The following things are all called holonomic or holonomy:


1. 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 $\ell$)
2. 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)
3. A smooth function is holonomic if it satisfies a homogeneous linear ODE with polynomial coefficients.
4. 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?



What is the relationship between various things called holonomic?
Not a complete answer, but you might wish to look at Historical Remarks at the beginning of Bryant's [Recent advances in the theory of holonomy](http://arxiv.org/pdf/math/9910059v2), which seems to offer a relationship between (1) and (4).


