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The Two-Body Problem - UC Santa Barbara

The Two-Body ProblemIn the previous lecture, we discussed a variety of conclusions we could makeabout the motion of an arbitrary collection of particles, subject only to a fewrestrictions. Today, we will consider a much simpler, very well-known problemin physics - an isolated system of two particles which interact through a centralpotential. This model is often referred to simply as thetwo- body the case of only two particles, our equations of motion reduce simply tom1 r1=F21;m2 r2=F12(1)A famous example of such a system is of course given by Newton s Law ofGravitation, where the two particles interact through a potential energy givenbyU12(|r1 r2|) =U21(|r2 r1|) =Gm1m2|r1 r2|2,(2)whereGis Newton s constant,G= 10 11N m2/kg2.

the functional form of r(t), we can nd the angular coordinate by simply inte-grating (t) = 0 + l m Z t 0 dt0 r2 (t0): (35) The combination of linear momentum, angular momentum, and energy con-servation in our system has led to a dramatic simpli cation - a system of two particles in three-dimensional space has been reduced to a problem of nding

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