On isochronous cases of the Cherkas system and Jacobi's last multiplier

Choudhury, A. Ghose; Guha, Partha

doi:10.1088/1751-8113/43/12/125202

Cited by 23 publications

(30 citation statements)

References 22 publications

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“…In [9], we constructed a Lagrangian for a system of the Liénard type using the Jacobi Last Multiplier (JLM) [10,11] and derived the corresponding Hamiltonian function. By constructing an obvious transformation of variables, we could easily map the Hamiltonian to that of the linear harmonic oscillator or to an isotonic potential, which enables us to study isochronous systems using the JLM.…”

Section: Introductionmentioning

confidence: 99%

The Jacobi Last Multiplier and Isochronicity of Liénard Type Systems

Guha

Choudhury²

2013

Rev. Math. Phys.

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We present a brief overview of classical isochronous planar differential systems focusing mainly on the second equation of the Liénard typeẍ + f (x)ẋ 2 + g(x) = 0. In view of the close relation between Jacobi's last multiplier and the Lagrangian of such a second-order ordinary differential equation, it is possible to assign a suitable potential function to this equation. Using this along with Chalykh and Veselov's result regarding the existence of only two rational potentials which can give rise to isochronous motions for planar systems, we attempt to clarify some of the previous notions and results concerning the issue of isochronous motions for this class of differential equations. In particular, we provide a justification for the Urabe criterion besides giving a derivation of the Bolotin-MacKay potential. The method as formulated here is illustrated with several well-known examples like the quadratic Loud system and the Cherkas system and does not require any computation relying only on the standard techniques familiar to most physicists.

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Section: Introductionmentioning

confidence: 99%

The Jacobi Last Multiplier and Isochronicity of Liénard Type Systems

Guha

Choudhury²

2013

Rev. Math. Phys.

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“…As shown in [12] (and references therein) this second-order equation admits a Lagrangian description via the existence of a Jacobi last multiplier (JLM) M(x), of the form…”

Section: Explicit Determination Of Isochronous Systems and The Jacobimentioning

confidence: 98%

“…4 ) + y 2 It is easy to verify that the condition stated in Proposition 3.2 is indeed satisfied by this system. The potential V (x) may be expressed as 1/8(Q 2 + Q −2 ) with Q = −(1 + x) −1 [12].…”

Section: The Role Of the Jacobi Last Multiplier And Isochronous Systemsmentioning

confidence: 99%

The role of the Jacobi last multiplier and isochronous systems

Guha

Choudhury²

2011

Pramana - J Phys

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“…However, the study of the isochronicity conditions is non-trivial, and the technique required considerable computational effort. The same problem was re-examined in [4,8] using the JLM to derive the conditions for isochronous solution behavior much more directly and with far less computational effort. Here we shall follow this latter approach to examine (10) for possible isochronous behavior.…”

Section: Search For Isochronous Behavior Via the Jlmmentioning

confidence: 99%

Lagrangian dynamics and possible isochronous behavior in several classes of non-linear second order oscillators via the use of Jacobi last multiplier

Tanriver

Choudhury

Gambino

2015

International Journal of Non-Linear Mechanics

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In this paper, we employ the technique of Jacobi Last Multiplier (JLM) to derive Lagrangians for several important and topical classes of non-linear second-order oscillators, including systems with variable and parametric dissipation, a generalized anharmonic oscillator, and a generalized Lane–Emden equation. For several of these systems, it is very difficult to obtain the Lagrangians directly, i.e., by solving the inverse problem of matching the Euler–Lagrange equations to the actual oscillator equation. In order to facilitate the derivation of exact solutions, and also investigate possible isochronous behavior in the analyzed systems, we next invoke some recent theoretical results and attempt to map the potential term to either the simple harmonic oscillator or the isotonic potential for specific values of the coefficient parameters of each non-linear oscillator. We find non-trivial parameter sets corresponding to isochronous dynamics in some of the considered systems, but none in others. Finally, the Lagrangians obtained here are coupled to Noether׳s theorem, leading to non-trivial conservation laws for several of the oscillators

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On isochronous cases of the Cherkas system and Jacobi's last multiplier

Cited by 23 publications

References 22 publications

The Jacobi Last Multiplier and Isochronicity of Liénard Type Systems

The Jacobi Last Multiplier and Isochronicity of Liénard Type Systems

The role of the Jacobi last multiplier and isochronous systems

Lagrangian dynamics and possible isochronous behavior in several classes of non-linear second order oscillators via the use of Jacobi last multiplier

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