Acta Applicandae Mathematica

, Volume 66, Issue 1, pp 25–39

The Inverse Problem of the Calculus of Variations for Sixth- and Eighth-order Scalar Ordinary Differential Equations

  • Martin Juráš
Article

Abstract

On the equation manifold of the 2nth-order scalar ordinary differential equation, n≥3,
$$\frac{{\partial ^{2_n } u}}{{\partial x^{2_n } }} = f\left( {x,u,\frac{{\partial u}}{{\partial x}}, \ldots ,\frac{{\partial ^{2_{n - 1} } u}}{{\partial x^{2_{n - 1} } }}} \right),$$
we construct a contact two-form Π such that dΠ≡0 modΠ, if and only if Equation (1) admits a nondegenerate Lagrangian of order n. We show that the space of all nondegenerate Lagrangians for (1) is at most one-dimensional. The necessary and sufficient conditions for sixth-order and eighth-order scalar ordinary differential equation to admit a variational multiplier are found in terms of vanishing of a certain set of functions. The exact relationship between the Lie algebra of the classical infinitesimal contact symmetries of a variational Equation (1) and its the Lie subalgebra of infinitesimal divergence symmetries is established.
inverse problem of the calculus of variations variational principles for scalar ordinary differential equations variational bicomplex divergence symmetries 

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References

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    Anderson, I. M.: The Variational Bicomplex, Academic Press, New York.Google Scholar
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    Anderson, I. M. and Thompson, G.: The inverse problem of the calculus of variations for ordinary differential equations, Mem. Amer. Math. Soc. 98 (473) (1992).Google Scholar
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    Darboux, G.: Leçon sur la théorie générale des surfaces, Vol. III, Gauthier-Villars, Paris, 1894.Google Scholar
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    Fels, M. E.: The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations, Trans. Amer. Math. Soc. 384 (1996), 5007–5029.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Martin Juráš
    • 1
  1. 1.Department of MathematicsNorth Dakota, State UniversityFargoU.S.A.

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