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speed of the wave. Indeed, larger waves travel faster—a typical observation in many
wave propagation phenomena. Note also that the amplitude function,
represents propagation at the speed which is precisely the group speed, see
§4.4. The initial data for this solution must take the form

where k is a given constant (rather than appear in the more general form sin
and for a suitable amplitude function.

This concludes the set of examples that have been taken from a rather broad spectrum
of simple mechanical and electrical systems. We will now consider the more specialised
branch of classical mechanics.

5.2 CELESTIAL MECHANICS
We present three typical problems that arise from planetary, or related, motions: E5.8
The Einstein equation (for Mercury); E5.9 Planetary rings; E5.10 Slow decay of a
satellite orbit.

E5.8 The Einstein equation (for Mercury)
Classical Newtonian (Keplerian) mechanics leads to an equation for a single planet
around a sun of the form

where is the polar angle of the orbit, u is inversely proportional to the radial coordinate
of the orbit and h measures the angular momentum of the planet. However, when a
correction based on Einstein's theory of gravitation is added, the equation becomes

where is a small parameter (about for Mercury, the planet for which the
equation was first introduced). The plan is to find an asymptotic solution of equation
(5.60), using the method of multiple scales (cf. E4.2), subject to

(It happens that equation (5.60) can be integrated as it stands, in terms of Jacobian
elliptic functions, but this tends to obscure the character of the solution and is therefore;
hardly worth the effort since is so small.)