A Generalized Nonlinear Oscillator From Non-Standard Degenerate Lagrangians and Its Consequent Hamiltonian Formalism

Abstract

Nonlinear oscillators play an important role in science and engineering and they exist in many types depending on the physical problem and its corresponding dynamical system. Two important classes of nonlinear oscillators are (1) those characterized by the unusual property “a frequency completely independent of the amplitude and is the same as that of the linear harmonic oscillator” due to its significant impacts in nonlinear dynamics, chaos theory and pattern formation and (2) those characterized by a “position-dependent mass” as introduced by Mathews and Lakshmanan due to their relevance in describing many fundamental properties in science like quantum optics and metal clusters. These oscillators are also described by non-standard Lagrangians which is an important class of functions existing in some group of dissipative dynamical systems. Deriving the equations of motion for systems holding non-standard Lagrangians is a very motivating open problem and requires particular awareness. Motivated by these features, the main goal of this work is to show that a comparable nonlinear differential equation of a non-linear oscillator with periodic solutions to the one studied by Mathew and Lakshmanan may be derived as well from non-standard degenerate complexified Lagrangians. More precisely, we will prove that the equation of motion for such types of nonlinear oscillators with periodic solutions may be derived as well from non-standard degenerate Lagrangians which are linear in velocities. The generalized Hamiltonian formulation for the case of non-standard degenerate Lagrangians and based on Dirac’s theory for degenerate systems was also discussed.