The classical variational principles in Mechanics, such as Hamilton’s principle and the principle of d'Alembert-Lagrange, hold for conservative systems with perfect bilateral constraints. Several attempts have been made in literature to generalise Hamilton’s principle for mechanical systems with perfect unilateral constraints involving impulsive motion. This has led to a number of different variants of Hamilton’s principle, some expressed as variational inequalities. Up to now, the connection between these different principles has been missing. The aim of this research project is to put these different principles of Hamilton, as well as other variational principles in Mechanics, in a unified framework by using the concept of weak and strong extrema. The difference between weak and strong variations of the motion is studied in detail. Each type of variation leads to a variant of the principle of Hamilton in the form of a variational inequality. Each type of variation leads to different necessary and sufficient conditions on the impact law. The principle of Hamilton with strong variations is valid for perfect unilateral constraints with a completely elastic impact law, whereas the weak form of Hamilton’s principle only requires perfect unilateral constraints and no condition on the energy.

Configuration manifold with constraint

Weak variations (left) and strong variations (right)