Question
Verify the virial theorem for the case of periodic motion of two particles gravitationally bound to one another.
Step 1
For a system of particles interacting through gravitational forces, the virial theorem is given by: \[ 2 \langle T \rangle + \langle V \rangle = 0 \] Show more…
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8.17 ** If you did Problem 4.41 you met the virial theorem for a circular orbit of a particle in a central force with U = kr^n. Here is a more general form of the theorem that applies to any periodic orbit of a particle. (a) Find the time derivative of the quantity G = r · p and, by integrating from time 0 to t, show that (G(t) - G(0)) / t = 2⟨T⟩ + ⟨F · r⟩ where F is the net force on the particle and ⟨f⟩ denotes the average over time of any quantity f. (b) Explain why, if the particle’s orbit is periodic and if we make t sufficiently large, we can make the left-hand side of this equation as small as we please. That is, the left side approaches zero as t → ∞. (c) Use this result to prove that if F comes from the potential energy U = kr^n, then ⟨T⟩ = n⟨U⟩/2, if now ⟨f⟩ denotes the time average over a very long time.
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