A Lagrangian Approach to Dry Friction via Orthogonal Virtual Coordinates

This work presents the Orthogonal Virtual Coordinate (OVC) method for incorporating Coulomb dry friction into the Lagrangian framework. We introduce a virtual coordinate z_star(q), orthogonal to the constraint surface in the reduced kinetic metric. The Euler–Lagrange equation in z_star, evaluated at the constrained limit, yields the normal reaction N = - m * (dz*^2/dt^2). Once N is determined, tangential friction follows by covariant projection of the Coulomb law. The approach is justified from Newtonian mechanics and from the geometry of constrained motion, with curvature effects accounted for via the second fundamental form. Three canonical examples are developed step by step: the inclined plane, the paraboloid of revolution, and motion on a rotating Earth. Each begins with the metric and its inverse, derives z* from the gradient, and verifies Newtonian equivalence through tangential checks. The method is then related to Lagrange–d’Alembert, Kane’s method, and Gauss’s principle, and possible extensions are outlined.