NON-ASSESSED MAPLE COURSEWORK 3

(Version 1.0, 24 March 2000)

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.