A Lagrangian Approach to Dry Friction via Orthogonal Virtual Coordinates

We present a novel method for incorporating dry friction—proportional to the normal force—into the Lagrangian formalism. The approach introduces orthogonal virtual coordinates (OVC) that extend the configuration space with geometrically meaningful, non-physical degrees of freedom. These coordinates yield the normal force via virtual acceleration derived directly from the equations of motion, allowing friction to be consistently added as a generalized force. The method is tested on a wide variety of mechanical systems: flat and curved surfaces, rotating frames, potentials with explicit constraint coupling, and multi-constraint environments. In all cases, the results match classical Newtonian solutions while preserving the full analytical elegance of Lagrangian mechanics. This makes the OVC formalism a powerful addition to the toolbox of theoretical and applied mechanics.