Euler’s Equation via Lagrangian Dynamics with Generalized Coordinates

Dennis S. Bernstein, Ankit Goel, Omran Kouba

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Category: math.DS
Journal tier: Note/Short/Other
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper’s Proposition 1 correctly proves (a)⇔(b) and (b)⇔(c) and shows (c)⇒(20), relying on the identities [A2×A3 A3×A1 A1×A2]x×=(Ax)×A and AT(Ax)×A=(det A)x× (its Lemmas 2 and 1, respectively). However, the proof as written does not explicitly provide the missing converse (20)⇒(c), despite later using it to conclude (15)–(16) from (20) for Euler parameters. This converse follows immediately by combining (20) with the purely algebraic identity AT[A2×A3 A3×A1 A1×A2]=(det A)I3 and the nonsingularity of S, but this step is not shown in the paper. The candidate model solution explicitly supplies this missing direction, so it completes the equivalence chain that the paper claims. See Proposition 1 and its proof, including equations (15)–(20) and Lemmas 1–2 in the PDF .

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} note/short/other

\textbf{Justification:}

The contribution is a compact, instructive derivation connecting generalized-coordinate parameterizations to Euler’s equation, with a clean algebraic Proposition. The main algebra is correct and useful pedagogically. The only substantive issue is a small gap: the converse (20)⇒(c) under nonsingularity is not shown in the proof though it is later used. Adding a brief argument invoking the cofactor identity closes the loop. With that fix, the note is sound and suitable for publication as a short technical piece.