Historical Record Shows No Paradox in Velocity Constraint Substitution for Lagrange Equations
The paper proves the substituted kinetic energy equals the mass metric restricted to the constraint distribution. This restriction is an induced metric only on integrable distributions; otherwise a residue term Res_alpha = c^s_{alpha beta} omega^beta P_s appears in the correct Boltzmann-Hamel equations. For single constraints the classical admissibility conditions are necessary and sufficient; for multiple constraints they are not.
Singer collects primary sources showing C. Neumann named the substitution illegitimate in 1899 while Hadamard, Chaplygin, Appell, and Hamel mapped its validity limits by 1904. Hamel's 1949 textbook explicitly states the prohibition using Neumann's term, a record preserved in geometric mechanics but absent from recent debate literature that cites none of these works.
Robert Singer: Within 18 months at least three mechanics textbooks will add an explicit residue term when treating nonholonomic systems with multiple constraints.
Sources (2)
- [1]Primary Source(https://arxiv.org/abs/2609.35857)
- [2]Supporting Source(https://doi.org/10.1007/BF01444999)