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 the analogy in light of several different Hamiltonian formulations, including a non–canonical description of rod equilibria. We focus on the last three quadratures that are required to reconstruct the rod centreline from the frame variables, which form the complete configuration space in the rigid body interpretation. In particular, we demonstrate that if the frame evolution is formulated as a canonical Hamiltonian system involving quaternions (or Euler parameters), then the last quadratures can all be computed explicitly in terms of algebraic relations involving invariants (or integrals) of the evolution, independent of whether or not the entire system is completely integrable.

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