A New Twisting Somersault

By Holger R. Dullin

Platform and springboard diving are among the most beautiful Olympic sports. A fascinating outcome from our study of aerial motion and its mathematical description is the suggestion of a new twisting somersault [5] with five full twists, called “513XD.” The Fédération internationale de natation (FINA) diving code states that 513XD has three half-somersaults and \(X=10\) half-twists. This dive has not yet been performed, but we believe that it is humanly possible and divers will perform it at world-class competitions in the future. “Bodies in Space,” the ARC-funded research project that supported this work, is being carried out jointly with the New South Wales Institute of Sports in Sydney, Australia.

What mathematics led to the discovery of this tricky new dive?

Non-rigid Body Dynamics

Differential equations describing the motion of the human body have been used in biomechanics for a long time. Biomechanists typically model the body as a collection of rigid pieces coupled by joints. Inspired by geometric mechanics, our approach exploits the symmetry of the equations and, most importantly, clearly separates the shape of the athlete from the overall orientation. It all starts with Euler’s equation

\[\mathbf{\dot{L}} = \mathbf{L} \times \mathbf{\Omega}, \tag1 \]

where \(\mathbf{L} = \in \mathbb{R}^3\) is the angular momentum vector and \(\mathbf{\Omega} \in \mathbb{R}^3\) is the angular velocity vector, both in a body frame. An orthogonal matrix \(R \in SO(3)\) describes the boy's orientation. The matrix \(R\) transforms vectors from the body frame to the space-fixed frame, so that \(R \mathbf{L}\) gives the constant angular momentum in space. Differentiating the constant vector \(R \mathbf{L}\) with respect to time yields \(\dot{R} \mathbf{L} + R \mathbf{\dot{L}} = 0\), and solving for \(\mathbf{\dot{L}}\) gives Euler’s equation \((1)\). Differentiating \(R^t R = id\) demonstrates that \(R^t \dot R\) is antisymmetric. The usual hat-map from \(\mathbb{R}^3\) to \(so(3)\) identifies the angular momentum as \(\hat \Omega = R^t \dot R\). The essential point is that \((1)\) is true even for a shape-changing body, as long as angular momentum conservation holds.

Computing the angular momentum for a shape-changing body [3] gives

\[\mathbf{L} = I \Omega + \mathbf{A}, \tag2 \]

where \(I\) is the body’s tensor of inertia and \(\mathbf{A} \in \mathbb{R}^3\) is a momentum shift vector. The difference between rigid and non-rigid body dynamics now appears. When the shape is constant, the vector \(\mathbf{A}\) is zero and Euler’s rigid body equations are thus recovered. When the shape is changing, the symmetric moment of inertia tensor \(I(t)\) and the momentum shift vector \(\mathbf{A}(t)\) encompass all the complexity of a particular coupled rigid body model for the human body. Given a particular shape change, one can compute \(I(t)\) and \(\mathbf{A}(t)\); explicit formulas for this are available in [3]. Maurice Raymond Yeadon first derived similar equations for the description of the twisting somersault, as described in his collected papers [6].

Let us first describe a somersault. Fix a coordinate system in the body’s trunk where the \(x\)-axis points out of the chest, the \(y\)-axis points to the left, and the \(z\)-axis points towards the head. By definition, a somersault is a motion where \(R\) is a rotation about a fixed \(\mathbf{L} = (0, l, 0)^t\), which is an eigenvector of \(I\). When the diver pulls into a tuck position, the corresponding eigenvalue of \(I\) decreases and the angular momentum \(\mathbf{\Omega}\) increases, as determined by \((2)\). While the shape changes, the momentum shift \(\mathbf{A}\) is non-zero but parallel to \(\mathbf{L}\), so that all three vectors remain parallel throughout the dive and—in the body frame—the solution remains at the equilibrium point \(\mathbf{\dot{L}} = 0\).

