利用对称性和母函数计算和最小化转动惯量
Using symmetries and generating functions to calculate and minimize moments of inertia
摘要：
本文给出了旋转体和等高线图生成体转动惯量的计算公式。为此，利用图形的对称性和生成函数。生成函数和对称性的结合使用大大简化了许多实际计算。此外，生成函数的显式使用给出了将转动惯量写成泛函的可能性，这反过来又允许利用变分法来获得关于这个基本量的一些性质的新见解。特别是，在某些限制下的惯性矩的最小化是可能的，通过使用变分方法。
英文摘要：
This paper presents some formulae to calculate moments of inertia for solids of revolution and for solids generated by contour plots. For this, the symmetry properties and the generating functions of the figures are utilized. The combined use of generating functions and symmetry properties greatly simplifies many practical calculations. In addition, the explicit use of generating functions gives the possibility of writing the moment of inertia as a functional, which in turn allows to utilize the calculus of variations to obtain a new insight about some properties of this fundamental quantity. In particular, minimization of moments of inertia under certain restrictions is possible by using variational methods.
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