A brief note on the analytical solution of Meshchersky's equation within the inverse problem of Lagrangian mechanics

Meshchersky's equation is a basic differential equation in the mechanics of variable-mass particles. This note particularly considers the case in which a one-dimensional and position-dependent mass particle is under the action of a potential force. The absolute velocity of mass ejection (or accretion) is supposed to be a linear function of the particle velocity. Within the formulation of the inverse problem of Lagrangian mechanics, an analytical solution of Meshchersky's equation is here derived. The solution method follows from applying the concept of constant of motion of an extremum problem, which is a fundamental ground in the theory of invariant variational principles.