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

Article

DOI: 10.1023/A:1010660232439

- Cite this article as:
- Juráš, M. Acta Applicandae Mathematicae (2001) 66: 25. doi:10.1023/A:1010660232439

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## Abstract

On the equation manifold of the 2 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*th-order scalar ordinary differential equation,*n*≥3,$$\frac{{\partial ^{2_n } u}}{{\partial x^{2_n } }} = f\left( {x,u,\frac{{\partial u}}{{\partial x}}, \ldots ,\frac{{\partial ^{2_{n - 1} } u}}{{\partial x^{2_{n - 1} } }}} \right),$$

*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 variationsvariational principles for scalar ordinary differential equationsvariational bicomplexdivergence symmetries

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© Kluwer Academic Publishers 2001