Curvature-Induced Normal Dynamics and Detachment Conditions for Constrained Constant-Speed Motion on 2DManifolds

The dynamic interaction between self-propelled constant-speed motion and surface topology is a fundamental problem in active matter physics and the control of surface-climbing robotics. While tangential trajectories on curved manifolds are well-documented, the curvature-coupled dynamics normal to the surface remain underexplored. In this paper, we present a rigorous analytical framework for the normal dynamics of a point mass constrained to move with a strictly constant tangential projection speed (\( V \)) over a smooth two-dimensional manifold. By applying the Weingarten map (Shape Operator) within a Newtonian framework, we derive the governing equation for the normal distance \( D: D'' \approx V^2 k_n - F_N/M \), where \( k_n \) is the normal curvature along the instantaneous trajectory and \( F_N \) is the applied normal force (e.g., gravity or adhesion). This reveals a purely geometry-induced inertial lift term, \( +V^2 k_n \), generated by the non-holonomic constraint of maintaining constant speed on a curved path. We establish the analytical threshold for surface detachment (\( V^2 k_n > F_N/M \)) and demonstrate that this effect is highly anisotropic on non-spherical surfaces. The core kinematic identity linking the normal acceleration to the inner product of velocity and the normal vector's derivative is formally verified using the Lean 4 theorem prover. Our findings provide a generalized mathematical tool for predicting the lift-off of active particles and calculating the minimum adhesion requirements for autonomous robots navigating complex topological surfaces.