CENTRAL FORCE PROBLEMS - Reed College

1 4 CENTRAL FORCE is where it all began. Newton sMathematical Principles ofNatural Philosophy( )waswrittenandpublishedattheinsistanceof youngEnmundHalley( )sothattheworldatlargemightbecomeacquain tedwith the general theory of motion that had permitted Newton to assure Halley(on occasion of the comet of ) that comets trace elliptical orbits (andthat in permitted Halley to identify that comet now known as Halley scomet withthecometsof and ,andtopredictitsreturnin ). Itwaswitharemarkablydetailedbutradically innovativeaccountofthequantumtheory of hydrogen that Schr odinger signaled his invention of wave mechanics,a comprehensive quantum theory of motion-in-general. In both instances,description of the general theory was preceded by demonstration of the successof its application to the problem of two bodies in CENTRAL ButIwouldliketo preface our work with some remarks intended to place that problem in itslarger often yield, as classical physicists, to an unexamined predisposition tosuppose thatallinteractions are necessarily 2-body interactions.

2 To describe asystem of interacting particles we find it natural to writemi xxxi=FFFi+ j FFFijFFFij= FORCE onithbyjth1It is interesting to note that thePrincipiabegins with a series of eightDefinitions, of which the last four speak about centripetal FORCE . 2 CENTRAL FORCE problemsand to writeFFFij= FFFji: action = reaction ( )asanexpressionofNewton s3rdLaw. Butitisentirelypossibletocontemplate3-bo dy forcesFFFi,jk= FORCE onjthdue to membership in i, j, k and to consider it to be a requirement of an enlarged 3rdLaw that2 FFFi,jk+FFFj,ki+FFFk,ij=000( )Forthedynamicsofasystemofinteractingpar ticlestoadmitofLagrangianformulation the interparticle forces must be conservative (derivable from apotential).

3 For a 2-particle system we would introduceUinteraction(xxx1,xxx2) and,to achieve compliance with ( ), require( 1+ 2)Uinteraction(xxx1,xxx2)=000( )For a 3-particle system we introduceUinteraction(xxx1,xxx2,xxx3) and require( 1+ 2+ 3)Uinteraction(xxx1,xxx2,xxx3)= ( )The multi-particle interaction idea finds natural accommodation within suchascheme,butthe3rdLawisseentoimposeas evereconstraintuponthedesignof the interaction expect the interparticle FORCE system to be insensitive to grosstranslation of the particle population:Uinteraction(xxx1+aaa,xxx2+aa a,..,xxxn+aaa)=Uinteraction(xxx1,xxx2,.. ,xxxn)This, pretty evidently, requires that the interaction potential depend upon itsarguments only through their differencesrrrij=xxxi rrrjof which there are an antisymmetric 1,n : total ofN=12n(n 1) suchrrr s2 Though illustrated here as it pertains to 3-body forces, the idea extendsstraightforwardly ton-body forces, but the action/reaction language seemsin that context to lose some of its naturalness.

4 For discussion, seeclassicalmechanics( / ), page insensitivityVinteraction(Rrrr12,Rrrr13, ..)=Vinteraction(rrr12,rrr13,..)prettyev identlyrequiresthatthetranslationallyinv ariantinteractionpotentialdepend upon its argumentsrrrijonly through their dot productsrij,kl=rrrij rrrkl=xxxi xxxk xxxi xxxl xxxj xxxk+xxxj xxxlof which there is a symmetric array with a total of12N(N+1)=12(n2 n+2)(n 1) (n2 n+2)(n 1)n0162155120 Table1:Number of arguments that can appear in a translationallyand rotationally invariant potential that describesn-body the casen=2 one hasU(r12,12)=U([rrr1 rrr2] [rrr1 rrr2]) giving 1U=+2U (r12,12) [rrr1 rrr2] 2U= 2U (r12,12) [rrr1 rrr2]whence 1U+ 2U=000 Weconcludethatconservative interaction forces, if translationally/rotationallyinvariant, are automatically CENTRAL , automatically conform to Newton s3rdLaw.

5 Inthecasen=3onehasU(r12,12,r12,13,r12,23 ,r13,13,r13,23,r23,23)and consigning the computational labor toMathematica finds that compliancewith the (extended formulation of) the 3rdLaw is again automatic: 1U+ 2U+ 3U=000I am satisfied, even in the absence of explicit proof, that a similar result holdsin every (sun/comet,sun/planet, earth/moon) of what he reasonably construed to be instances ofthe 2-body problem , though it was obvious that they became so by dismissing4 CENTRAL FORCE problemsFigure1:At left:a classical scattering process into which twoparticles enter, and from which(after some action-at-a-distance hasgone on)two particles depart. Bound interaction can be thought ofas an endless sequence of scattering processes.

