Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 219the configuration space for the rigid–body and ‘reduce out’ translations toarrive at SO(3) as the configuration space (see [Marsden and Ratiu (1999)]).The equations of motion for a rigid body with a fixed point in a gravitationalfield give an interesting example of a system that is Hamiltonian (see(3.12.3.2)) relative to a Lie–Poisson bracket (see (3.13.2)). The underlyingLie algebra consists of the algebra of infinitesimal Euclidean motions in R 3 .The basic phase–space we start with is again T ∗ SO(3), parameterized byEuler angles and their conjugate momenta. In these variables, the equationsare in canonical Hamiltonian form. However, the presence of gravity breaksthe symmetry, and the system is no longer SO(3) invariant, so it cannotbe written entirely in terms of the body angular momentum p. One alsoneeds to keep track of Γ, the ‘direction of gravity’ as seen from the body.This is defined by Γ = A −1 k, where k points upward and A is the elementof SO(3) describing the current configuration of the body. The equationsof motion areṗ 1 = I 2 − I 3I 2 I 3p 2 p 3 + Mgl(Γ 2 χ 3 − Γ 3 χ 2 ),ṗ 2 = I 3 − I 1I 3 I 1p 3 p 1 + Mgl(Γ 3 χ 1 − Γ 1 χ 3 ),ṗ 3 = I 1 − I 2I 1 I 2p 1 p 2 + Mgl(Γ 1 χ 2 − Γ 2 χ 1 ),and ˙Γ = Γ × Ω,where Ω is the body’s angular velocity vector, I 1 , I 2 , I 3 are the body’s principalmoments of inertia, M is the body’s mass, g is the acceleration ofgravity, χ is the body fixed unit vector on the line segment connectingthe fixed point with the body’s center of mass, and l is the length of thissegment.The Euclidean Group and Its Lie AlgebraAn element of SE(3) is a pair (A, a) where A ∈ SO(3) and a ∈ R 3 .The action of SE(3) on R 3 is the rotation A followed by translation by thevector a and has the expression(A, a) · x = Ax + a.Using this formula, one sees that multiplication and inversion in SE(3) aregiven by(A, a)(B, b) = (AB, Ab + a) and (A, a) −1 = (A −1 , −A −1 a),