3.1: Variations and Perturbations and their Implications for the Virial Theorem

Perturbation analysis is truly an old mechanism in which one explores the behavior of a system in a known state by assuming there are small variations of the independent variables describing the system and determining the individual variation in the independent variables. The vehicle for determining the independent variable changes is found in the very Equations which describe the initial state of the system. The Equations usually chosen for this type of analysis are the Equations of motion for the system. For example, consider the Equations of motion for an object moving under the influence of a point potential Φ.

\[\frac{\mathrm{d}^2 \mathbf{r}}{\mathrm{dt}^2}=-\nabla \Phi .\label{3.1.1}\]

Assume a solution \(\mathbf{r}_{\mathrm{o}}(\mathrm{t})\) is known, which satisfies Equation \ref{3.1.1} for a particular potential \(\Phi_{\mathrm{o}}\). Since Equation \ref{3.1.1} is valid for any system where Φ is known, one could define

\[\left.\begin{array}{l}
\Phi=\Phi_0+\delta \Phi \\[4pt]
\mathbf{r}=\mathbf{r}_0+\delta \mathbf{r}
\end{array}\right\}\label{3.1.2}\]

and Equation \ref{3.1.1} would require

\[\frac{\mathrm{d}^2(\mathbf{r}+\delta \mathbf{r})}{\mathrm{dt}^2}=-\nabla(\Phi+\delta \Phi) .\label{3.1.3}\]

However, since both ∇ and d/dt are linear operators, Equation \ref{3.1.3} becomes

\[\frac{\mathrm{d}^2 \mathbf{r}_0}{\mathrm{dt}^2}+\frac{\mathrm{d}^2 \delta \mathbf{r}}{\mathrm{dt}^2}=-\nabla \Phi-\nabla \delta \Phi,\label{3.1.4}\]

but we already know that

\[\frac{\mathrm{d}^2 \mathbf{r}_0}{\mathrm{dr}^2}=-\nabla \Phi_0,\label{3.1.5}\]

so that by subtracting Equation \ref{3.1.5} from Equation \ref{3.1.4}, we get

\[\frac{\mathrm{d}^2 \delta \mathbf{r}}{\mathrm{dr}^2}=-\nabla \delta \Phi,\label{3.1.6}\]

which we called the perturbed Equations of motions where δΦ is the perturbation that involves the perturbation δr. A short approach which leads to the same result is to "take the variation" of Equation \ref{3.1.1} wherein the operator δ is not affected by time or space derivatives. This technique "works" because the time and space operators in the Equation of motion are linear, hence any linear perturbation or departure from a given solution will produce the sum of the original Equations of motion on the perturbed Equations of motion. In general, I shall use the variational operator δ in this sense, that is, it represents a small departure of a variable from the value it had which satisfies the Equations governing the system. It is not necessary that one perturbs the Equations of motion in order to gain information about the system. Clearly any Equations which describe the structure of the system are subject to this type of analysis. Thus, if taking variations of the Equations of motion produces useful results, might not the variational form of the moments of those Equations also be expected to contain interesting information? It was in this spirit that Paul Ledoux developed the variational form of the scalar virial theorem¹, and was able to predict the pulsational period of a star.

The variational approach yields differential Equations which describe parameter relationships for a system disturbed from an initial state. If that state happens to be an equilibrium state, variational analysis of the Equations of motion would yield a description of the system motion about the equilibrium configuration. Variational analysis of spatial moments could then be expected to yield macroscopic properties of that motion. This is indeed the case, as Ledoux¹ demonstrated by determining perhaps the most obvious macroscopic property of such motion, the pulsational period of the system. Chandrasekhar² found the tensor form of the virial theorem useful in determining non-radial modes of oscillation of stars. In addition, he and Fermi³ investigated the effects of a magnetic field on the pulsation of a star. An additional macroscopic property closely connected with the pulsational period, with which this approach deals, is the global stability of the system. We shall examine this aspect of the analysis later. For now, let us be content with observing in some detail how the variational approach yields the pulsational periods of stars.