Abstract
On the equation manifold of the 2nth-order scalar ordinary differential equation, n≥3,
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.
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Juráš, M. The Inverse Problem of the Calculus of Variations for Sixth- and Eighth-order Scalar Ordinary Differential Equations. Acta Applicandae Mathematicae 66, 25–39 (2001). https://doi.org/10.1023/A:1010660232439
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DOI: https://doi.org/10.1023/A:1010660232439