A Lagrangian Approach to Dry Friction via Orthogonal Virtual Coordinates

A geometric Lagrangian formulation for systems with dry (Coulomb) friction is presented. The method introduces an orthogonal virtual coordinate defined along the normal to the constraint surface in the kinetic metric. From the Euler–Lagrange equation in this coordinate, evaluated on the constraint, a closed analytic expression for the normal acceleration is obtained. This relation determines the reaction force and allows the frictional dynamics to be formulated entirely within the variational framework. The approach unifies constraint, curvature and dissipation through the kinetic metric, showing that dry friction admits an exact geometric representation. Classical examples such as the inclined plane, the paraboloid and the rotating Earth illustrate the simplicity and generality of the formulation.