Integrability and Chaos in Figure Skating
Integrability and Chaos in Figure Skating
复制标题
花样滑冰中的可积性和混沌
DOI：
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发表时间：
2018
影响因子：
3
通讯作者：
V. Putkaradze
中科院分区：
文献类型：
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作者：
Vaughn Gzenda;V. Putkaradze
我们推导并分析了一个花样滑冰运动员的三维模型。我们将滑冰者建模为一个在空间中运动的三维体，受非完整约束和连续接触冰面的完整约束。对于一个静态的（非铰接的）滑冰者，我们证明了系统是可积的当且仅当质心在滑冰者方向上的投影与与冰的接触点一致，以及对惯量轴方向的一些温和的（和现实的）假设。通过证明两个新的动量线性运动常数的存在性，证明了系统的可积性，给出了可积非完整力学系统的一个新的、高度非平凡的例子。我们还考虑了质心在滑板方向上的投影与接触点不重合的情况，并通过研究附近轨迹的散度证明了这种不可积情况表现出明显的混沌行为。我们还证明了从可积到混沌过渡过程中的复杂行为。我们的模型显示了现实滑冰的许多特征，特别是花样滑冰，我们推测现实滑冰运动员可能会直观地使用系统的力学特性来控制冰上的表现。
We derive and analyze a three-dimensional model of a figure skater. We model the skater as a three-dimensional body moving in space subject to a non-holonomic constraint enforcing movement along the skate’s direction and holonomic constraints of continuous contact with ice and pitch constancy of the skate. For a static (non-articulated) skater, we show that the system is integrable if and only if the projection of the center of mass on skate’s direction coincides with the contact point with ice and some mild (and realistic) assumptions on the directions of inertia’s axes. The integrability is proved by showing the existence of two new constants of motion linear in momenta, providing a new and highly non-trivial example of an integrable non-holonomic mechanical system. We also consider the case when the projection of the center of mass on skate’s direction does not coincide with the contact point and show that this non-integrable case exhibits apparent chaotic behavior, by studying the divergence of nearby trajectories. We also demonstrate the intricate behavior during the transition from the integrable to chaotic case. Our model shows many features of real-life skating, especially figure skating, and we conjecture that real-life skaters may intuitively use the discovered mechanical properties of the system for the control of the performance on ice.