Nonstandard Lagrangians and branching: The case of some nonlinear Liénard systems
10 May 2019-Modern Physics Letters A (World Scientific Publishing Company)-Vol. 34, Iss: 14, pp 1950110
TL;DR: In this paper, a new form of Lagrangian that shows branching for the associated Hamiltonian was proposed and has relevance to the quadratic type and mixed linear-quadratic Lienard-type nonlinear dynamical equations.
Abstract: We propose a new form of Lagrangian that shows branching for the associated Hamiltonian and has relevance to the quadratic type and mixed linear-quadratic Lienard-type nonlinear dynamical equations...
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TL;DR: In this article, the physics of plasma waves and magnetohydrodynamic (MHD) equilibrium of sunspots based on the concept of non-standard Lagrangians were studied. But the results were limited to the case of a single sunspot.
Abstract: In this work, we study the physics of plasma waves and magnetohydrodynamic (MHD) equilibrium of sunspots based on the concept of non-standard Lagrangians which play an important role in several bra...
14 citations
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TL;DR: In this paper the dynamics models based on non-standard Birkhoffians, including exponentialBirkhoffian, power law Birkhoffsian, and logarithm Birkh offian, are proposed, which are called non- standard Birkh Offian systems.
7 citations
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01 Apr 2020TL;DR: In this paper, the basic representations of position and momentum in a quantum mechanical system, that are guided by a generalized uncertainty principle and lead to a corresponding one-parameter eigenvalue problem, can be interpreted in terms of an extended Schrodinger equation embodying momentum-dependent mass.
Abstract: We show in this paper that the basic representations of position and momentum in a quantum mechanical system, that are guided by a generalized uncertainty principle and lead to a corresponding one-parameter eigenvalue problem, can be interpreted in terms of an extended Schrodinger equation embodying momentum-dependent mass Some simple consequences are pointed out
3 citations
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TL;DR: In this paper, the basic representations of position and momentum in a quantum mechanical system, that are guided by a generalized uncertainty principle and lead to a corresponding one-parameter eigenvalue problem, can be interpreted in terms of an extended Schrodinger equation embodying momentum-dependent mass.
Abstract: We show in this paper that the basic representations of position and momentum in a quantum mechanical system, that are guided by a generalized uncertainty principle and lead to a corresponding one-parameter eigenvalue problem, can be interpreted in terms of an extended Schrodinger equation embodying momentum-dependent mass. Some simple consequences are pointed out.
2 citations
References
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TL;DR: In this paper, a sufficient condition for monotonicity of the period function T of a center O of the title's equation is given, and a characterization of isochronous centers is given when f and g are odd and analytic.
76 citations
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TL;DR: In this paper, an exact quantization of a PT-symmetric (reversible) Lienard-type one-dimensional nonlinear oscillator both semiclassically and quantum mechanically was carried out.
Abstract: We carry out an exact quantization of a PT-symmetric (reversible) Lienard-type one-dimensional nonlinear oscillator both semiclassically and quantum mechanically. The associated time-independent classical Hamiltonian is of nonstandard type and is invariant under a combined coordinate reflection and time reversal transformation. We use the von Roos symmetric ordering procedure to write down the appropriate quantum Hamiltonian. While the quantum problem cannot be tackled in coordinate space, we show how the problem can be successfully solved in momentum space by solving the underlying Schrodinger equation therein. We explicitly obtain the eigenvalues and eigenfunctions (in momentum space) and deduce the remarkable result that the spectrum agrees exactly with that of the linear harmonic oscillator, which is also confirmed by a semiclassical modified Bohr–Sommerfeld quantization rule, while the eigenfunctions are completely different.
51 citations
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TL;DR: In this article, an exact quantization of a PT symmetric (reversible) Li-nard type one dimensional nonlinear oscillator both semiclassically and quantum mechanically was carried out.
Abstract: We carry out an exact quantization of a PT symmetric (reversible) Li\'{e}nard type one dimensional nonlinear oscillator both semiclassically and quantum mechanically. The associated time independent classical Hamiltonian is of non-standard type and is invariant under a combined coordinate reflection and time reversal transformation. We use von Roos symmetric ordering procedure to write down the appropriate quantum Hamiltonian. While the quantum problem cannot be tackled in coordinate space, we show how the problem can be successfully solved in momentum space by solving the underlying Schr\"{o}dinger equation therein. We obtain explicitly the eigenvalues and eigenfunctions (in momentum space) and deduce the remarkable result that the spectrum agrees exactly with that of the linear harmonic oscillator, which is also confirmed by a semiclassical modified Bohr-Sommerfeld quantization rule, while the eigenfunctions are completely different.
34 citations
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TL;DR: In this paper, a necessary and sufficient condition for the isochronicity of a point of (E ) to be an isochronous center is given. But this condition is not applicable to the case where the center is analytic (not necessary odd).
32 citations