### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 004, 21 pages      arXiv:1208.2300      https://doi.org/10.3842/SIGMA.2013.004
Contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”

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

Sara Cruz y Cruz a and Oscar Rosas-Ortiz b
a) SEPI-UPIITA, Instituto Politécnico Nacional, Av. IPN No. 2580, Col. La Laguna Ticomán, C.P. 07340 México D.F. Mexico
b) Physics Department, Cinvestav, A.P. 14740, México D.F. 07000, Mexico

Received July 31, 2012, in final form January 12, 2013; Published online January 17, 2013

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.

Key words: Pöschl-Teller potentials; dissipative dynamical systems; Poisson algebras; classical generating algebras; factorization method; position-dependent mass.

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