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Both involve some 'averaging' over energies (kinetic and potential) and make some prediction about their mean values. As far as the least action principles, one could think of them as saying that the actual path is one that makes an equipartition between the two kinds of energies...

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I don't have the chops to properly evaluate it, but The virial theorem for action-governed theories looks promising. – dfan Jul 26 '12 at 20:44
up vote 13 down vote

As that lovely article linked by dfan says the virial theorem comes from varying the action $S[x]$ by $x\rightarrow(1+\epsilon)x$

$$\frac{1}{T}\delta S = \frac{1}{T}\epsilon\int_{0}^{T} dt\{m\dot{x}^2 -x\frac{\partial V}{\partial x}\}$$

This is a variation of the action and therefore must vanish up to some boundary terms if $x$ is a solution of the equations of the motion. But the equation $\delta S=0$ is just the virial theorem:

$$2\langle T\rangle = \langle x\cdot F\rangle$$

where the angle brackets mean time average.

The only remaining issue is neglect of the boundary terms. This enforces the condition on the virial theorem that the motion be bounded and that I take a long enough time average. If both of these conditions are true, then I can take $T\rightarrow \infty$. Since everything is bounded the boundary terms remain finite as $T\rightarrow \infty$ and therefore there contribution to $\frac{\delta s}{T}$ goes to zero. Leaving us with the virial theorem.

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