Central Forces and Orbits

Abstract

Central forces are of considerable importance in both classical and quantum mechanics so we briefly review their properties. In this chapter we deal with two bodies of masses m1 and m2. We ignore the motion of the center-of-mass so that only the relative interparticle coordinates r and θ are treated, as will be discussed below. The effective mass for motion in \(\left (r,\theta \right )\) is the reduced mass μ = m1m2∕(m1+m2). The central forces , \(\boldsymbol{F}\left (r\right )\), and the potentials that produce them, \(U\left (r\right )\), depend only upon the spherical coordinate r. The angular momentum vector \(\boldsymbol{L} =\boldsymbol{ r} \times \boldsymbol{ p}\), where \(\boldsymbol{r}\) is the position vector and \(\boldsymbol{p}\) is the linear momentum vector, is conserved for all central potentials. This will be proven in Problem 1 of this chapter. Because \(\boldsymbol{L}\) is conserved, the motion under the action of a central force takes place in a plane so that only two dimensions are required to describe it. Plane polar coordinates \(\left (r,\theta \right )\) are almost always used in central force problems (see Appendix B.2). When written in polar coordinates, the total energy of a system under the influence of a central potential contains an angular momentum term, \(\ell^{2}/\left (2\mu r^{2}\right )\) where l is the magnitude of the angular momentum \(\boldsymbol{L}\). For constant l this term acts as an additional potential, which is usually referred to as the centrifugal potential. Thus, we may define the “effective potential ” as $$\displaystyle{ U_{\text{eff}}\left (r\right ) = U\left (r\right ) + \frac{\ell^{2}} {2\mu r^{2}} }$$