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 smooth planar curve. By adopting arc-length parametrization and the Frenet–Serret frame, GQM eliminates the coordinate singularities inherent in Cartesian systems, which fail at vertical tangents or multi-valued trajectory regions. The method operates as a three-pass pipeline. Pass 1 delivers the spatial speed field and normal contact force in exact closed form for any curve with an analytic height function—no quadrature, no approximation, including the full rail loading map and liftoff conditions. Pass 2 confines the time-of-flight computation, which is irreducibly non-elementary by Liouville’s theorem, to a single adaptive quadrature evaluated to machine precision. Pass 3 recovers phase-stable long-time trajectories for conservative systems by exact half-period tiling, bounding accumulated phase error many orders of magnitude below standard explicit or symplectic ODE integrators. The pipeline is anchored on a speed-field kernel encoding the force model in modular form: redefining the kernel alone accommodates friction, quadratic drag, and rotating-frame potentials, leaving the quadrature pipeline intact. Six such kernels are derived. Numerical validation includes a simple pendulum benchmarked against RK4, with symplectic (St？rmer-Verlet) behaviour discussed analytically via its known error scaling, and a cubic curve with an inflection point and Coulomb friction, demonstrating that the continuously signed Frenet frame handles curvature sign changes without frame-flip artefacts; the latter result is cross-checked against an independent direct time-domain integration of the same friction model. A unified table reduces eight standard curve families to this single framework.