Dynamics | Part IA, 2003

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

The particles undergo a motion for which xi(t)=a(t)ai\mathbf{x}_{i}(t)=a(t) \mathbf{a}_{i}, where the vectors ai\mathbf{a}_{i} are independent of time tt. Show that the equations of motion will be satisfied as long as the function a(t)a(t) satisfies


where Λ\Lambda is a constant and the vectors ai\mathbf{a}_{i} satisfy

Λmiai=Gi=jiGmimj(aiaj)aiaj3\Lambda m_{i} \mathbf{a}_{i}=\mathbf{G}_{i}=\sum_{j \neq i} \frac{G m_{i} m_{j}\left(\mathbf{a}_{i}-\mathbf{a}_{j}\right)}{\left|\mathbf{a}_{i}-\mathbf{a}_{j}\right|^{3}}

Show that ()(*) has as first integral


where kk is another constant. Show that

Gi=iW\mathbf{G}_{i}=\nabla_{i} W

where i\boldsymbol{\nabla}_{i} is the gradient operator with respect to ai\mathbf{a}_{i} and

W=ij<iGmimjaiaj.W=-\sum_{i} \sum_{j<i} \frac{G m_{i} m_{j}}{\left|\mathbf{a}_{i}-\mathbf{a}_{j}\right|} .

Using Euler's theorem for homogeneous functions (see below), or otherwise, deduce that

iaiGi=W.\sum_{i} \mathbf{a}_{i} \cdot \mathbf{G}_{i}=-W .

Hence show that all solutions of ()(* *) satisfy

ΛI=W\Lambda I=-W


I=imiai2I=\sum_{i} m_{i} \mathbf{a}_{i}^{2}

Deduce that Λ\Lambda must be positive and that the total kinetic energy plus potential energy of the system of particles is equal to k2I\frac{k}{2} I.

[Euler's theorem states that if

f(λx,λy,λz,)=λpf(x,y,z,)f(\lambda x, \lambda y, \lambda z, \ldots)=\lambda^{p} f(x, y, z, \ldots)


xfx+yfy+zfz+=pf.]\left.x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}+z \frac{\partial f}{\partial z}+\ldots=p f .\right]

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