Shape modes of the \ (^4\) and \ (^6\) kinks: Effective Hamiltonian from a symplectic reduction

We present a rigorous derivation of the effective Hamiltonian for a slowly moving kink with a small‑amplitude shape‑mode excitation in \ (^4\) and \ (^6\) models. Starting from the field‑theoretic symplectic form, we perform its pullback to a four‑dimensional collective‑coordinate manifold parametrised by the kink position \ (a\), a collective momentum \ (P\), the shape‑mode amplitude \ (Q\) and its conjugate momentum \ (\). The expansion is controlled by a single small parameter \ (\), and all steps are carried out systematically. We obtain the symplectic form \ (= ₀ + ₁ + O (^3) \), where \ (₀ = dP da + d dQ\) (of order \ (\) and \ (^2\) respectively under the natural scaling) and \ (₁\) contains the non‑canonical corrections of order \ (^2\) and \ (^3\). An explicit near‑identity Darboux transformation is constructed that removes \ (₁\), leaving the symplectic form canonical through \ (O (^2) \). The Hamiltonian, evaluated on the ansatz and expanded to the same order, becomes \ H = M + P^22M + 12^2 + ₒ₇^₂2Q^2 + O (^3), \ a free non‑relativistic particle plus a decoupled harmonic oscillator. All steps are self‑contained, the approximations are controlled by a single small parameter \ (\), and the result provides a solid foundation for quantisation.