Infinite-dimensional Lagrange–Dirac systems with boundary energy flow I: Foundations

The paper’s Theorem 4.18 states the equivalence of (a) the Lagrange–d’Alembert variational principle, (b) the forced Euler–Lagrange equations with boundary term, (c) the Lagrange–d’Alembert–Pontryagin principle, (d) its stationarity equations, and (e) the Lagrange–Dirac inclusion, for V = Ω^k(M) with the restricted dual pairing that includes boundary contributions. Propositions 4.14, 4.16 and 4.10 supply (a)⇔(b), (c)⇔(d), and (d)⇔(e), respectively, using the canonical one-form and symplectic two-form on T*V, the Dirac differential d_DL, and the Lagrangian force field F̃; see Theorem 4.18, Proposition 4.14, Proposition 4.16, and Proposition 4.10, together with the definitions of Θ, Ω and the canonical Dirac structure as graph(Ω^♭) and the explicit form of F̃ (and FL) . The candidate’s solution reproduces these steps: (a)⇔(b) by varying ∫ L(ϕ, ϕ̇, dϕ) and using the restricted pairing and Stokes’ theorem (matching Proposition 4.14), (c)⇔(d) by independent variations of (ϕ, ν, α, α_∂) to get ϕ̇=ν, α=∂L/∂ν, α_∂=0, and the bulk/boundary evolutions (matching Proposition 4.16), and (d)⇔(e) by equating pairings with graph(Ω^♭) and using d_DL and F̃, including the basepoint matching that forces α=∂L/∂ν and α_∂=0 (as in Proposition 4.10). Sign conventions and the (-1)^k term, the boundary condition F_∂ = -ι^*(∂L/∂ζ), and the explicit formula for F̃ all agree with the paper’s definitions and results . Hence both are correct and essentially follow the same proof strategy.