关于变分法的反问题
On the inverse problem of calculus of variations
摘要：
我们表明，给定一个四阶常微分方程，它可能是可能的，以确定拉格朗日如果三阶导数是缺席（或消除）的方程。这是费尔斯条件[M. E. Fels，标量四阶常微分方程变分法的逆问题，Trans.Amer.Math.Soc.348（1996）5007-5029]，其确保了在四阶方程的情况下拉格朗日量的存在性和唯一性。关键是Jacobi最后一个乘子，就像二阶方程一样。两个方程从数论论文霍尔，一个二阶和一个四阶，将被用来验证的方法。雅可比最后乘子和李对称性之间的已知联系也被利用。最后用同样的方法求出了两个物理学中的四阶方程的拉格朗日量。
英文摘要：
We show that given an ordinary differential equation of order four, it may be possible to determine a Lagrangian if the third derivative is absent (or eliminated) from the equation. This represents a subcase of Fels'conditions [M. E. Fels, The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations, Trans. Amer. Math. Soc. 348 (1996) 5007-5029] which ensure the existence and uniqueness of the Lagrangian in the case of a fourth-order equation. The key is the Jacobi last multiplier as in the case of a second-order equation. Two equations from a Number Theory paper by Hall, one of second and one of fourth order, will be used to exemplify the method. The known link between Jacobi last multiplier and Lie symmetries is also exploited. Finally the Lagrangian of two fourth-order equations drawn from Physics are determined with the same method.
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