The Application of Lagrange Equations to Mechanical Systems With Mass Explicitly Dependent on Position

Abstract

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.