, (pe)/GPB, (5.33)

where GPOs the isotropy subgroup of pe with respect tothe coadjoint action of SO(2) ( = S) by cotangent liftsT*aRb on T*( (0, cu X S). In fact, again by the cotan-gent bundle reduction theorem, one has

r*((O,co)XS)~o~T*(O,co). (5.34)The reduced phase space has the canonical symplecticstructure dr A dp,The reduced Hamiltonian H,,o,ppen T* (0, CO satisfies

HPo.pe(A = P:Wqm2/h + m2)) + VPo,Pe() (5.35)where the amended potential Vpwpes given by

VP*P~(r)=vPO(Ilrll)2(m,m,/($m2))rz= V(Ilrll) + /IPoll2(ml+ m2>

P2e+ 2(mlm2/(ml + m2))r * (5.36)After two reductions, the kinetic energy of the originalHamiltonian H appears as the sum of three decoupledterms: the translational kinetic energy llpol12/2(ml+ m2) corresponding to the total linear momentum, theoverall rotational kinetic energy

Pi2(mpn2/(ml + m2))r *

corresponding to the total angular momentum, and thevibrational kinetic energy

Pf

F. J. Lin and J. E. Marsden: Symplectic reduction and topology 1289

2hm2/h + m2)) *There is no Coriolis term in the reduced Hamiltonian.Hamiltons equations in the newest reduced systembecome

Pr= - ar 9a4&9(I=-= 1ape (mlm2/(ml + m2))r *Pe=w (5.39)

(5.37)

(5.38)

aHP,,P,Ji*= - 7=0 (5.40)and 8 is a cyclic coordinate. Again, the dynamics can bedescribed by the first two equations, and the dynamicshas dropped to the reduced phasespace (r,p,). This is anespecially simple case of a general construction.C. Reduced phase spaces and Hamiltons equations

The preceding two-step procedure exhibits two appli-cations of the symplectic reduction theorem and the con-sequent heorem that the (reduced) dynamics is describ-able on the reduced phasespace.Further, it is an exampleof a reduction by stages, in which the phase space isreduced by a semidirect product group. In this case, hesemidirect product group is the Euclidean group,E( 2) = R2@ SO( 2). We now present he coordinate-freestatements ollowed by their corresponding coordinatizedforms.Symplectic reduction theorem:18 et (P,Q) be a sym-plectic manifold on which there is a Hamiltonian leftaction of a Lie group G with equivariant momentum mapJ : P-t .9 *. Assume that ,UE~* is a regular value of J andthat the isotropy subgroup GPunder the coadjoint actionacts freely and properly on J- t(p). Then the reducedphasespacePcL= J - (p)/G, is a manifold with symplec-tic form J& satisfying

7Tpp = p, (5.41)where rP : J-(p) -+ Pp is the canonical projection andiF : J-(p) + P is inclusion.Theorem on dynamics on reduced phasespaces:18 etH : T*Q-+R be a G-invariant Hamiltonian, i.e.,HOT*@* = H for all gEG. The flow Ft of the Hamiltonianvector field X, leaves he set J - l(p) invariant and com-mutes with the (induced) G,-action on J - (p), so itinduces a Hamiltonian flow fl on (T*Q), by .R,oF~



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= flop. The reduced Hamiltonian HP : ( T*Q), + R isdefined (as above) by Hporp = Ho$,. Further, the Hamil-tonian vector fields X, and XHPare related by the canon-ical projection rfP.Next, we apply these coordinate-free ideas to derivethe corresponding results in curvilinear coordinates: Let(P,R) be a symplectic manifold and H : P-R be a givenC function cal led the Hamiltonian. Define the Hamil-tonian vector field X, : P- TP by the conditioni,,,R = dH, (5.42)

where x&l is the interior product of XH and fl, and dHis the exterior derivative of H. Let (ql ,..., 8,pl ,...,pN) becanonical coordinates for n. ThenN

a= C dqAdpii=l (5.43)and

xH=(g,-$). (5.44)Further, an integral curve of X, is a solution (q(t) ,p( t))of Hamiltons equations

aHi=&jj, i= I,2 ,..., V, (5.45)aHii= - g, i= I,2 ,..., N. (5.46)

