The derivation and application of the Lagrange equations of motion to systems with mass varying explicitly with position are discussed. Two perspectives can be followed: systems with a material type of source, attached to particles continuously gaining or losing mass, and systems for which the variation of mass is of a nonlinear control volume type, mass trespassing a control surface. This is the case if, for some theoretical or practical reason, a partition into subsystems is considered. An important class of problems in which the extended Lagrange equations turn to be useful emerges from “hydromechanics,” whenever a finite number of generalized coordinates can be used, under the concept of the added mass tensor. A particular and interesting one is addressed in the present paper: the classical hydrodynamic impact of a rigid body against a liquid free surface.
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Celso P. Pesce (2003) studied this question.