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# First Integrals and Integrating Factors of Second-Order Autonomous Systems

[+] Author and Article Information
Tamás Kalmár-Nagy

Department of Fluid Mechanics,
Faculty of Mechanical Engineering,
Budapest University of
Technology and Economics,
Budapest 1111, Hungary
e-mail: jcnd@kalmarnagy.com

Balázs Sándor

Department of Hydraulic and
Water Resources Engineering,
Faculty of Civil Engineering,
Budapest University of
Technology and Economics,
Water Management Research Group—
Budapest 1111, Hungary
e-mail: sandor.balazs@epito.bme.hu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 8, 2017; final manuscript received May 21, 2018; published online July 26, 2018. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 13(9), 090901 (Jul 26, 2018) (7 pages) Paper No: CND-17-1409; doi: 10.1115/1.4040410 History: Received September 08, 2017; Revised May 21, 2018

## 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.

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