= Breathing mode
{title2=$u_i\propto r_i-R_{\rm cm}$}
A breathing mode is a collective <normal mode> in which a structure expands and contracts radially while preserving its overall symmetry. For a uniform scaling pattern, the <displacement> of point $i$ is proportional to its position relative to the <center of mass>, as displayed above. In a central <spring> framework this pattern is a <normal mode> only when applying the <stiffness matrix> returns a multiple of the same pattern with the appropriate <mass matrix>; radial motion alone does not establish an <eigenvector>.
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