Dynamical representations of constrained multicomponent nonlinear Schrödinger equations in arbitrary dimensions

Task A: The paper’s Section 3.1 differentiates Gj(Ψ)=cj−⟨Ψ,KjΨ⟩ twice, substitutes the DFPM equation, and derives the linear system Aλ=b with A_{jk}=⟨P_D^⊥KkΨ, KjΨ⟩+⟨Ψ, KjP_D^⊥KkΨ⟩ and b_j=kj(cj−⟨Ψ,KjΨ⟩)+⟨P_D^⊥(δE/δΨ†),KjΨ⟩+⟨Ψ,KjP_D^⊥(δE/δΨ†)⟩+2⟨Ψ_τ,KjΨ_τ⟩. The candidate reproduces A correctly but carries a minus sign on the quadratic τ-velocity term in b; the paper’s derivation gives a plus sign when the λ-terms are kept on the left and all other terms are moved to the right to form b (see Eqs. (17)–(18), then (24)–(26) for the construction of A and b) . Task B: Both the paper and the candidate note that at a stationary point the dynamic constraints force Gj=0 and, with Ψ_τ=0, equation (17) reduces to P_D^⊥(δE/δΨ†+∑λj δGj/δΨ†)=0, which implies the constrained Euler–Lagrange condition P_Ĝ^⊥(δE/δΨ†)=0 (compare the necessary condition in Eq. (12) with the projected dynamics in Eqs. (15)–(17)) . Note: later specialized formulae for b in Section 4 (only dynamically damped constraints) show a sign pattern for the velocity term that looks inconsistent with the general result, likely a presentation/normalization quirk in that subsection; the general derivation in Section 3.1 supports the plus sign in Eq. (25) .