For the general N-body problem, the reduced Hamil-tonian HP on the reduced phase space (q,,pp)eP,= J - t (p)/G, can be determined by employing an ex-plicit generalization of the momentum map J defined inthe reduced energy-momentum method, as follows:H,&,)=fl~& + V&i), (5.47)

where g is the relevant fundamental metric tensor,pp isthe shifted momentumP/i =P - PJ,

VP is the amended potential(5.48)

V&p) = V(qJ + -b-f- (qJp9 (5.49)J is now the momentum map corresponding to the rele-vant total momentum of the N-body system, and.Y( qp) is the relevant mass actor. For the first reduction,p is the total linear momentum, J is the momentum mapcorresponding to the total linear momentum, fl is thetotal mass,p is the total momentum, and pJ is the total

linear momentum (cf. Sections V B and V A). For thesecond eduction, p is the total angular momentum, J isthe momentum map corresponding to the total angularmomentum, Y is the instantaneousmoment of inertia, pis the total momentum minus the total linear momentum,and pJ is the total angular momentum (cf. Sects. 111Band V B).Define the reduced Hamiltonian vector field X, :PP,, -+ TPpbyixH fir = dH,,P (5.50)

where R, and H,, are determined by the symplectic re-duction theorem. Further, HP can be explicitly deter-mined as just described. Then an integral curve(qp( t),p,( t)) of XHcl s a solution of the reduced Hamil-tons equations

aHpQI=apri) i= 1,2 ...,N - 1,a4ppi=w i= 1,2 ...,N - 1.

1290 F. J. Lin and J. E. Marsden: Symplecti c reduction and topology

(5.51)

As shown by the examplesof the dynamics relevantto atomic clusters and the dynamics relevant to atom-atom differential cross sections described n the previoussections, these techniques apply both to the dynamics ofbound molecular systemsand to the dynamics of molec-ular collisions. A framework for describing two-body mo-lecular dynamics and including the dynamics of a(bound) diatomic molecule and the dynamics of atom-atom scattering as special cases s described in the nextsection.Vi. GLOBAL TOPOLOGY OF THE ENERGY-MOMENTUM MAP FOR A MOLECULARINTERACTION POTENTIAL

We follow the approach of Smale,29who examinedthe Kepler problem and made a global topological studyof the energy-momentum map (H x Jo) Te : TQ + R2defined on the tangent bundle TQ. We will consider thecorresponding energy-momentum map (H X Jo) F*Q :7-Q-+ R2 defined on the cotangent bundle T*Q. Thesemaps are related by the Legendre transformation FL :TQ-+ T*Q; specifically,

(HXJ) TQ= (HxJ) pQm. (6.1)For simplici ty of notation, in the following we will omitthe subscript r*Q on the energy-momentum map. Wenow consider the energy-momentum map defined on thecotangent bundle PQ and start with the original phasespaceand make a global topological study of the energy-momentum map H X J,g X J : T*(M2 X R2) + R



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X R X R2 from the original phase space to the spacewhere the energy map Hand the momentum mapsJe andJ take their values E, p& and po, respectively. Here wehave written M2 for R2\{O}. We consider the inverseimages (or integral manifolds or fibers) IE,pBpO (HXJex J)-(B ,p&po) with ( E,papo)cR4. Then the bifur-cation set B in R4 is defined to be the set of points inR4 over which the energy-momentum map fails (in thedifferentiable sense) to be locally trivial. The bifurcationset includes the critical values of the energy-momentummap, and on the bifurcation set the topological type ofI E,pepohanges.The dynamical system on T*(M2 x R2)induces a dynamical sys tem on the reduced ntegral man-

ifold IE,po = IE,papo /GPO. his is a consequence f the the-orem on the dynamics on reduced phase spaces.* Theisotropy subgroup GPO f p. is R2 for all values ofpoeR . Hence, we have ,#& = I E,pe R2 or all valuesofpo, and the topologtcal type of IE,pgpOs independent ofPO.The problem has now been simplified to the study ofthe reduced Integral manrfold IE,pe and the (reduced)energy-momentum map H x Jo : T*M2 -+ R x R. Wecan write the bifurcation set B as B = Zc C R2. We nowchoose the potential V to be a molecular interaction po-tential. To be specific, we choose V to be the Morse po-tential in dimensionless units with a* = 5. Then the bi-furcation set Ze in RX R is given by

&J=&U22U23,with

(6.2)

~l={(E,pe)~R~:V~~(r~i~)=E and O< lpel Qem,,19(6.3)

