On the complete integrability of a nonlinear oscillator from group theoretical perspective

Abstract

In this paper, we investigate the integrability aspects of a physically important nonlinear oscillator which lacks sufficient number of Lie point symmetries but can be integrated by quadrature. We explore the hidden symmetry, construct a second integral and derive the general solution of this oscillator by employing the recently introduced $\lambda$-symmetry approach and thereby establish the complete integrability of this nonlinear oscillator equation from a group theoretical perspective.

15 pages

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• A Quantum Quasi-Harmonic Nonlinear Oscillator with an Isotonic Term
• Quantization of Hamiltonian systems with a position dependent mass: Killing vector fields and Noether momenta approach
• Isochronous $n$-dimensional nonlinear PDM-oscillators: linearizability, invariance and exact solvability
• n-dimensional PDM non-linear oscillators: Linearizability and Euler-Lagrange or Newtonian invariance
• $n$-dimensional PDM-damped harmonic oscillators: Linearizability, and exact solvability
• On the symmetries of a nonlinear non-polynomial oscillator
• Dynamics of the Modified Emden and pseudo-Modified Emden Equations: position-dependent mass, invariance and exact solvability
• A Class of Isochronous and Non-Isochronous Nonlinear Oscillators
• A study on Darboux polynomials and their significance in determining other integrability quantifiers: A case study in third-order nonlinear ordinary differential equations