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To distinguish the moment quantity from the gravitational constant, write it as $\mathcal{G}={\sum }_i{\mathbf{p}}_i\cdot {\mathbf{r}}_i$. Newton's equations give ${\dot{\mathbf{p}}}_i={\mathbf{F}}_i$ and ${\dot{\mathbf{r}}}_i={\mathbf{p}}_i/m_i$. Differentiating provesAverage from $0$ to $\tau$, divide by $\tau$, and apply the supplied long-time boundary condition. The result isThe PDF's intermediate formula omits this minus sign. It cannot follow from the preceding derivative identity: an attractive inverse-square circular orbit already has $\mathbf{F}\cdot \mathbf{r}<0$ and $T>0$, contradicting the printed positive-sign version. The later homogeneous-potential relation is consistent with the corrected sign.

Interpret $W$ as the total pair potential energy, with each pair counted once, for example $W={\sum }_{i<j}C|{\mathbf{r}}_i-{\mathbf{r}}_j{|}^{p+1}$. The printed pair sum leaves one particle index unspecified; an ordered-pair convention with the corresponding factor absorbed into $C$ has the same degree of homogeneity. Under a common dilation of all positions,Differentiating with respect to $\lambda$ at one is the Euler theorem for homogeneous functions. Since ${\mathbf{F}}_i=-{\nabla }_{i}W$, it givesSubstitution into the correctly signed averaged identity proves the virial theorem for a homogeneous pair potential:For $p\ne -1$, the conserved total energy can be evaluated using these averages:Use the kinetic temperature convention $⟨T⟩=\kappa \Theta$, where $\kappa =3N{k}_B/2>0$ for three-dimensional thermalized particles, or the same positive coefficient as a kinetic-temperature definition. At fixed particle number and interaction parameters,Thus the equilibrium heat capacity is negative precisely whensubject to existence of the bounded equilibrium sequence assumed by the virial theorem. At $p=-3$ the virial energy is zero and this derivative vanishes. At $p=-1$ the written power-law potential is constant, its force vanishes, and the bounded-motion virial condition excludes nonzero kinetic energy; division by $p+1$ is not permissible. The Newtonian case $p=-2$ recovers $E_{\mathrm{tot}}=-⟨T⟩$.

The virial theorem relates the average value of a function along a trajectory to the forces that bind it. For an isolated system of particles interacting through Newtonian gravitational potential energy, it gives $2⟨K⟩+⟨U⟩=0$: the average kinetic energy is half the magnitude of the average negative potential energy. An eccentric Kepler orbit makes the distinction vivid: kinetic energy and potential energy vary throughout the orbit, although their sum is constant. The relation concerns their time averages, rather than their values at every instant. It underlies the virial mass estimate and the negative heat capacity of a virialized gravitational system.

Use constant positive particle masses $m_i$, positions ${\mathbf{r}}_i$, mechanical momenta ${\mathbf{p}}_i=m_i{\dot{\mathbf{r}}}_i$, and total kinetic energy $K={\sum }_i{\mathbf{p}}_i^2/(2m_i)$. The auxiliary quantity $G(t)$ is a sum of dot products, not the gravitational constant, which we denote here by $G_N$:This finite-$T$ identity is exact wherever Newton's second law holds along a regular trajectory; ${\mathbf{F}}_i$ must include every force. The endpoint term is what a finite observation window leaves behind. For the infinite-time theorem, suppose the averages on the left exist and $(G(T)-G(0))/T\to 0$. A sufficient condition is uniformly bounded positions and velocities for finitely many particles. More generally $G(T)=o(T)$ suffices, even without those individual bounds. Bounded positions alone do not supply a bound on momenta, especially near a singular potential energy; nor does a small endpoint term prove thermodynamic equilibrium. Periodic motion with $G(T)=G(0)$ gives the identity without an endpoint term at that finite $T$. For an isolated system, use an inertial centre of mass frame so translational drift does not contribute bulk kinetic energy.^[ref]

For a differentiable potential energy that is a homogeneous function of degree $q$, choose its additive constant so that the scaling law really holds. With conservative forces, the Euler theorem for homogeneous functions givesThus a harmonic oscillator has $q=2$ and $⟨K⟩=⟨U⟩$. An isolated system with Newtonian gravitational potential energy hasEach interacting pair occurs once, and $U=0$ at infinite separation. Here the total energy $E=K+U$ is conserved. An external field or a mixture of different scaling powers requires its own force term; one cannot assign the whole potential energy a single degree $q$. The virial theorem for a homogeneous pair potential states the corresponding general energy relation.