1.2: The Classical Derivation of the Virial Theorem

The virial theorem is often stated in slightly different forms having slightly different interpretations. In general, we shall repeat the version given by Claussius and express the virial theorem as a relation between the average value of the kinetic and potential energies of a system in a steady state or a quasi-steady state. Since the understanding of any theorem is related to its origins, we shall spend some time deriving the virial theorem from first principles. Many derivations of varying degree of completeness exist in the literature. Most texts on stellar or classical dynamics (e.g. Kurth¹) derive the theorem from the Lagrange identity. Landau and Lifshitz² give an eloquent derivation appropriate for the electromagnetic field which we shall consider in more detail in the next section. Chandrasekhar³ follows closely the approach of Claussius while Goldstein⁴ gives a very readable vector derivation firmly rooted in the original approach and it is basically this form we shall develop first. Consider a general system of mass points m_i with position vectors r_i which are subjected to applied forces (including any forces of constraint) f_i. The Newtonian Equations of motions for the system are then

\[\dot{\mathbf{p}}_{\mathrm{i}}=\frac{\mathrm{d}\left(\mathrm{m}_{\mathrm{i}} \mathbf{v}_{\mathrm{i}}\right)}{\mathrm{dt}}=\mathbf{f}_{\mathrm{i}}.\label{1.2.1}\]

Now define

\[\mathrm{G}=\sum_{\mathrm{i}} \mathbf{p}_{\mathrm{i}} \cdot \mathbf{r}_{\mathrm{i}}=\sum_{\mathrm{i}} \mathrm{m}_{\mathrm{i}} \frac{\mathrm{d} \mathbf{r}_{\mathrm{i}}}{\mathrm{dt}} \cdot \mathbf{r}_{\mathrm{i}}=\frac{1}{2} \sum_{\mathrm{i}} \mathrm{m}_{\mathrm{i}} \frac{\mathrm{d}\left(\mathbf{r}_{\mathrm{i}} \cdot \mathbf{r}_{\mathrm{i}}\right)}{\mathrm{dt}}=\frac{1}{2} \frac{\mathrm{d}}{\mathrm{dt}}\left(\sum_{\mathrm{i}} \mathrm{m}_{\mathrm{i}} \mathrm{r}_{\mathrm{i}}^2\right).\label{1.2.2}\]

The term in the large brackets is the moment of inertia (by definition) about a point and that point is the origin of the coordinate system for the position vectors r_i. Thus, we have

\[\mathrm{G}=\frac{1}{2} \frac{\mathrm{dI}}{\mathrm{dt}},\label{1.2.3}\]

where I is the moment of inertia about the origin of the coordinate system.

Now consider

\[\frac{\mathrm{dG}}{\mathrm{dt}}=\sum \dot{\mathbf{r}}_{\mathrm{i}} \cdot \mathbf{p}_{\mathrm{i}}+\sum_{\mathrm{i}} \dot{\mathbf{p}}_{\mathrm{i}} \cdot \mathbf{r}_{\mathrm{i}},\label{1.2.4}\]

but

\[\sum_{\mathrm{i}} \dot{\mathbf{r}}_{\mathrm{i}} \cdot \mathbf{p}_{\mathrm{i}}=\sum_{\mathrm{i}} \mathrm{m}_{\mathrm{i}} \dot{\mathbf{r}}_{\mathrm{i}} \cdot \dot{\mathbf{r}}_{\mathrm{i}}=\sum_{\mathrm{i}} \mathrm{m}_{\mathrm{i}} \mathrm{v}_{\mathrm{i}}^2=2 \mathrm{T},\label{1.2.5}\]

where T is the total kinetic energy of the system with respect to the origin of the coordinate system. However, since \(\dot{\mathbf{p}}_{\mathrm{i}}\) is really the applied force acting on the system (see Equation 1.2.1), we may rewrite Equation \ref{1.2.4} as follows:

\[\frac{\mathrm{dG}}{\mathrm{dt}}=2 \mathrm{T}+\sum_{\mathrm{i}} \mathrm{f}_{\mathrm{i}} \cdot \mathrm{r}_{\mathrm{i}}.\label{1.2.6}\]

The last term on the right is known as the Virial of Claussius. Now consider the Virial of Claussius. Let us assume that the forces f_i obey a power law with respect to distance and are derivable from a potential. The total force on the ith particle may be determined by summing all the forces acting on that particle. Thus

\[\mathbf{f}_{\mathrm{i}}=\sum_{\mathrm{j} \neq \mathrm{i}} \mathbf{F}_{\mathrm{ij}},\label{1.2.7}\]

where F_ij is the force between the ith and jth particle. Now, if the forces obey a power law and are derivable from a potential then,

\[\mathbf{F}_{\mathrm{ij}}=\nabla_{\mathrm{i}} \mathrm{m}_{\mathrm{i}} \Phi\left(\mathrm{r}_{\mathrm{ij}}\right)=-\nabla_{\mathrm{i}} \mathrm{a}_{\mathrm{ij}} \mathrm{r}_{\mathrm{ij}}^{\mathrm{n}}.\label{1.2.8}\]

