A particle of reduced mass and angular momentum moves in the central potential V(r) =…

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Gravitation and central-force motion · Gravitation and central-force motion

Action and Symmetry, Ch The Kepler problem redone: reduction, quadrature and the Runge-Lenz vector

Problem

A particle of reduced mass

\mu

and angular momentum

\ell

moves in the central potential

V(r) = -\frac{k}{r} - \frac{\gamma_2}{r^2}, \qquad k \gt 0,\ \gamma_2 \gt 0,

an inverse-square attraction with an extra attractive

1/r^2

term, with

2\mu\gamma_2 \lt \ell^2

so that the effective potential is still repulsive at short range and the motion is bound between a pericentre and an apocentre. ((a)) Write the effective potential

V_{\rm eff}(r)

and show it has exactly the form of the inverse-square effective potential with the term

\ell^2/2\mu r^2

replaced by

\ell'^{\,2}/2\mu r^2

, where

\ell' = \sqrt{\ell^2 - 2\mu\gamma_2}

. Carry the orbit quadrature through (substitute

u=1/r

and complete the square, exactly as for

V=-k/r

) to show that the orbit is the precessing conic

r = \frac{p'}{1+e'\cos[\kappa(\varphi-\varphi_1)]}, \qquad \kappa = \frac{\ell'}{\ell},

for constants

p'

e'

\varphi_1

fixed by

E

\ell

\mu

k

, and give

p'

. ((b)) Hence show that the apsidal angle is

\Delta\varphi = \frac{\pi}{\sqrt{1-2\mu\gamma_2/\ell^2}},

and evaluate it in degrees for

2\mu\gamma_2/\ell^2 = 0.19

. ((c)) Define

\mathbf{A}_{\rm LRL} = \mathbf{p}\times\mathbf{L} - \mu k\hat{\mathbf{r}}

, the Runge-Lenz vector built from the inverse-square part of the force alone. The full radial force is

f(r) = -k/r^2 - 2\gamma_2/r^3

. Using

\dot{\mathbf p} = f(r)\hat{\mathbf r}

and the identity

\hat{\mathbf r}\times\mathbf L = -\mu r^2\dot{\hat{\mathbf r}}

, show that

\dot{\mathbf A}_{\rm LRL} = \frac{2\mu\gamma_2}{r}\dot{\hat{\mathbf r}},

and say why this is consistent with the precession found in part (b).

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