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

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 Pöschl-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.

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Cited by in corpus (16)

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• Group classification of (1+3)-dimensional Schrödinger equations with position dependent mass
• n-dimensional PDM non-linear oscillators: Linearizability and Euler-Lagrange or Newtonian invariance
• Position-dependent mass Dirac equation and local Fermi velocity
• Coherent and squeezed states: introductory review of basic notions, properties and generalizations
• Position-dependent mass systems: Classical and quantum pictures
• Quantum Smooth Boundary Forces from Constrained Geometries
• Quantum and semi-classical aspects of confined systems with variable mass
• $n$-dimensional PDM-damped harmonic oscillators: Linearizability, and exact solvability
• So (2, 1) algebra, local Fermi velocity, and position-dependent mass Dirac equation
• Symmetries of the Schroedinger-Pauli equation for neutral particles
• Exact solvability of PDM systems with extended Lie symmetries