The subscript on the ∇-operator implies that the gradient is to be taken in a coordinate system having the ith particle at the origin. Carrying out the operation implied by Equation \ref{1.2.8}, we have

\[\mathbf{F}_{\mathrm{ij}}=-n \mathrm{a}_{\mathrm{ij}} \mathrm{r}_{\mathrm{ij}}^{(\mathrm{n}-2)}\left(\mathbf{r}_{\mathrm{i}}-\mathbf{r}_{\mathrm{j}}\right).\label{1.2.9}\]

Now since the force acting on the ith particle due to the jth particle may be paired off with a force exactly equal and oppositely directed, acting on the jth particle due to the ith particle, we can rewrite Equation \ref{1.2.7} as follows:

\[\mathbf{f}_i=\sum_j \mathbf{F}_{i j}=\sum_{j>i} \mathbf{F}_{i j}+\mathbf{F}_{j i}.\label{1.2.10}\]

Substituting Equation \ref{1.2.10} into the definition of the Virial of Claussius, we have

\[\sum_{\mathrm{i}} \mathbf{f}_{\mathrm{i}} \cdot \mathbf{r}_{\mathrm{i}}=\sum_{\mathrm{i}} \sum_{\mathrm{j}>\mathrm{i}} \mathbf{F}_{\mathrm{ij}} \cdot \mathbf{r}_{\mathrm{i}}+\mathbf{F}_{\mathrm{ji}} \cdot \mathbf{r}_{\mathrm{j}}.\label{1.2.11}\]

It is important here to notice that the position vector r_i, which is 'dotted' into the force vector, bears the same subscript as the first subscript on the force vector. That is, the position vector is the vector from the origin of the coordinate system to the particle being action upon. Substitution of Equation \ref{1.2.9} into Equation \ref{1.2.11} and then into Equation \ref{1.2.6} yields:

\[\frac{\mathrm{dG}}{\mathrm{dt}}=2 \mathrm{T}-\mathrm{n} \mathcal{U},\label{1.2.12}\]

where \(\mathcal{U}\) is the total potential energy.^1.1 For the gravitational potential n = -1, and we arrive at a statement of what is known as Lagranges’ Identity:

\[\frac{\mathrm{dG}}{\mathrm{dt}}=\frac{1}{2} \frac{\mathrm{d}^2 \mathrm{I}}{\mathrm{dt}^2}=2 \mathrm{T}+\Omega.\label{1.2.13}\]

To arrive at the usual statement of the virial theorem we must average over an interval of time (T₀). It is in this sense that the virial theorem is sometimes referred to as a statistical theorem. Therefore, integrating Equation \ref{1.2.12}, we have

\[\frac{1}{\mathrm{T}_0} \int_0^{\mathrm{T}_0} \frac{\mathrm{dG}}{\mathrm{dt}} \mathrm{dt}=\frac{2}{\mathrm{T}_0} \int_0^{\mathrm{T}_0} \mathrm{T}(\mathrm{t}) \mathrm{dt}-\frac{\mathrm{n}}{\mathrm{T}_0} \int_0^{\mathrm{T}_0} U(\mathrm{t}) \mathrm{dt}.\label{1.2.14}\]

and, using the definition of average value we obtain:

\[\frac{1}{\mathrm{T}_0}\left[\mathrm{G}\left(\mathrm{T}_0\right)-\mathrm{G}(0)\right]=2\overline{\mathrm{T}}-\mathrm{n} \bar{\mathcal{U}}.\label{1.2.15}\]

If the motion of the system over a time T₀ is periodic, then the left-hand side of Equation \ref{1.2.15} will vanish. Indeed, if the motion of the system is bounded [i.e., G(t) < ∞], then we may make the left hand side of Equation \ref{1.2.15} as small as we wish by averaging over a longer time. Thus, if a system is in a steady state the moment of inertia (I) is constant and for systems governed by gravity

\[2 \overline{\mathrm{T}}+\overline{\Omega}=0.\label{1.2.16}\]

It should be noted that this formulation of the virial theorem involves time averages of indeterminate length. If one is to use the virial theorem to determine whether a system is in accelerative expansion or contraction, then he must be very careful about how he obtains the average value of the kinetic and potential energies.