# 14.4: Center of Mass Oscillations


Transforming the coordinates into the center of mass of the two oscillating masses elucidates an interesting feature of the normal modes for the two-coupled linear oscillator. As illustrated in Figure $$(14.2.1)$$, the center-of-mass coordinate for the two mass system is

\begin{align*} 2R_{cm} &= l + x_1 + l + l^{\prime} + x_2 \\[4pt] &= 2l + l^{\prime} + \eta_2 \end{align*}

while the relative separation distance is

$r = (l + l^{\prime} + x_2) − (l + x_1) = l^{\prime} − \eta_1\notag$

That is, the two normal modes are

\begin{align} \eta_1 = l^{\prime} − r \\ \eta_2 = 2 R_{cm} − 2l − l^{\prime} \notag\end{align}

The $$\eta_1$$ mode, which has angular frequency $$\omega_1 = \sqrt{\frac{\kappa +2\kappa^{\prime}}{M}}$$ corresponds to an oscillations of the relative separation $$r$$, while the center-of-mass location $$R_{cm}$$ is stationary. By contrast, the $$\eta_2$$ mode, with angular frequency $$\omega_2 = \sqrt{\frac{\kappa}{M}}$$ corresponds to an oscillation of the center of mass $$R_{cm}$$ with the relative separation $$r$$ being a constant.

Figure $$\PageIndex{1}$$ illustrates the decoupled center-of-mass $$R_{cm}$$, and relative motions $$r$$ for both normal modes of the coupled double-oscillator system. The difference in angular frequencies and amplitudes is readily apparent. It is of interest to consider the special case where the spring constant $$\kappa = 0$$ for the two outside springs. Then the angular frequencies are $$\omega_1 = \sqrt{\frac{2\kappa^{\prime}}{M}}$$ and $$\omega_2 = 0$$ for the two normal modes. When $$\kappa = 0$$ the $$\eta_2$$ mode is a spurious center-of-mass mode since it corresponds to an oscillation with $$\omega_2 = 0$$ in spite of the fact that there are no forces acting on the center of mass. That is, the center-of-mass momentum must be a constant of motion. This spurious center-of-mass oscillation is a consequence of measuring the displacements $$(x_1, x_2)$$ with respect to an arbitrary external reference that is not related to the center of mass of the coupled system. Spurious center-of-mass modes are encountered frequently in many-body coupled oscillator systems such as molecules and nuclei. In such cases it is necessary to project out the center-of-mass motion to eliminate such spurious solutions as will be discussed later.

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