Lagrangian dynamics and possible isochronous behavior in several classes of non-linear second order oscillators via the use of Jacobi last multiplier. (September 2015)

• Record Type: Journal Article
• Authors: Tanriver, U.; Roy Choudhury, S.; Gambino, G.
• Is Part Of: International journal of non-linear mechanics. Volume 74(2015)
• Page Start: 100
• Page End: 107
• Publication Date: 2015-09
• Subjects: Lagrangians -- Jacobi Last Multiplier (JLM) -- Isochronous dynamics -- Simple harmonic oscillator -- Isotonic potentials -- Conservation laws; Nonlinear mechanics -- Periodicals; Mécanique non linéaire -- Périodiques; Nonlinear mechanics; Periodicals; 531
• DOI: 10.1016/j.ijnonlinmec.2015.04.006
• Languages: English
• ISSNs: 0020-7462

Abstract

Abstract: 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. Abstract : Highlights: Topical classes of non-linear second-order oscillators (with variable and parametric dissipation and generalized anharmonic oscillator). Derivation of the Lagrangian via Jacobi Last Multiplier. Search of isochronous behavior using the Jacobi Last Multiplier. Conservation laws via Noether׳s theorem.