17. 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 a four‑parameter moving wobbling kink ansatz in the double sine–Gordon model, we embed a comoving oscillating mode into the field phase space and compute the exact pullback of the canonical symplectic form $\Omega = \int (\delta\pi\wedge\delta\phi)\,dx$ without any approximation. The resulting $4\times4$ symplectic matrix contains the expected free translational block, the free internal oscillator block (with the correct relativistic factor $1/\gamma$), and off‑diagonal coupling terms that mix the two sectors at orders $O(v)$, $O(A)$ and $O(vA)$.  All coefficients are expressed through overlap integrals of the kink profile and the shape mode. The work supplies the rigorous classical geometric data required for a future quantisation of the fully coupled translational–internal dynamics within the effective ansatz framework.