Poisson brackets formulation for the dynamics of a position-dependent mass particle

Abstract

Following the history of mechanics (Dugas in A history of mechanics, Dover, New York, 1988), we read that Poisson’s theorem figured in the 1811 edition of the celebrated Mecanique Analytique. Indeed, as it can be observed in the classical textbooks, Poisson brackets formulation is one of the cardinal chapters of analytical mechanics. Given the natural motivation to accommodate variable-mass systems at the level of classical analytical mechanics, we will herein direct our attention toward Poisson brackets formulation. To wit, considering the case of a position-dependent mass particle, in which we will assume that the absolute velocity of mass ejection or aggregation is a linear function of the generalized velocity, we will endeavor to provide such position-dependent mass problems with an appropriate Poisson brackets formulation. To our very best knowledge, this means an original contribution to the research field of the analytical mechanics of variable-mass systems. We will start establishing the Poisson brackets definition for the dynamics of a position-dependent mass particle, which will be posited in harmony with the classical mathematical portrait of analytical mechanics. Therefrom, we will demonstrate consequent results which give rise to the required formulation, namely, Jacobi’s identity, canonical equations expressed by means of such Poisson brackets and Poisson’s theorem. Last, we will apply Poisson’s theorem to evaluate the relationship between the conservation laws which are at our disposal in the domain of position-dependent mass problems.