Using the geometric phase to optimize planar somersaults

William Tong*, Holger R. Dullin

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We derive the equations of motion for the planar somersault, which consist of two additive terms. The first is the dynamic phase that is proportional to the angular momentum, and the second is the geometric phase that is independent of angular momentum and depends solely on the details of the shape change. Next, we import digitized footage of an elite athlete performing 3.5 forward somersaults off the 3m springboard and use the data to validate our model. We show that reversing and reordering certain sections of the digitized dive can maximize the geometric phase without affecting the dynamic phase, thereby increasing the overall rotation achieved. Finally, we propose a theoretical planar somersault consisting of four shape-changing states, where the optimization lies in finding the shape-change strategy that maximizes the overall rotation of the dive. This is achieved by balancing the rotational contributions from the dynamic and geometric phases, in which we show the geometric phase plays a small but important role in the optimization process.

Original languageEnglish
Pages (from-to)44-62
Number of pages19
JournalIMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)
Issue number1
Publication statusPublished - Feb 2019


  • somersault
  • nonrigid body dynamics
  • geometric phase
  • biomechanics


Dive into the research topics of 'Using the geometric phase to optimize planar somersaults'. Together they form a unique fingerprint.

Cite this