How are constraint forces represented in Lagrangian mechanics?

Question

Suppose we try to obtain the movement equation for a particle sliding on a sphere (no friction, ideal bodies...). The only forces acting on the particle are its weight and - here's my problem - a force that keeps the particle attached to the sphere*. How I am supposed to represent mathematically this kind of forces? As a restriction on the coordinates of the particle? As a side note: My teacher refers to this, in Spanish, as a "fuerza de ligadura". I don't have a clue on its correct translation; does ligature force make any sense?

Answer 1

Constraints are all that' needed. Just make sure the generalized coordinates are selected so that the constraints ie in this case use spherical coordinates. So ρ= constant and all the derivatives with respect to ρ are zero. You can calculate the actual constraint force after you solve the equations of motion.

Jim