Lagrangians for simple systems with variable mass

Abstract

The Lagrangian formulation of classical mechanics and its applications figure prominently in the educational literature. Yet systems with variable mass are summarily excluded from this formulation and discussed in terms of Newtonian theory only. This omission is neither technically justified nor desirable from a pedagogical point of view, because it might suggest to the student that such systems are beyond the scope of the Lagrangian approach. Therefore, the authors show that the formalism can be readily extended to include those variable mass systems that are treated in the textbooks in a Newtonian fashion, and its application is illustrated by means of three instructive examples.

Particle dynamics with variable mass and charge

“…In one degree of freedom, with F independent of v, Darboux showed [3,6] that a Lagrangian can always be constructed as the solution of a partial differential equation. In general, however, it is not evident how a set of canonical variables can be found such that (1) is satisfied.…”

“…Fig. 3 is an enlargement of the regular orbit, illustrating the close fit of the adiabatic boundary, ∂ μ given by (6). Just as a Poincaré section provides a visual guide to regular and chaotic regions in phase space, so these orbit plots can reveal the presence of hidden invariants.…”

“…The dynamics of a body with variable mass has a long and distinguished history [1][2][3][4][5][6]. Most of the early work centered on the evolution of stars, which throughout their lives lose mass by electromagnetic radiation and corpuscular emission.…”

Dynamics of body separation—analytical procedure

“…The nonholonomic properties of the system are very often included into consideration [20,21]. The conservation laws [22][23][24] of the variable mass system are widely used for describing the phenomena.…”

Global Invariants for Variable-Mass Systems

“…Darboux [7] showed how to construct a Lagrangian from a given equation of motion, but only for one-dimensional systems without velocity-dependent potentials. Leubner and Krumm [8] applied Darboux's method to a few elementary examples. For velocity-dependent potentials, which occur for example in charged particle motion in magnetic fields [9], we were unable to find a set of canonical variables, but were able to construct an exact invariant directly from the non-Hamiltonian equations of motion.…”