Article

Acta Applicandae Mathematica

, Volume 66, Issue 1, pp 25-39

First online:

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

  • Martin JurášAffiliated withDepartment of Mathematics, North Dakota, State University

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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