TY - JOUR

T1 - Non-holonomic geometric structures of rigid body system in Riemann-Cartan space

AU - Yajima, Takahiro

AU - Yamasaki, Kazuhito

AU - Nagahama, Hiroyuki

N1 - Publisher Copyright:
© 2018 The Author(s).

PY - 2018/8

Y1 - 2018/8

N2 - 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.

AB - 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.

KW - Free rigid body

KW - Non-holonomic system

KW - Riemann-Cartan geometry

KW - Topological singularity

KW - Torsion tensor

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U2 - 10.1088/2399-6528/aad600

DO - 10.1088/2399-6528/aad600

M3 - Article

AN - SCOPUS:85078358620

SN - 2399-6528

VL - 2

JO - Journal of Physics Communications

JF - Journal of Physics Communications

IS - 8

M1 - 085008

ER -