The Kick Approximation

A twisting somersault dive begins with a somersault, followed by a shape change for which \(\mathbf{A}\) is not parallel to \(\mathbf{L}\). This motion moves \(\mathbf{L}\) away from the equilibrium point and the body starts twisting, with vector \(\mathbf{L}\) revolving about the \(z\)-axis. To understand the dynamics, let’s consider a fast shape change, which makes \(\mathbf{A}\) arbitrarily large in the kick-limit. In this limit, \((1)\) and \((2)\) yield

\[\mathbf{\dot{L}} \approx -\mathbf{L} \times I^{-1} \mathbf{A}, \tag3 \]

a linear, time-dependent differential equation. The shape change is simple to integrate when it occurs such that the direction of \(I^{-1} \mathbf{A}\) remains constant, and the solution is a rotation about that direction. Moving a stretched arm in the \(yz\)-plane produces a rotation \(R_x\) about the \(x\)-axis. In a typical twisting somersault, the first arm motion starts the twisting and—following a full number of twists—reversing the arm motion stops it.

We observed that when the second arm motion is performed after a half twist instead of a full twist, it has the opposite effect: instead of stopping the twist, it speeds it up. The reason for this is geometrically simple. Let \(\alpha\) be the amount of rotation generated by the arm motion. While \(R_x(\alpha) R_z(2\pi) R_x(-\alpha)\) is the identity, \(R_x(\alpha) R_z(\pi) R_x(-\alpha)= R_x(2\alpha) R_z(\pi)= R_z(\pi) R_x(-2\alpha)\) is not. So the initial \(\mathbf{L} = (0,l,0)^t\) moves closer to the pole in the second case. The twisting motion speeds up as \(\mathbf{L}\) gets closer to the pole, and thus the second arm kick after a half-twist increases the twisting speed (see Figure 1).

The Full Model

This argument is no doubt approximate, and the true motion is more complicated than the analysis indicates. However, it does convey the central idea. More details and a full numerical simulation, in which the arm motions are performed with realistic speed, confirm the mechanism’s validity [5].¹ When the second arm motion involves both arms—one down, and one up, like the wings of a windmill (see Figure 2)—it achieves an effect roughly twice as big. We amplify that effect in [1], where a rotating disc replaces the arms. A reverse arm motion occurring a full number of twists later stops the high-speed twist, and a fourth arm motion stops the twisting altogether, as in Figure 1.

But how much does the body rotate in space? A diver must perform a half-integer number of somersaults for the dive to be successful overall. Symmetry reduction to the body frame eliminates the rotation about the fixed angular momentum vector in space. But geometric mechanics teaches us that one can recover the missing somersault angle as a combination of a geometric phase and a dynamic phase from data of the reduced equations alone. Richard Montgomery [4] does this for rigid bodies, and Alejandro Cabrera and La Plata [2] do the same for non-rigid bodies. We extend these formulas to our setting in [3], and show that within a certain limit, the ratio of the number of somersaults to the number of twists is a rotation number of the integrable Euler top. All of this leads to a good theoretical understanding of the twisting somersault; now we hope to find a volunteer athlete to try the new 513XD dive—with five full twists—in practice!

¹ View an animation of the 513XD dive here.

References
[1] Bharadwaj, S., Duignan, N., Dullin, H.R., Leung, K., & Tong, W. (2016). The diver with a rotor. Indagationes Mathematicae, 27, 1147-1161.
[2] Cabrera, A., & Plata, L. (2007). A generalized montgomery phase formula for rotating self-deforming bodies. Journal of Geometry and Physics, 57, 1405-1420.
[3] Dullin, H.R., & Tong, W. (2016). Twisting somersault. SIAM Journal on Applied Dynamical Systems, 15, 1806-1822.
[4] Montgomery, R. (1991). How much does the rigid body rotate? A Berry-phase from the 18th century. American Journal of Physics, 59, pp. 394-398.
[5] Tong, W., & Dullin, H.R. (2015). A new twisting somersault - 513XD. Preprint, arxiv:1612.06455.
[6] Yeadon, M.R. (2015). Twisting Somersaults. SB & MC.

Holger R. Dullin is a professor of applied mathematics at the University of Sydney in Australia. His research interests are Hamiltonian dynamical systems, including integrable systems, fluid dynamics, (non-) rigid body dynamics, and semiclassical quantization.