First Integrals and Integrating Factors of Second-Order Autonomous Systems

Abstract

We present a new approach to the construction of first integrals for second-order autonomous systems without invoking a Lagrangian or Hamiltonian reformulation. We show and exploit the analogy between integrating factors of first-order equations and their Lie point symmetry and integrating factors of second-order autonomous systems and their dynamical symmetry. We connect intuitive and dynamical symmetry approaches through one-to-one correspondence in the framework proposed for first-order systems. Conditional equations for first integrals are written out, as well as equations determining symmetries. The equations are applied on the simple harmonic oscillator and a class of nonlinear oscillators to yield integrating factors and first integrals.