G2 Group in Mechanical Problems

Author: Irakli Mantidze
Keywords: Algebra, Group, Octonion.
Annotation:

Consider two balls of different sizes, rolling on each other, without slipping or spinning. (noslip/no-spin condition). System has symmetry group SO3×SO3 arising from the isometry groups of each ball, but for balls whose radii are in the ratio 3:1, and only for this ratio: the local symmetry group of the system increases from SO3×SO3 to G2, a 14-dimensional Lie group. G2 is interesting mathematical object, which can be realized as the automorphism group of the octonions. Many problems can be stated and solved differently using octonions.