The equations of Lagrange written for a non-material volume
The authors consider a continuous body whose motion is dependent on a finite set of generalised coordinates. Then they extend classical Lagrange equations involving the total kinetic energy of the body with respect to a non-material control volume which coincides instantaneously with the material volume. The authors employ the well-known formalism of continuum mechanics to show that the extended Lagrange equations must additionally contain some integrals over the boundary of the body which are originated by differences of velocities on coinciding material and non-material surfaces. The Lagrange equations so obtained are employed to derive the equations governing the rectilinear motion of a rocket and the falling of a folded string hanged vertically down from a fixed point.
scientific article; zbMATH DE number 1757439