Dynamical equations, invariants and spectrum generating algebras of mechanical systems with position-dependent mass

Sara Cruz Y Cruz, Oscar Rosas-Ortiz

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17 Citations (Scopus)

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.
Original languageAmerican English
JournalSymmetry, Integrability and Geometry: Methods and Applications (SIGMA)
DOIs
StatePublished - 26 Aug 2013

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Mechanical Systems
Algebra
Invariant
Dependent
Integrals of Motion
Canonical Transformation
First Integral
Lagrange
Euler
Factorization
Lie Algebra
Motion
Form

Cite this

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title = "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 P{\"o}schl-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.",
author = "{Cruz Y Cruz}, Sara and Oscar Rosas-Ortiz",
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T1 - Dynamical equations, invariants and spectrum generating algebras of mechanical systems with position-dependent mass

AU - Cruz Y Cruz, Sara

AU - Rosas-Ortiz, Oscar

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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.

U2 - 10.3842/SIGMA.2013.004

DO - 10.3842/SIGMA.2013.004

M3 - Article

JO - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

JF - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SN - 1815-0659

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