Dynamics | Part IA, 2002

Write down the equations of motion for a system of nn gravitating point particles with masses mim_{i} and position vectors xi=xi(t),i=1,2,,n\mathbf{x}_{i}=\mathbf{x}_{i}(t), i=1,2, \ldots, n.

Assume that xi=t2/3ai\mathbf{x}_{i}=t^{2 / 3} \mathbf{a}_{i}, where the vectors ai\mathbf{a}_{i} are independent of time tt. Obtain a system of equations for the vectors ai\mathbf{a}_{i} which does not involve the time variable tt.

Show that the constant vectors ai\mathbf{a}_{i} must be located at stationary points of the function

i19miaiai+12jijGmimjaiaj\sum_{i} \frac{1}{9} m_{i} \mathbf{a}_{i} \cdot \mathbf{a}_{i}+\frac{1}{2} \sum_{j} \sum_{i \neq j} \frac{G m_{i} m_{j}}{\left|\mathbf{a}_{i}-\mathbf{a}_{j}\right|}

Show that for this system, the total angular momentum about the origin and the total momentum both vanish. What is the angular momentum about any other point?

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