Dynamical Equations, Invariants and Spectrum Generating Algebras of Mechanical Systems with Position-Dependent Mass
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 requi...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
National Academy of Science of Ukraine
2013-01-01
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.3842/SIGMA.2013.004 |
| _version_ | 1818252483573579776 |
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| author | Sara Cruz y Cruz Oscar Rosas-Ortiz |
| author_facet | Sara Cruz y Cruz Oscar Rosas-Ortiz |
| author_sort | Sara Cruz y Cruz |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-12T16:24:54Z |
| format | Article |
| id | doaj.art-1c438bddbede479eab0ecd8b548bc2c3 |
| institution | Directory Open Access Journal |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2024-12-12T16:24:54Z |
| publishDate | 2013-01-01 |
| publisher | National Academy of Science of Ukraine |
| record_format | Article |
| series | Symmetry, Integrability and Geometry: Methods and Applications |
| spelling | doaj.art-1c438bddbede479eab0ecd8b548bc2c32022-12-22T00:18:53ZengNational Academy of Science of UkraineSymmetry, Integrability and Geometry: Methods and Applications1815-06592013-01-019004Dynamical Equations, Invariants and Spectrum Generating Algebras of Mechanical Systems with Position-Dependent MassSara Cruz y CruzOscar Rosas-OrtizWe 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.http://dx.doi.org/10.3842/SIGMA.2013.004Pöschl-Teller potentialsdissipative dynamical systemsPoisson algebrasclassical generating algebrasfactorization methodposition-dependent mass |
| spellingShingle | Sara Cruz y Cruz Oscar Rosas-Ortiz Dynamical Equations, Invariants and Spectrum Generating Algebras of Mechanical Systems with Position-Dependent Mass Symmetry, Integrability and Geometry: Methods and Applications Pöschl-Teller potentials dissipative dynamical systems Poisson algebras classical generating algebras factorization method position-dependent mass |
| title | Dynamical Equations, Invariants and Spectrum Generating Algebras of Mechanical Systems with Position-Dependent Mass |
| title_full | Dynamical Equations, Invariants and Spectrum Generating Algebras of Mechanical Systems with Position-Dependent Mass |
| title_fullStr | Dynamical Equations, Invariants and Spectrum Generating Algebras of Mechanical Systems with Position-Dependent Mass |
| title_full_unstemmed | Dynamical Equations, Invariants and Spectrum Generating Algebras of Mechanical Systems with Position-Dependent Mass |
| title_short | Dynamical Equations, Invariants and Spectrum Generating Algebras of Mechanical Systems with Position-Dependent Mass |
| title_sort | dynamical equations invariants and spectrum generating algebras of mechanical systems with position dependent mass |
| topic | Pöschl-Teller potentials dissipative dynamical systems Poisson algebras classical generating algebras factorization method position-dependent mass |
| url | http://dx.doi.org/10.3842/SIGMA.2013.004 |
| work_keys_str_mv | AT saracruzycruz dynamicalequationsinvariantsandspectrumgeneratingalgebrasofmechanicalsystemswithpositiondependentmass AT oscarrosasortiz dynamicalequationsinvariantsandspectrumgeneratingalgebrasofmechanicalsystemswithpositiondependentmass |