旋转体的转动惯量与变分法
Moments of inertia for solids of revolution and variational methods
摘要：
本文用生成旋转体的函数给出了旋转体的转动惯量的一些公式。这些表达式的开发利用了这些对象的圆柱对称性，避免了明确使用多重积分，提供了一种简单的教学方法。显式地使用生成实体的函数使得将转动惯量写成泛函成为可能，这反过来又使我们能够利用变分法来获得对这个基本量的某些性质的新认识。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：
We present some formulae for the moments of inertia of homogeneous solids of revolution in terms of the functions that generate the solids. The development of these expressions exploits the cylindrical symmetry of these objects, and avoids the explicit use of multiple integration, providing an easy and pedagogical approach. The explicit use of the functions that generate the solid gives the possibility of writing the moment of inertia as a functional, which in turn allows us to utilize the calculus of variations to obtain a new insight into 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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