TY - JOUR
T1 - Dynamical equations, invariants and spectrum generating algebras of mechanical systems with position-dependent mass
AU - Cruz Y Cruz, Sara
AU - Rosas-Ortiz, Oscar
PY - 2013
Y1 - 2013
N2 - 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.
AB - 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.
KW - Classical generating algebras
KW - Dissipative dynamical systems
KW - Factorization method
KW - Poisson algebras
KW - Position-dependent mass
KW - Pöschl-Teller potentials
UR - http://www.scopus.com/inward/record.url?scp=84872821929&partnerID=8YFLogxK
U2 - 10.3842/SIGMA.2013.004
DO - 10.3842/SIGMA.2013.004
M3 - Artículo
SN - 1815-0659
VL - 9
JO - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
JF - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
M1 - 004
ER -