# What is the relationship between various things called holonomic?

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?

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Also, a Poisson manifold is called 'holonomic' if it has finitely many symplectic leaves. The explanation here is that the corresponding Hamiltonian vector fields collectively determine a holonomic system iff there are finitely many symplectic leaves. –  Greg Muller Nov 22 '09 at 20:02

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, which seems to offer a relationship between (1) and (4).

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Thanks, that's actually quite helpful. And it leads to the connection with 2 and 3 via looking at things as constraints of solutions to subspaces. –  Charles Siegel Nov 23 '09 at 13:17

(In (2) you want the solutions to be a finite dimensional vector space locally.)

(3) is related to (2) in that the $D$-module corresponding the the ODE which has that function as a solution is holonomic (this is pretty obvious in terms of how you expressed the condition that the solution space be finite dimensional locally: that is always true for ODEs) In general 'holonomic' in (2) and (3) means maximally overdetermined.

There aren't any other connections between the four meanings.

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Ahh, I knew finite dimensional, just forgot to write it, have edited it into the question. And if there's no other connections, I'm curious as to why the name "holonomy" or "holonomic" was chosen for all of them. –  Charles Siegel Nov 23 '09 at 4:06

I don't know the connection between the four points, but it may help to redefine holonomy in your first point. A constraint is generally a regular distribution (in the sense of a subbundle of the tangent bundle of the configuration manifold).

If that distribution is integrable, then the constraint is said to be holonomic. And indeed, by restricting the system to one particular integral submanifold, one obtains the definition of holonomy in your first point.

The point is, in mechanics, "holonomic" is just another word for "integrable distribution". If the constraint distribution is not integrable, the system is called "nonholonomic".

Maybe that may help to make a connection with point 4.?

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And also with point (2) ? –  Emerton Feb 10 '10 at 20:16