Euler–Poincaré reduction and the Kelvin–Noether theorem for discrete mechanical systems with advected parameters and additional dynamics

The paper’s Theorem 6 states and proves the left-left discrete Euler–Poincaré equations with advected parameters and additional dynamics using a group difference map τ, its inverse right-trivialized tangent dτ−1, the diamond operators, and fixed-endpoint variations; the key identities, definitions, and the variation formulas (notably δξk = dτ−1_{−hξk}(η_{k+1}) − dτ−1_{hξk}(η_k) and δnk, δsk, δak) are given and then assembled into the stationarity conditions producing equations (42)–(44) . The candidate solution follows the same route: it derives the δξk-identity from Definition 5 and property (35) of dτ−1, uses the variations of nk, sk, ak from the left actions, invokes the diamond pairings and τ∗(·), telescopes the ξ-terms via (dτ−1)∗, and isolates the δqk-terms to obtain ∂ℓd/∂nk = −τ∗(hξk−1)∂ℓd/∂sk−1, along with the complementary updates nk+1 = τ(−hξk)sk and ak+1 = τ(−hξk)ak. These steps coincide with the paper’s proof after equations (47)–(49) and the definitions of τ, dτ, dτ−1 in (32)–(35) . No missing hypotheses or logical gaps are apparent beyond the standard local assumptions on τ; thus both are correct and essentially the same proof.