1.3: Velocity Dependent Forces and the Virial Theorem
- Page ID
- 141441
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)There is an additional feature of the virial theorem as stated in Equation \ref{1.2.16} that should be mentioned. If the forces acting on the system include velocity dependent forces, the result of the virial theorem is unchanged. In order to demonstrate this, consider the same system of mass points mi subjected to forces fi which may be divided into velocity dependent (\(\overline{\mathbf{w}}_{\mathrm{i}}\)) and velocity independent forces (zi). The Equations of motion may be written as:
\[\dot{\mathbf{p}}_{\mathrm{i}}=\mathbf{f}_{\mathrm{i}}=\overline{\mathbf{w}}_{\mathrm{i}}+\mathbf{z}_{\mathrm{i}}.\label{1.3.1}\]
Substituting into Equation \ref{1.2.6}, we have
\[\frac{\mathrm{dG}}{\mathrm{dt}}-\sum_{\mathrm{i}} \overline{\mathbf{w}}_{\mathrm{i}} \cdot \mathbf{r}_{\mathrm{i}}=2 \mathrm{T}+\sum_{\mathrm{i}} \mathbf{z}_{\mathrm{i}} \cdot \mathbf{r}_{\mathrm{i}}.\label{1.3.2}\]
Remembering that the velocity dependent forces may be rewritten as
\[\overline{\mathbf{w}}_{\mathrm{i}}=\alpha_{\mathrm{i}} \mathbf{v}_{\mathrm{i}}=\alpha_{\mathrm{i}} \frac{\mathrm{d} \mathbf{r}_{\mathrm{i}}}{\mathrm{dt}}.\label{1.3.3}\]
We may again average over time as in Equation \ref{1.2.12}. Thus
\[\frac{1}{\mathrm{T}_0} \int_0^{\mathrm{T}_0} \frac{\mathrm{dG}}{\mathrm{dt}} \mathrm{dt}-\frac{1}{\mathrm{T}_0} \int_0^{\mathrm{T}_0} \sum_{\mathrm{i}} \alpha_{\mathrm{i}} \frac{\mathrm{d} \mathbf{r}_{\mathrm{i}}}{\mathrm{dt}} \cdot \mathrm{r}_{\mathrm{i}} \mathrm{dt}=2 \overline{\mathrm{T}}-\mathrm{n} \overline{\mathcal{U}},\label{1.3.4}\]
where \(\overline{\mathcal{U}}\) is the average value of the potential energy for the "non-frictional" forces. Carrying out the integration on the left hand side we have
\[\frac{1}{\mathrm{T}_0}\left[\mathrm{G}\left(\mathrm{T}_0\right)-\mathrm{G}(0)\right]+\frac{1}{2 \mathrm{T}_0} \sum_{\mathrm{i}} \alpha_{\mathrm{i}}\left[\mathrm{r}_{\mathrm{i}}^2\left(\mathrm{T}_0\right)-\mathrm{r}_{\mathrm{i}}^2(0)\right]=2 \overline{\mathrm{T}}-\mathrm{n} \overline{\mathcal{U}}.\label{1.3.5}\]
Thus, if the motion is periodic, both terms on the left hand side of Equation \ref{1.3.5} will vanish in a time T0 equal to the period of the system. Indeed both terms can be made as small as required providing the "frictional" forces \(\overline{\mathrm{w}}_\mathrm{i}\) do not cause the system to cease to be in motion over the time for which the averaging is done. This apparently academic aside has the significant result that we need not worry about any Lorentz forces or viscosity forces which may be present in our subsequent discussion in which we shall invoke the virial theorem.