6 At right:mediatedinteraction, as contemplated in relativistic includethe theory of elementary particles. Primitive scattering events arelocal and have not four legs but three:one particle enters and twoemerge, else two enter and one emerges. In the figure the time axisruns .spectatorbodiesas irrelevant (atleastinleadingapproximation3). Thatonly2-body interactions contribute to the dynamics of celestial many-body systemsis a proposition enshrined in Newton suniversal law of gravitational interaction,which became the model for many-body interactions ofalltypes. The implicitclaim here, in the language we have adopted, is that the interaction potentialsencountered in Nature possess the specialized structureUinteraction(xxx1,xxx2.)

7 ,xxxn)= pairsUij(rij)whererij rij,ij= rrrij rrrijand from which arguments of the formrij,kl(three or more indices distinct) are absent. To create a many-body celestialmechanics Newton would, in effect, have us setUij(rij)= Gmimj1rijwhenceFFFij= iUij= Gmimj1r2ij rrrij3 Halley, in constructing his predicted date ( ) of cometary return, tookinto account a close encounter with to the equivalent 1-body problem : Jacobi coordinates5where, it will be recalled,rrrij xxxi xxxj= rrrjiis directedxxxi xxxjandFFFijrefers to the forceimpressed theoryofcollisions, the CENTRAL FORCE problem our present concern was conceived by Newtonto involveinstantaneous action at a distance, a concept to which many ofhis contemporaries (especially in Europe) took philosophical exception,4andconcerning his use of which Newton himself appears to have been defensivelyapologetic.

8 He insisted that he did not philosophize, was content simplyto the pertinence of his calculations was beyond in the 20thCentury action-at-a-distance ran afoul conceptually, if notunder ordinary circumstances practically of the Principle of Relativity, withits enforced abandonment of the concept of distant simultaneity. Physicistsfound themselvesforcedto adopt the view thatall interaction is local, andall remote action mediated, whether by fields or by real/virtual particles. SeeFigure1and its caption for remarks concerning this major conceptual Reduction of the 2-body problem to the equivalent 1-body to be the case that particlesm1andm2are subject to no forces except forthe conservative CENTRAL forces which they exert upon each other.

9 Proceedingin reference to a Cartesian inertial frame,5we writem1 xxx1= 1U (xxx1 xxx2) (xxx1 xxx2) m2 xxx2= 2U (xxx1 xxx2) (xxx1 xxx2) (3)A change of variables renders this system of equations much more amenable tosolution. Writingm1xxx1+m2xxx2=(m1+m2)XXXxxx1 xxx2=RRR( )we havexxx1=XXX+m2m1+m2 RRRxxx2=XXX m1m1+m2 RRRwhencerrr1=+m2m1+m2 RRRrrr2= m1m1+m2 RRR( )and the equations (3) decouple:M XXX=000( ) RRR= 1m1+1m2 U(R)( )Equation ( ) says simply that in the absence of externally impressed forces4 They held to the so-called Principle of Contiguity, according to whichobjects interact only by would be impossible to talk about the dynamics of two bodies in aworld that containsonlythe two bodies.

10 The subtle presence of a universefull of spectator bodies appears to be necessary to lend physical meaning tothe inertial frame was emphasized first by E. Mach, and later byA. S. FORCE problemsm1rrr1xxx1rrr2m2 XXXxxx2 Figure2:Coordinatesxxxiposition the particlesmiwith respect toan inertial frame,XXXlocates the center of mass of the 2-body system,vectorsrrridescribe particle position relative to the center of mass. RRRXXXF igure3:Representation of the equivalent one-body motion of the center of mass is unaccelerated. Equation ( ) says that thevectorRRR xxx1 xxx2=rrr1 rrr2moves as though it referred to the motion of aparticle of reduced mass 1 =1m1+1m2 =m1m2m1+m2(6)Reduction to the equivalent 1-body problem : Jacobi coordinates7in an impressed CENTRAL FORCE fieldFFF(RRR)= U(R)= U (R) RRR(7) = Gm1m2R2 RRRin the gravitational caseFigure3illustrates the sense in which the one-body problem posed by ( )is equivalent to the two-body problem from which it :6 Suppose vectors{xxx1,xxx2.}