The following things are all called holonomic or holonomy:
- A holonomic constraint on a physical system is one where the constraint gives a relationship between coordinates that doesn't involve the velocities at all. (ie, $r=\ell$ for the simple pendulum of length $\ell$)
- A D-module is holonomic (I might not have this one quite right) if the solutions are locally a finite dimensional vector space, rather than something more complicated depending on where on the manifold you look (this is apparently some form of being very over-determined)
- A smooth function is holonomic if it satisfies a homogeneous linear ODE with polynomial coefficients.
- On a smooth manifold $M$, with vector bundle $E$ with connection $\nabla$, the holonomy of the connection at a point $x$ is the subgroup of $GL(E_x)$ given by the transformations you get by parallel transporting vectors along loops via the connection.
Now, all of these clearly have something to do with differential equations, and I can see why 2 and 3 are related, but what's the common thread? Is 4 really the same type of phenomenon, or am I just looking for connections by terminological coincidence?