The effects of shear on the potential energy landscape are examined with regard to the mechanical properties of ductile glasses. Shear strain is found to cause the disappearance of local potential energy minima, rendering the system mechanically unstable and forcing the system to move to alternate local minima. These mechanical instabilities are characterized by the decrease to zero of several features of the potential energy landscape: a barrier height, the curvature at both the barrier and the local minimum, and the distance from the barrier to the local minimum. From an analysis of the results of different size systems, the mechanical instabilities are shown to be localized to a small number of atoms. The consequences of the mechanical instabilities on the mechanical properties are stress drops which give rise to the yielding and plastic-flow behavior characteristic of ductile glasses; these stress drops are discontinuous at the molecular level, but become continuous at the macroscopic level. A simple model is developed which demonstrates that different stress-strain behaviors are obtained as the characteristics of the mechanical instabilities change. This simple model is also used to elucidate the effects of temperature on the mechanical properties.