2,={(E,p,+R2:E=O}, (6.4)83=C(E,pe)ER2:VPg(rmax) E), (6.5)

where r,,,i, and (when it exists) r,,,,, are functions of pedefined as follows:rmin=min{~R:V&(r)=O, V;O(r) >O, and r< CO}r>0 (6.6)rmax= min {dkV&(r) =0, F$(r) ~0, and r< CO},

r> min (6.7)and Pe,,,,, s determined numerically to be 1.413 for thecase a* = 5. The bifurcation set Z. divides theE,pBplane into four regions (including one region with

two components) with integral manifolds I&,g of differ-ent topological types. The bifurcation set s plotted in Fig.1. The real line R acts on lE,peas the dynamical groupby restricting the dynamics on T*M2 because he energyH is an integral. The group SO(2) acts on M2 by rota-tions and induces an action by cotangent lifts on T*M2.The induced action of SO(2) on T*M2 leaves all of thefunctions V, H, and Je invariant as well as the integralmanifolds IE,#O. Th e angular momentum Jo is also anintegral.) The (induced) action of SO( 2) commutes withthe action of R, so the action of the product group SO( 2)XR is defined on T*M2 and on IE,p, for each(E,pokR2.Proposition: Let M2 = R2 (0). Let V be the Morsepotential in dimensionless units with a* = 5. Let(E,pe)~R2 and p&O. An integral manifold IE,*,C T*M2 is the (not necessarilydisjoint) union of homo-geneous spaces of SO (2) XR, i.e., spaces of the form(SO(2) XR)/G, where i = 1, 2 ...,n.Prooj Given (E,pe)ER2 and p&O, SO(2) XR actstransitively on each component of IE,p, C T*M2. We thenapply an elementary heorem characterizing transitive ac-tions (see, e.g., Ref. 40, p. 75): Let the group G acttransitively on a set S and let Gibe the isotropy subgroupof Xi for x,&i . Then the action of G on S is equivalent tothe action of G on the coset space G/G, (i = 1, 2 ..., n).Letting i label the ComponentsOf IE,#e,we get IE,po =U ;= *G/G,. Q.E.D.Thus a component of an integral manifold is of one ofthe following topological types depending on the value of(E,pe)eR2. In order of increasing value of the energy E,the possib le topological types of the integral manifoldsIE,pO n the different regions of the (E,pe)-plane are (i)the empty set (when n = 0) :I -Oe 4on Rl=C(E7pe)ER2:E< VPe(rmin)


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2.0

1.0

P 0.0

-1.0

-2.0

FIG. 1. The bifurcation set POin the Q+plane for the case ofa Morse potential in dimension-less units with a* = 5.

. Sigma-3ipma-.-.-.-.- Sigma-2

,=s' xs'

on R~={(E,P~)ER* : Vp,(rmin>


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a scattering trajectory. This is consistent with the classi-fication scheme of Goldstein.t4More specifically, scattering trajectories can be clas-sified as grazing, (eventually) orbiting, or rebounding. Agrazing trajectory lies in the noncompact component ofan integral manifold of type (vi). An (eventually) orbit-ing trajectory initially lies on the cylinder and eventuallylies on the circle in the integral manifold corresponding o&. A rebounding trajectory lies in the noncompact inte-gral manifold of type (viii). This classification scheme sconsistent with that familiar to molecular dynamicists(see, for example, Ref. 3, Fig. 12; Ref. 6, p. 556; or Ref.7, p. 55; who each discussed he dynamics in terms of theLennard-Jones potential). Further, our bifurcation setshown in Fig. 1 for the ( E,pe)-plane for a Morse potentialis consistent with the description of the topology of scat-tering in the first quadrant of the (p&E)-plane for theLennard-Jones potential as plotted by Ford and Wheeler3Fig. 13) in terms of the dimensionlessangular momen-tum and the dimensionlessenergy.Finally, we restate the above results in terms of re-duced phase spaces:A leaf IE,P/GPO f the reduced spaceJP, (ps)/GPB is the (not necessarily disjoint) union ofcoset spacesof the form ((SO(2) X R)/GXi)/S1, i.e., apoint, a circle, or a line.For a fixed value of pe, the reduced space [(JX Je) - (po,pe) /GPO]/GPes foliated by leaves,which areparameterized by E and are the union of components ofthe following topological types: a point, a circle, or a line.(i) If E < VPe(r,in), then the leaf of the reducedphase space s the empty set.(ii) If E = VPO(rmin), then the leaf of the reducedphase space s topologically a point.(iii) If V,,(r,i,) < E < 0, then the leaf of the re-duced phase space s topologically a circle.(iv) If E = 0 and VP,( rmax) > 0, then the leaf of thereduced phasespace s topologically the disjoint union ofa circle and a point.(v) If 0 < E < V,,,(r,,), then the leaf of the re-duced phasespace s topologically the disjoint union of acircle and a line.(vi) If E = V,,(r,,), then the leaf of the reducedphase space s topologically the union of a circle and atangent line.(vii) If E > Vpg(rmax), then the leaf of the reducedphase space s topologically a line.A leaf topologically equivalent to (a) a point corre-sponds to a rigid diatomic molecule in case (ii) and totwo stationary distantly separatedatoms in case (iv), (b)a circle corresponds to the phase space trajectory of avibrating diatom in cases (iii) and (v), (c) a line corre-sponds to the phase space rajectory of an elastic atom-atom collision: a grazing collision in case (v), an orbiting

