MAPLE WORK 3

(Version 2.0, 21/02/00)

This coursework requires some hand calculation, which should be completed in the usual way. All the Maple tasks have been explicitly indicated.

1. Show that in the two-body problem the angular momentum with respect to the centre of the field is conserved.

[MAPLE TASK] Verify this result numerically by plotting graphs of the angular momentum versus time for both a bounded and unbounded orbit.

2(a). (MAPLE TASK) Set up the differential equations in Maple for motion under the force law, f(r) = r^{3}. For the initial condition

r(0) = 1.0, θ(0) = 0.0, ṙ(0) = 0.0, θ̇(0) = 1.0,

generate a numerical solution and plot it in the (r, ṙ) and (x, y) planes.

2(b). By putting r̈ = 0 (which implies for this system that θ̇ is a constant), find the circular orbit of constant radius. Generate the numerical solution corresponding to this orbit when r(0) = 1.0, and plot a graph of r against time t to verify that r is constant. In addition, plot a graph of the orbit in the (x, y) plane to verify that it is circular.