Discrete Dirac structures and discrete Lagrange–Dirac dynamical systems in mechanics

The paper’s Theorem 5.16 is proved by a direct specialization of the general discrete Dirac-structure construction (Theorem 5.7) from an arbitrary manifold M to M=T*Q, together with the specific (+/−) discrete flat maps and lifted constraints; see the statement and proof of Theorem 5.16 and the general result in Theorem 5.7 . The candidate solution follows the same route: it (i) recalls the general theorem, (ii) identifies the twisted finite-difference maps and the resulting discrete canonical flat maps (Ωd±)♭ on T*Q in local form, (iii) computes the annihilators (Δd±_{T*Q})^∘ using the discrete pairing, and (iv) applies the general theorem to conclude maximal isotropy and hence a discrete Dirac structure. These steps match the paper’s definitions and local formulas for (Ωd±_{T*Q})♭ (i.e., ((q0,p0),(q1,p1)) ↦ (q1,p1,−p̂+_1,q̂−_1) and (q0,p0,−p̂−_0,q̂+_0)) and for the annihilators ((Δd+_{T*Q})^∘(q1,p1) = {(η,ξ): η∈Δ^∘_Q(q1), ξ=0}, and similarly for −), as well as the discrete dual pairing used to derive them . Consequently, both the paper and the model establish the same result by substantially the same proof.