Ivancevic_Applied-Diff-Geom

218 Applied Differential Geometry: A Modern Introductionor, in coordinates [Marsden and Ratiu (1999)],Ad (Rθ ,a)(ξ, v) = (ξ, ξJa + R θ v). (3.54)In proving (3.54) we used the identity(R θ J = JR θ . Identify the dualµalgebra, se(2) ∗ , with matrices of the form2 J 0 ), via the nondegenerateα 0pairing given by the trace of the product. Thus, se(2) ∗ is isomorphic to R 3via( µ2 J 0 )∈ se(2) ∗ ↦−→ (µ, α) ∈ R 3 ,α 0so that in these coordinates, the pairing between se(2) ∗ and se(2) becomes〈(µ, α), (ξ, v)〉 = µξ + α · v,that is, the usual dot product in R 3 . The coadjoint group action is thusgiven byAd ∗ (R θ ,a) −1(µ, α) = (µ − R θα · Ja + R θ α). (3.55)Formula (3.55) shows that the coadjoint orbits are the cylinders T ∗ S 1 α ={(µ, α)| ‖α‖ = const} if α ≠ 0 together with the points are on the µ−axis.The canonical cotangent bundle projection π : T ∗ S 1 α → S 1 α is defined asπ(µ, α) = α.Special Euclidean Group in the 3D SpaceThe most common group structure in human–like biodynamics is thespecial Euclidean group in 3D space, SE(3). It is defined as a semidirect(noncommutative) product of 3D rotations and 3D translations, SO(3)✄R 3 .The Heavy TopAs a starting point consider a rigid body (see (3.12.3.2) below) movingwith a fixed point but under the influence of gravity. This problem still has aconfiguration space SO(3), but the symmetry group is only the circle groupS 1 , consisting of rotations about the direction of gravity. One says thatgravity has broken the symmetry from SO(3) to S 1 . This time, eliminatingthe S 1 symmetry mysteriously leads one to the larger Euclidean groupSE(3) of rigid motion of R 3 . Conversely, we can start with SE(3) as