A Lagrangian, Covariant Treatment of Dry Friction via Orthogonal Virtual Coordinates

This work presents a detailed Lagrangian and geometrically transparent formulation of dry (Coulomb) friction using Orthogonal Virtual Coordinates (OVC). A virtual coordinate is introduced along the kinetic-unit normal to a possibly time-dependent constraint surface. Evaluating the Euler–Lagrange equation in this normal coordinate yields a closed expression for the normal acceleration and, consequently, for the reaction force, without the need for Lagrange multipliers. Tangential dynamics with Coulomb friction follow from a covariant projection onto the tangent bundle of the constraint, producing the exact mechanical dissipation law. The derivation is carried out explicitly, first for general moving constraints and then specialized to static surfaces. Several classical and time-dependent examples are worked out in detail. A dedicated section clarifies how this variational approach differs from projection-based methods such as Udwadia–Kalaba and the extension of Wu, Wang and Udwadia.