Holonomic constraints

Let \(\textbf{u} = (u_1,...,u_n)\) be the generalized coordinates of some mechanical system. A holonomic constraint is an equation of the form \(f(\textbf{u})=0\), (or \(f(\textbf{u},t)=0\)) for some function \(f\) which constrain the equations of motion. For e.g. the motion of a particle moving on the surface of a sphere is subject to a holonomic constraint.

A system may involve many such constraints, let us denote them in vector form as \(\textbf{f}(\textbf{u})=\textbf{0}\). Differentiating both sides results in the Pfaffian constraints. Note that some Pfaffian constraints result in non-holonomic constraints, reducing the space of possible of velocities without affecting the position space.