Dynamical Equations, Invariants and Spectrum Generating Algebras of Mechanical Systems with Position-Dependent Mass

TL;DR

This paper investigates classical systems with position-dependent mass, deriving their dynamical equations, conserved quantities, and algebraic structures, and introduces new solvable potentials related to Lie algebras.

Contribution

It constructs the Lagrangian and Hamiltonian for variable mass systems, finds a canonical transformation to constant mass equations, and discovers new solvable potentials linked to Lie algebras.

Findings

Derived modified Euler-Lagrange and Hamilton's equations for variable mass systems.

Identified new solvable Pöschl-Teller potentials with algebraic structures.

Mapped variable mass equations to constant mass form through canonical transformation.

Abstract

We analyze the dynamical equations obeyed by a classical system with position-dependent mass. It is shown that there is a non-conservative force quadratic in the velocity associated to the variable mass. We construct the Lagrangian and the Hamiltonian for this system and find the modifications required in the Euler-Lagrange and Hamilton's equations to reproduce the appropriate Newton's dynamical law. Since the Hamiltonian is not time invariant, we get a constant of motion suited to write the dynamical equations in the form of the Hamilton's ones. The time-dependent first integrals of motion are then obtained from the factorization of such a constant. A canonical transformation is found to map the variable mass equations to those of a constant mass. As particular cases, we recover some recent results for which the dependence of the mass on the position was already unnoticed, and find new solvable potentials of the P\"oschl-Teller form which seem to be new. The latter are associated to either the su(1,1) or the su(2) Lie algebras depending on the sign of the Hamiltonian.