## The Inverse Problem of the Calculus of Variations for Ordinary Differential Equations, Issue 473 (Google eBook)This monograph explores various aspects of the inverse problem of the calculus of variations for systems of ordinary differential equations. The main problem centres on determining the existence and degree of generality of Lagrangians whose system of Euler-Lagrange equations coicides with a given system of ordinary differential equations. The authors rederive the basic necessary and sufficient conditions of Douglas for second order equations and extend them to equations of higher order using methods of the variational bicomplex of Tulcyjew, Vinogradov, and Tsujishita. The authors present an algorithm, based upon exterior differential systems techniques, for solving the inverse problem for second order equations. a number of new examples illustrate the effectiveness of this approach. |

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### Contents

1 | |

EQUATIONS | 12 |

3 FIRST INTEGRALS AND THE INVERSE PROBLEM FOR SECOND ORDER EQUATIONS | 27 |

4 THE INVERSE PROBLEM FOR FOURTH ORDER ORDINARY DIFFERENTIAL EQUATIONS | 35 |

5 EXTERIOR DIFFERENTIAL SYSTEMS AND THE INVERSE PROBLEM FOR SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS | 50 |

6 EXAMPLES | 64 |

7 THE INVERSE PROBLEM FOR TWO DIMENSIONAL SPRAYS | 88 |

### Common terms and phrases

alternative Lagrangians arbitrary functions calculus of variations Cartan characters Cartan test Cartan-Kahler theorem closed form coefficients compute connection contact forms contact one forms coordinates defined Department of Mathematics dependent variables dH closed differential conditions differential ideal differential operator dimension Douglas equation manifold equivalent Euler-Lagrange equations Euler-Lagrange operator example exterior derivatives exterior differential systems finite dimensional formula fourth order functions F Helmholtz conditions Henneaux higher order implies independent invariant inverse problem jet bundle LEMMA linear Math matrix modulo non-singular order Lagrangian order ordinary differential ordinary differential equations paper partial differential equations Problem 5.1 PROOF PROPOSITION prove Ricci tensor satisfy second order equations second order ordinary skew-symmetric skew-symmetric Ricci solve subspace symmetric symmetric matrix system of equations system of fourth system of second tableau tensor is recurrent Theorem 2.6 torsion total derivative variational bicomplex variational multiplier problem variational principle