Посилання:Dynamical Equations, Invariants and Spectrum Generating Algebras of Mechanical Systems with Position-Dependent Mass / Sara Cruz y Cruz, Oscar Rosas-Ortiz // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 46 назв. — англ.
Підтримка:This paper is a contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”. The full collection is available at http://www.emis.de/journals/SIGMA/SESSF2012.html.
The authors thank the anonymous referees for their comments to improve the presentation and motivation of the paper. The financial support of CONACyT-Mexico (project 152574), MICINN-Spain (project MTM2009-10751), IPN grant COFAA and projects SIP20120451, SIPSNIC-2011/04, is acknowledged.
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.