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Euler's rigid rotators, Jacobi elliptic functions, and the Dzhanibekov or tennis racket effect
American Journal of Physics ( IF 0.9 ) Pub Date : 2021-03-22 , DOI: 10.1119/10.0003372
Christian Peterson 1 , William Schwalm 1
Affiliation  

In this paper, the torque-free rotational motion of a general rigid body is developed analytically and is applied to the flipping motion of a T-handle spinning in zero gravity that can be seen in videos on the internet. This flipping motion is known both as the Dzhanibekov effect (after the cosmonaut who reported it) and more recently the tennis racket effect. The presentation is self-contained, accessible to students, and is complementary to the treatment found in most texts in that it involves a time-dependent analytical solution in terms of elliptic functions as opposed to a development based on conservation laws. These two complementary approaches are interesting and useful in different ways. In the present approach, the Euler rigid-body equations are derived and then solved as differential equations that are satisfied by Jacobi elliptic functions. This is analogous to solving the spring–mass harmonic oscillator problem by turning Newton's laws into differential equations that are satisfied by sine and cosine functions. The Jacobi functions are closely related to these trigonometric functions and are only slightly more complicated. They are defined as geometrical ratios on a reference ellipse and developed geometrically without reference to power series or complex variables. However, because these functions are less familiar, they are introduced in a short Appendix where their main properties are derived. Also, a link is provided to a Mathematica script for animating the analytical solution to the present problem.

中文翻译:

欧拉的刚性旋转器,雅可比椭圆函数以及Dzhanibekov或网球拍效果

在本文中,分析了一般刚体的无转矩旋转运动,并将其应用于以零重力旋转的T型手柄的翻转运动,这可以在互联网上的视频中看到。这种翻转动作既被称为Dzhanibekov效应(在举报的宇航员之后),最近被称为网球拍效应。该演示文稿是独立的,可供学生使用,并且是对大多数教科书中的处理方法的补充,因为它涉及椭圆函数方面的时变分析解决方案,而不是基于保护法则的开发。这两种互补的方法很有趣,并且以不同的方式有用。在目前的方法中,推导出欧拉刚体方程,然后将其求解为雅可比椭圆函数所满足的微分方程。这类似于通过将牛顿定律转化为正弦和余弦函数满足的微分方程来解决弹簧-质量谐振子问题。Jacobi函数与这些三角函数密切相关,只是稍微复杂一点。它们被定义为参考椭圆上的几何比率,并且在几何上展开而未参考幂级数或复杂变量。但是,由于这些功能不太熟悉,因此在简短的附录中对其进行了介绍,并在其中导出了它们的主要属性。另外,提供了指向Mathematica脚本的链接,以使当前问题的分析解决方案具有动画效果。这类似于通过将牛顿定律转化为正弦和余弦函数满足的微分方程来解决弹簧-质量谐振子问题。Jacobi函数与这些三角函数密切相关,只是稍微复杂一点。它们被定义为参考椭圆上的几何比率,并且在几何上展开而未参考幂级数或复杂变量。但是,由于这些功能不太熟悉,因此在简短的附录中对其进行了介绍,并在其中导出了它们的主要属性。另外,提供了指向Mathematica脚本的链接,以使当前问题的分析解决方案具有动画效果。这类似于通过将牛顿定律转化为正弦和余弦函数满足的微分方程来解决弹簧-质量谐振子问题。Jacobi函数与这些三角函数密切相关,只是稍微复杂一点。它们被定义为参考椭圆上的几何比率,并且在几何上展开而未参考幂级数或复杂变量。但是,由于这些功能不太熟悉,因此在简短的附录中对其进行了介绍,并在其中导出了它们的主要属性。另外,提供了指向Mathematica脚本的链接,以使当前问题的分析解决方案具有动画效果。Jacobi函数与这些三角函数密切相关,只是稍微复杂一点。它们被定义为参考椭圆上的几何比率,并且在几何上展开而未参考幂级数或复杂变量。但是,由于这些功能不太熟悉,因此在简短的附录中对其进行了介绍,并在其中导出了它们的主要属性。另外,提供了指向Mathematica脚本的链接,以使当前问题的分析解决方案具有动画效果。Jacobi函数与这些三角函数密切相关,只是稍微复杂一点。它们被定义为参考椭圆上的几何比率,并且在几何上展开而未参考幂级数或复杂变量。但是,由于这些功能不太熟悉,因此在简短的附录中对其进行了介绍,并在其中导出了它们的主要属性。另外,提供了指向Mathematica脚本的链接,以使当前问题的分析解决方案具有动画效果。
更新日期:2021-03-23
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