On the Jacobi Last Multiplier, integrating factors and the Lagrangian formulation of differential equations of the Painlevé-Gambier classification

A. Ghose Choudhury, Partha Guha, Barun Khanra

Research output: Contribution to journalArticlepeer-review

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Abstract

We use a formula derived almost seventy years ago by Madhav Rao connecting the Jacobi Last Multiplier of a second-order ordinary differential equation and its Lagrangian and determine the Lagrangians of the Painlevé equations. Indeed this method yields the Lagrangians of many of the equations of the Painlevé-Gambier classification. Using the standard Legendre transformation we deduce the corresponding Hamiltonian functions. While such Hamiltonians are generally of non-standard form, they are found to be constants of motion. On the other hand for second-order equations of the Liénard class we employ a novel transformation to deduce their corresponding Lagrangians. We illustrate some particular cases and determine the conserved quantity (first integral) resulting from the associated Noetherian symmetry. Finally we consider a few systems of second-order ordinary differential equations and deduce their Lagrangians by exploiting again the relation between the Jacobi Last Multiplier and the Lagrangian.

Original languageBritish English
Pages (from-to)651-664
Number of pages14
JournalJournal of Mathematical Analysis and Applications
Volume360
Issue number2
DOIs
StatePublished - 15 Dec 2009

Keywords

  • First integral
  • Jacobi's Last Multiplier
  • Lagrangian
  • Painlevé equations

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