ELASTIC RODS, RIGID BODIES, QUATERNIONS AND THE LAST QUADRATURE

The Kirchhoff kinetic analogy relates the governing equations for the statics of elastic rods and the dynamics of rigid bodies. We discuss t he analogy in light of several different Hamiltonian formulations, inc luding a non-canonical description of rod equilibria. We focus on the last three quadratures that are required to reconstruct the rod centre line from the frame variables, which form the complete configuration s pace in the rigid body interpretation. In particular, we demonstrate t hat if the frame evolution is formulated as a canonical Hamiltonian sy stem involving quaternions (or Euler parameters), then the last quadra tures can all be computed explicitly in terms of algebraic relations i nvolving invariants (or integrals) of the evolution, independent of wh ether or not the entire system is completely integrable.