Coupled Translational and Internal Modes of Solitons: Exact symplectic pullback for the moving wobbling kink in the double sine–Gordon model

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.