Non-holonomic geometric structures of rigid body system in Riemann-Cartan space

Takahiro Yajima, Kazuhito Yamasaki, Hiroyuki Nagahama

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


A theory of non-Riemannian geometry (Riemann-Cartan geometry) can be applied to a free rotation of a rigid body system. The Euler equations of angular velocities are transformed into equations of the Euler angle. This transformation is geometrically non-holonomic, and the Riemann-Cartan structure is associated with the system of the Euler angles. Then, geometric objects such as torsion and curvature tensors are related to a singularity of the Euler angle. When a pitch angle becomes singular ±π/2, components of the torsion tensor diverge for any shape of the rigid body while components of the curvature tensor do not diverge in case of a symmetric rigid body. Therefore, the torsion tensor is related to the singularity of dynamics of the rigid body rather than the curvature tensor. This means that the divergence of the torsion tensor is interpreted as the occurrence of the gimbal lock. Moreover, attitudes of the rigid body for the singular pitch angles ±π/2 are distinguished by the condition that a path-dependence vector of the Euler angles diverges or converges.

Original languageEnglish
Article number085008
JournalJournal of Physics Communications
Issue number8
Publication statusPublished - 2018 Aug


  • Free rigid body
  • Non-holonomic system
  • Riemann-Cartan geometry
  • Topological singularity
  • Torsion tensor

ASJC Scopus subject areas

  • Physics and Astronomy(all)


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