The Lagrangian equations of motion for an infinitely thin vibrating ring and shell are derived, starting from the stretching and bending interactions of molecular force fields. The resulting spectral patterns form useful tools to classify the vibrational levels in annular and globular molecules. They provide a set of parent symmetry labels and predict the relative energetic ordering of the vibrational modes. The method is illustrated for the simple case of the skeletal modes in benzene. The results are compared with the classical treatments of elasticity theory, and with the tensor surface harmonic theory of cluster bonding.