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Lagrangians, Gauge Functions, and Lie Groups for Semigroup of Second‐Order Differential Equations

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  • Z. E. Musielak
  • N. Davachi
  • M. Rosario-Franco

Abstract

A set of linear second‐order differential equations is converted into a semigroup, whose algebraic structure is used to generate novel equations. The Lagrangian formalism based on standard, null, and nonstandard Lagrangians is established for all members of the semigroup. For the null Lagrangians, their corresponding gauge functions are derived. The obtained Lagrangians are either new or generalization of those previously known. The previously developed Lie group approach to derive some equations of the semigroup is also described. It is shown that certain equations of the semigroup cannot be factorized, and therefore, their Lie groups cannot be determined. A possible solution of this problem is proposed, and the relationship between the Lagrangian formalism and the Lie group approach is discussed.

Suggested Citation

  • Z. E. Musielak & N. Davachi & M. Rosario-Franco, 2020. "Lagrangians, Gauge Functions, and Lie Groups for Semigroup of Second‐Order Differential Equations," Journal of Applied Mathematics, John Wiley & Sons, vol. 2020(1).
    • Handle: RePEc:wly:jnljam:v:2020:y:2020:i:1:n:3170130
    DOI: 10.1155/2020/3170130
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    References listed on IDEAS

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    1. N. Wilson & A. H. Kara, 2012. "Equivalent Lagrangians: Generalization, Transformation Maps, and Applications," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-19, November.
    2. N. Wilson & A. H. Kara, 2012. "Equivalent Lagrangians: Generalization, Transformation Maps, and Applications," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    3. Musielak, Z.E., 2009. "General conditions for the existence of non-standard Lagrangians for dissipative dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 2645-2652.
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    Cited by:

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