# 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.

## Thoughts

• Relate this to differential forms.

Created: 2022-03-13 Sun 21:44

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