This paper presents the 3-D multi-scale analysis of a cylindrical liquid ligament subjected to a capillary instability. This analysis aims to investigate the evolution of the ligament interface paying a specific attention to the physical mechanisms involved at small scales. The capillary instability behavior is obtained from direct numerical simulations. Calculations are performed for several wavenumbers of the initial sinusoidal perturbation. During the capillary instability, the scale space is divided in two regions: the small-scale region where a thinning mechanism is identified and the large-scale region where a thickening mechanism is observed. Although the characteristic scale dmax of the large-scale region displays a dynamics that agrees with the Rayleigh linear theory, this agreement is lost for the characteristic scale d1of the small scale region showing that the non-linear effects mainly concentrate on the small scales. The dynamics of the characteristic scale d1 follows three successive regimes. The development of a simple model allows identifying the physical mechanisms related to these three regimes as well as their dependences with the wavenumber of the perturbation. Among other results it is found that the capillary contraction regime that develops when the breakup is approached is always preceded by an elongation mechanism whose effect is to increase the specific-surface-area of the ligament.