Nonlinear Effects in a Weakly Nonholonomic Systems With a Small Degrees of Freedom

For problem A (particle with ż = ε(ẋy − ẏx)), the paper derives ẍ = −(c/m)x, ÿ = −(c/m)y via Gibbs–Appell and shows ż is constant, yielding an O(1) transgression over t ~ 1/ε, which matches the model’s result obtained via Lagrange–d’Alembert/Chetaev with λ and z̈(1+ε²(x²+y²))=0 ⇒ z̈=0 (paper: equations (9)–(10) and subsequent integration of the constraint; see ). For problem B (almost holonomic pendulum), both perform the φ=ψ+arctan(ξ/r) reduction, obtain the first approximation yielding the catenary law for C (dx/dξ = r/√(ξ²+r²), dy/dξ = ξ/√(ξ²+r²)), and then a third-order normal-form expansion for ψ(t) and for x,y. The model’s third-order formulas for ψ and for x,y coincide with the paper’s explicit expressions (22)–(23), while the paper provides explicit oscillatory parts f_x,f_y and numerical evidence of O(ε⁴) accuracy over t ∈ [0,1/ε] ( ). The model’s proofs differ in technique for A and use a slightly different diagonalization for B, but the results agree.