We extend the geometric subsystem quantisation programme to configurations with simultaneously excited translational and internal degrees of freedom. Using the double sine–Gordon wobbling kink as a concrete example, we embed an ansatz with independent position a, velocity v, and shape‑mode amplitude A and phase into the field phase space. The exact pullback of the canonical symplectic form = \, dx is computed without approximation. The translational block dP da (with P = M v) and the internal block b I\, A\, dA d (with I = ^2\, dx) are exact. The off‑diagonal coupling terms between the two sectors are overlap integrals of the shape mode with derivatives of the kink profile; they are of order vA and vanish in the static limit. In the non‑relativistic small‑amplitude regime, a perturbative Darboux diagonalisation reduces the system to a free particle plus an independent oscillator. The paper provides the rigorous classical geometric data for the future quantisation of fully coupled soliton dynamics.
Timmermans et al. (Thu,) studied this question.