The Inverse Problem of the Calculus of Variations for Sixth- and Eighth-order Scalar Ordinary Differential Equations期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:

Martin Juráš

Affiliation:

(1) Department of Mathematics, North Dakota, State University, Fargo, ND 58105-5075, U.S.A.

Abstract:

On the equation manifold of the 2nth-order scalar ordinary differential equation, n

$frac{{partial ^{2_n } u}}{{partial x^{2_n } }} = fleft( {x,u,frac{{partial u}}{{partial x}}, ldots ,frac{{partial ^{2_{n - 1} } u}}{{partial x^{2_{n - 1} } }}} right),$

we construct a contact two-form Pgr

such that d Pgr

mod

, 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.

Keywords:

inverse problem of the calculus of variations variational principles for scalar ordinary differential equations variational bicomplex divergence symmetries

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