【摘 要】
：
We use Frolicher-Nijenhuis theory to reformulate the four Helmholtz conditions[3,4] for the inverse problem of the calculus of variations in terms of semi-b
【机 构】
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Faculty of Mathematics,“Al,I.Cuza” University
【出 处】
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The Third International Conference on Dynamics,Vibration and
论文部分内容阅读
We use Frolicher-Nijenhuis theory to reformulate the four Helmholtz conditions[3,4] for the inverse problem of the calculus of variations in terms of semi-basic 1-forms and linear partial differential operators, three of order one and one of order two[1].In the homogeneous case, which corresponds to projective metrizability and Finsler metrizability, we prove that some of the Helmholtz conditions dependent on the other ones.More exactly, we show that a spray is projectively metrizable if and only if two Helmholtz conditions are satisfied and it is Finsler metrizable if and only if three Helmholtz conditions are satisfied.We use Spencers technique [2] to address the formal integrabillity of partial differential operators that define the Helmholtz conditions.A special attention is paid to the formal integrability of these operators in the homogeneous case.
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