The inadmissibility of substituting a velocity constraint into kinetic energy before forming Lagrange's equations, and its exact limits of validity, were established in print between 1895 and 1904
Robert Singer's "The Myth of Hamel's Paradox" (arXiv:2609.35857) reconstructs a documented chain of named results predating the "Hamel's paradox" label (coined 2010; see claim-hamels-paradox-is-a-2010-coinage-not-hamels-own-question) by more than a century. A. Vierkandt (1892) first located the error in rolling-body problems. C. Neumann (1899) named the operation "the illegitimate form of the kinetic energy." Jacques Hadamard (1895), Sergei Chaplygin (1897), and Paul Appell (1899) each gave conditions under which the substitution is or is not valid. Hamel's own 1904 paper generalized these conditions to quasi-velocities — the same mechanism his 1949 textbook states directly at the example later used to pose the "paradox" (see claim-hamel-1949-textbook-states-prohibition-and-reason-at-knife-edge-example). Singer presents two tables, spanning 1892–1908 and 1949–2015, of sources he states he "examined at first hand," arguing the result then stayed continuously in print through the 20th century in specialist mechanics texts and in geometric mechanics, and that the Russian-language literature still studies the same limits today as the "Hadamard–Hamel problem."
This is the central historiographical thesis of Singer's paper. It is no longer single-sourced: Borisov, Mamaev & Bizyaev's 2016 historical review (a different author group, in a different venue, a decade earlier) independently documents substantially the same chain, though it centers the narrative on "Lindelöf's error" (1895) rather than crediting Vierkandt (1892) as the one who first located it — see claim-2016-russian-review-independently-corroborates-1895-1904-correction-chain for the corroboration and the framing discrepancy. The 1890s–1900s primary sources themselves (Vierkandt, Neumann, Hadamard, Chaplygin, Appell, Hamel's 1904 paper) still have not been independently re-read by this vault at first hand — only quoted and tabulated by Singer and, independently, narrated by Borisov et al.
Source
“The inadmissibility of the substitution was recognized in the 1890s [4–6], and C. Neumann gave the operation its classical name, the illegitimate form of the kinetic energy [7]. Hadamard [8], Appell [9] and Hamel [10] established its limits of validity; the Russian literature studies these limits to this day as the Hadamard–Hamel problem [11].”
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