Classical Mechanics (Kibble & Berkshire) · Problem 7.28

Non-linear oscillations and phase space — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · computer-assisted-problems

Problem

See the advance of the perihelion of Mercury It is possible to 'see' the advance of the perihelion of Mercury predicted by general relativity by direct numerical solution. Take Einstein's path equation (see Problem 7.13) in the dimensionless form

\frac{d^2\upsilon}{d\theta^2} + \upsilon = \frac{1}{1-e^2} + \eta\upsilon^2,

where

\upsilon = au

. Here

a

and

e

are the semi-major axis and eccentricity of the non-relativistic elliptic orbit and

\eta = 3MG/ac^2

is a small dimensionless parameter. For the orbit of Mercury,

\eta = 2.3 \times 10^{-7}

approximately. Solve this equation numerically with the initial conditions

r = a(1+e)

and

\dot{r} = 0

when

\theta = 0

; this makes

\theta = 0

an aphelion of the orbit. To make the precession easy to see, use a fairly eccentric ellipse and take

\eta

to be about 0.005, which speeds up the precession by a factor of more than

10^4

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