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Abstract: We show that given an ordinary differential equation of order four, it may bepossible to determine a Lagrangian if the third derivative is absent oreliminated from the equation. This represents a subcase of Fels-conditions M.E. Fels, The inverse problem of the calculus of variations for scalarfourth-order ordinary differential equations, Trans. Amer. Math. Soc. 3481996 5007-5029 which ensure the existence and uniqueness of the Lagrangianin the case of a fourth-order equation. The key is the Jacobi last multiplieras in the case of a second-order equation. Two equations from a Number Theorypaper by Hall, one of second and one of fourth order, will be used to exemplifythe method. The known link between Jacobi last multiplier and Lie symmetries isalso exploited. Finally the Lagrangian of two fourth-order equations drawn fromPhysics are determined with the same method.



Autor: M.C. Nucci, A.M. Arthurs

Fuente: https://arxiv.org/







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