Rigid body motions and Arnol’d’s theory of fronts on \(\mathbb S^2 \subset \mathbb R^3\). (English) Zbl 1021.70002
Summary: To any co-oriented curve (front) in the two-dimensional sphere \(\mathbb S^2\) we associate a rigid body motion together with an instantaneous axis of rotation. We prove that the pair of (antipodal) curves on the sphere determined by the instantaneous axis of rotation coincides with the envelope of the great circles normal to the original co-oriented curve (this envelope is called the caustic of the curve). The cusps of the caustic correspond to the points of the co-oriented curve for which the instantaneous axis of rotation is stationary. These results are stated and proved in the setting of Legendrian curves and contact geometry.
MSC:
| 70B10 | Kinematics of a rigid body |
| 70G45 | Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics |
Keywords:
rigid body motion; contact geometry; wave front; co-oriented curve; Legendrian curves; two-dimensional sphere; antipodal curves; instantaneous axis of rotation; caustic of curveReferences:
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