发表机构
FH Joanneum–University of Applied Sciences（FH Joanneum应用科学大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过历史文献考证和几何分析，证明Hamel悖论并非真正的悖论，并给出了约束代入动能合法性的精确条件，修正了Hamel 1904年论文中的必要性论断。
AI 中文摘要
在形成拉格朗日方程之前，将速度约束代入动能中，通常会产生不正确的运动方程。近期文献将这一失败称为“Hamel悖论”，该术语于2010年引入。我们证明并不存在悖论。这种代入的不合法性早在19世纪90年代就已得到认识；C. Neumann（1899）将这一操作命名为动能的非法形式，Hadamard（1895）、Chaplygin（1897）、Appell（1899）和Hamel（1904）确立了其有效性界限。Hamel于1949年出版的教科书（该悖论名义上的来源）明确禁止了这种做法，并使用了Neumann的术语；这一结果在专业文献和几何力学中一直持续刊印。现代争论未引用上述任何记录；我们从原始文献中收集这些记录，并以此衡量争论中的主张。几何分析将代入后的函数识别为质量度量在约束分布上的限制。若该分布可积，则此限制是积分流形的诱导度量，代入是合法的。若非可积，则正确嵌入的Boltzmann–Hamel方程与受限能量所得的方程相差一个余项，$\mathrm{Res}_\alpha=c^{s}_{\alpha\beta} \omega^\beta P_s$，该余项相对于捷径所选取的约束分布的补空间而言。当该余项系数的对称部分为零时，余项恰好消失。两种经典的容许性机制——质量度量的解耦和括号分量的消失——对此是充分的，并且对于单一约束也是必要的；对于多个约束则并非必要，这修正了Hamel 1904年论文中的一个必要性论断。
英文摘要
Substituting a velocity constraint into the kinetic energy before forming Lagrange's equations yields, in general, incorrect equations of motion. Recent literature calls this failure "Hamel's paradox," a term introduced in 2010. We show that there is no paradox. The inadmissibility of the substitution was recognized in the 1890s; C. Neumann (1899) named the operation the illegitimate form of the kinetic energy, and Hadamard (1895), Chaplygin (1897), Appell (1899) and Hamel (1904) established its limits of validity. Hamel's textbook of 1949, the nominal source of the paradox, states the prohibition and uses Neumann's term; the result remained in print in the specialist literature and in geometric mechanics. The modern debate cites none of this record; we collect it from the primary sources and measure the debate's claims against it. A geometric analysis identifies the substituted function as the restriction of the mass metric to the constraint distribution. If the distribution is integrable, the restriction is the induced metric of an integral manifold, and the substitution is legitimate. If not, the correctly embedded Boltzmann--Hamel equations differ from those of the restricted energy by a single residue, $\mathrm{Res}_α=c^{s}_{αβ} ω^βP_s$, relative to the complement of the constraint distribution that the shortcut selects. The residue vanishes exactly when the symmetric part of its coefficient vanishes. The two classical admissibility mechanisms, decoupling of the mass metric and vanishing bracket components, are sufficient for this and, for a single constraint, also necessary; for several constraints they are not, which corrects a necessity claim in Hamel's paper of 1904.
Comments32 pages, 4 figures, 2 tables. Also available at doi:10.5281/zenodo.22919510
