## 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 language | British English |
---|---|

Pages (from-to) | 651-664 |

Number of pages | 14 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 360 |

Issue number | 2 |

DOIs | |

State | Published - 15 Dec 2009 |

## Keywords

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