Fellow Mechi here. Yes, they can do that tau revolution without us. In the dynamic equations, tau is also used for the "generalized forces" (a vector/array which contains all the forces and torques that act on the generalized coordinates). At the same time, they are usually full of trigonometric functions (with 2*pi).

I can only recall one time in my academic career where someone used pi for something else than a time constant or circle constant. It made me feel so uneasy ...