The Geometric Quadrature Method (GQM): A Singularity-Free Spatial Formulation for Constrained Motion on Arbitrary Planar Curves

We present the Geometric Quadrature Method (GQM), a coordinate-free, spatial-domain framework for the dynamics of a particle constrained to an arbitrary smoothplanar curve. By adopting arc-length parametrization and the Frenet–Serret frame,GQM eliminates the coordinate singularities inherent in Cartesian systems, whichfail at vertical tangents or multi-valued trajectory regions.The method operates as a three-pass pipeline. Pass 1 delivers the spatial speedfield and normal contact force in exact closed form for any curve with an analyticheight function—no quadrature, no approximation, including the full rail loadingmap and liftoff conditions. Pass 2 confines the time-of-flight computation, which isirreducibly non-elementary by Liouville’s theorem, to a single adaptive quadratureevaluated to machine precision. Pass 3 recovers phase-stable long-time trajectoriesfor conservative systems by exact half-period tiling, bounding accumulated phaseerror many orders of magnitude below standard explicit or symplectic ODE integra-tors. The pipeline is anchored on a speed-field kernel encoding the force model inmodular form: redefining the kernel alone accommodates friction, quadratic drag,and rotating-frame potentials, leaving the quadrature pipeline intact. Six such ker-nels are derived.Numerical validation includes a simple pendulum benchmarked against RK4 andSt¨ormer–Verlet integrators, and a cubic curve with an inflection point and Coulombfriction, demonstrating that the continuously signed Frenet frame handles curvaturesign changes without frame-flip artefacts. A unified table reduces eight standardcurve families to this single framework.