collision in case (vi), and a rebounding collision in case(vii).VII. CONCLUDING REMARKS

For molecular N-body systemswith N)3, the frame-work outlined above provides a starting point for the dis-cussion of the topology of the dynamics, which may againbe classified n terms of bound or scattering trajectories.For the molecular three-body problem (we chooseN = 3 for simplicity), the scattering trajectories can beeither reactive, i.e.,A+BC+AB+C (7.1)

orA + BC+AC+ B,

or nonreactive, i.e.,(7.2)

A + BC+A + BC. (7.3)Within each of these categories, he scattering trajectorycan correspond to a collision that is either direct or com-plex (sometimes also called compound).8 In a direct re-active collision, the reaction takes place in a time shorterthan one rotational period of the combined system, i.e.,the time required for the reactants to rotate about eachother. In a complex reactive collision, the reaction pro-ceedsvia a long-li ved intermediate, i.e., a complex, whichexists for more than one classical rotational period.Hence, we make the following observations: Directcollisions generalize he scattering trajectories in the mo-lecular two-body systems hat are grazing trajectories orrebounding trajectories; complex collisions generalize hescattering trajectories in the two-body systems hat con-tain orbiting trajectories; and complex collisions can alsobe considered o be a generalization of bound trajectories.Application of the results regarding the decoupli ng ofthe reduced Hamiltonian at the linear level near a relativeequilibrium imply that the dynamics of a molecular N-body system can be drastically simplified at the linearlevel near a relative equilibrium. The phase curves in theoriginal phase space which project to equi librium posi-tions in the reduced system on the reduced phase spaceare called the relative equilibria of the original system.The results on simplifying the dynamics require the tra-jectory to be near a relative equilibrium, so we observethat physically the simplification of the molecular dynam-ics at the linear level requires the trajectory to be either abound trajectory (for N)2), an orbiting scattering tra-jectory (for N = 2), or a complex scattering trajectory(for N)3). (These conditions on the scattering trajecto-ries are not necessarilysufficient.)The geometric procedure outlined previously aboveprovides a prescription for deriving reduced phase spaces

F. J. Lin and J. E. Marsden: Symplectic reduction and topology i293



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that simplify the dynamics at the l inear level near a rel-ative equilibrium. It remains o extend his simplificationof t he dynamics to points other than relative equilibria.The reduction of the phasespace or a n N-body sys-tem with N>3 in R3 is more complicated than that dis-cussedabove or two-body systems n R* wh en the totalangular momentum J is nonzero.This paper has presented wo physical examples ofmolecular N-body systems for which symplectic reduc-tion is relevant. In addition, for the case of N = 2, wehave presented detailed, unified treatment of the planardynamics. We showed hat the two examplesof the ro-tating, vibrating diatomic molecu le and atom-atom elas-tic scattering are systemswhose nternal dynamics ie onreducedmanifolds of different topological types. This de-scription was unified by examinationof the bifurcation setin the energy, angular momentum plane. Further, theabovediscussion outlines the relevanceof this app roachfor the description of t he dynamics of larger N-body sys-tems.

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F. J. Lin and J. E. Marsden- Symplectic reduction and topology for applications in classical molecular dynamics