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In this paper the extended Lagrangian formalism for the rheonomic systems   (Dj. Mušicki, 2004), which began with the modification of the mechanics of   such systems (V. Vujičić, 1987), is extended to the systems with variable   mass, with emphasis on the corresponding energy relations. This extended   Lagrangian formalism is based on the extension of the set of chosen   generalized coordinates by new quantities, suggested by the form of   nonstationary constraints, which determine the position of the frame of   reference in respect to which these generalized coordinates refer. As a   consequence, an extended system of the Lagrangian equations is formulated,   accommodated to the variability of the masses of particles, where the   additional ones correspond to the additional generalized coordinates. By   means of these equations, the energy relations of such systems have been   studied, where it is demonstrated that here there are four types of energy   conservation laws. The obtained energy laws are more complete and natural   than the corresponding ones in the usual Lagrangian formulation for such   systems. It is demonstrated that the obtained energy laws, are in full   accordance with the energy laws in the corresponding vector formulation, if   they are expressed in terms of the quantities introduced in this formulation   of mechanics. The obtained results are illustrated by an example: the motion   of a rocket, which ejects the gasses backwards, while this rocket moves up a   straight line on an oblique plane, which glides uniformly in a horizontal   direction

Extended Lagrangian formalism for rheonomic systems with variable mass