The two momenta of an elastic rod: a Hamiltonian picture on framed Lie groups
The equilibrium equations of elastic rods can be obtained by balancing forces and moments, or by rendering a potential energy stationary. For complex filaments the energy route asks less of one's mechanical intuition. However, in the classical Hamiltonian picture, the components of the generalized momenta are postulated a priori and their physical meaning changes at the whim of the coordinate chart. Here, we draw on two ideas from mathematical physics. First, the extended tangent bundle of the configuration Lie group is framed by left-invariant vector fields. Second, we follow a one-dimensional reading of the Cartan--Lepage--Krupka theory of variational forms: in this setting the momenta are not postulated but forced. The Legendre transform becomes a linear change of frame, and a Poisson structure on the extended tangent bundle follows. For an isolated Cosserat rod, the momenta coincide, in every encoding of the rotation group, with the material force and moment familiar from rod theories. In the presence of interactions, two cases arise: interactions that depend only on the configuration, such as gravity, leave this identification intact; interactions that depend on the strains destroy it --- the conjugate momenta and the internal stresses part company. Because energies add, the momenta decompose into the internal stresses and an interaction contribution. Expressing the two factors of the Poisson bivector in different frames --- one adapted to the internal stresses, the other to the conjugate momenta --- then exposes the Hamiltonian flow directly in the internal variables. Applied to a tendon-actuated rod, where the standard passage to the Hamiltonian picture demands a nonlinear inversion, the construction delivers explicit equilibrium equations, the required inversion collapsing to a rank-one correction.