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The Math Behind Diving Can Be as Beautiful as Diving Itself

By Karthika Swamy Cohen

The mathematics of the twisting somersault. View referenced paper for details.
The Olympic sport of springboard and platform diving involves spectacular aerial acrobatics that can be a pleasure to watch. But have you ever thought that the many somersaults and twists performed in varied forms have beautiful math behind them as well? 

An athlete creates angular momentum during take-off and implements shape changes while airborne in order to accomplish the desired dive, both of which can be expressed as mathematical equations. 

In the case of somersaults in a tuck position with minimal inertial moments, the rotation axis and hence the angular velocity direction remain constant. While the values of the principal moments of inertia change due to the shape change, the principal axis does not change. Hence, from a mathematical standpoint, this is a simple dive. 

The mathematically more complex, and hence interesting, dive is the twisting somersault, since this includes a shape change which moves the principal axis. The dive thus generates a motion in which the rotational axis is not constant. Mathematical models can be used to obtain an understanding of the mechanics of twists produced and removed during such twisting somersaults.

In a recent paper published in the SIAM Journal on Applied Dynamical Systems, authors Holger Dullin and William Tong use modern tools from dynamical systems to understand the twisting somersault.

The mathematics of the twisting somersault. View referenced paper for details.
Using geometric mechanics methods, the authors utilize a kick-model, which allows one to analyze the principle requirements for a successful dive. 

In the simplest kick-model, the number of somersaults m, and the number of twists n are obtained through a rational rotation number W = m/n of a (rigid) Euler top. The authors first derive a version of the Euler equation for a shape-changing body. The equations are simple, deriving the explicit form of the important time-dependent terms for a system of coupled rigid bodies. 

They then take the case of a simple system of just two coupled rigid bodies—that is, the one-armed diver—and use their model to show how a twisting somersault can be achieved. In this simplified model, the kick changes the trajectory and the energy, but not the total angular momentum. 

The authors extend this to analyze the full model without the kick assumption, and derive the twisting-somersault formula, which factors in the airborne time of the diver, the time spent in various stages of the dive, the number of somersaults, the number of twists, the energy in the stages, and the angular momentum.

They use an analytically solvable approximation in which the shape change is instantaneous. The analysis sheds light on how a shape change generates a change in rotation in the presence of angular momentum.

By emphasizing the geometric mechanics, the translational and rotational symmetry of the problem is reduced. In this reduced definition, the amount of somersault—that is, the amount of rotation about the fixed angular momentum—is not present. Reconstruction allows recovery of this angle by solving an additional differential equation driven by the solution of the reduced equations. 

Numerical simulations for various dives are seen to validate the formula where realistic parameters are taken from actual observations of divers.

The team’s findings also apply to other sports—like aerial skiing—that involve general coupled rigid body dynamics. This can also be applied to spacecraft attitude control, which consists of two bodies mutually rotating around a common axis. 

Read the full version of this article now (available free for 90 days), and view other SIAM Nuggets.

  Karthika Swamy Cohen is the managing editor of SIAM News.

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