# 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...

6 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.

Abstract: The most common problems in nature are about non-conservative non-linearity. Non-conservative non-linear problems can be studied with variational problems of non-standard Lagrangians. Birkhoffian mechanics, as an extension of Hamiltonian mechanics naturally, is a sign that analytical mechanics has entered a new stage of development. Therefore, the study of dynamics based on non-standard Birkhoffians provides a new idea for solving non-conservative nonlinear dynamics problems. In this paper the dynamics models based on non-standard Birkhoffians, including exponential Birkhoffian, power law Birkhoffian, and logarithm Birkhoffian, are proposed, which are called non-standard Birkhoffian systems. Firstly, the Pfaff-Birkhoff principles with non-standard Birkhoffians are established, the differential equations of motion of non-standard Birkhoffian dynamics are also derived. Secondly, in accordance with the invariance of Pfaff action under the infinitesimal transformations, giving the definitions and criteria of Noether symmetric and quasi-symmetric transformations of non-standard Birkhoffian dynamics. And next, Noether’s theorems for non-standard Birkhoffian dynamics are proved, and the connections between Noether symmetries and conserved quantities of non-standard Birkhoffian dynamics are established; Finally, three examples are given to illustrate the applications of the results.

4 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

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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.

1 citations

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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.

Abstract: We study the period function T of a center O of the title's equation. A sufficient condition for the monotonicity of T , or for the isochronicity of O , is given. Such a condition is also necessary, when f and g are odd and analytic. In this case a characterization of isochronous centers is given. Some classes of plane systems equivalent to such equation are considered, including some Kukles’ systems.

67 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.

49 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.

32 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).

Abstract: In this work we study Eq. ( E ) x ¨ + f ( x ) x ˙ 2 + g ( x ) = 0 with a center at 0 and investigate conditions of its isochronicity. When f and g are analytic (not necessary odd) a necessary and sufficient condition for the isochronicity of 0 is given. This approach allows us to present an algorithm for obtained conditions for a point of ( E ) to be an isochronous center. In particular, we find again by another way the isochrones of the quadratic Loud systems ( L D , F ) . We also classify a 5-parameters family of reversible cubic systems with isochronous centers.

31 citations

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