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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—
Hungarian Academy of Sciences,
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

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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