Oscillating shape motion of a freely falling and bouncing water droplet has long fascinated and inspired scientists. We propose dynamic non-linear equations for closed, two-dimensional surfaces in gravity and apply it to analyze shape dynamics of freely falling and bouncing drops. The analytic and numerical solutions qualitatively well explain why drops oscillate among prolate/oblate morphology and display a number of features consistent with experiments. In addition, numerical solutions for simplified equations indicate nonlinear effects of nonperiodic/asymmetric motion and the growing amplitude in the surface density oscillations and well agree to previous experimental data.