# Internal symmetries of partial differential relation via the nonholonomic jet bundle

On a smooth n-dimensional Riemannian manifold $$M$$, suppose I have the kth order partial differential relation (PDR) written in the form:

$$\mathcal{R}=\mathcal{R}\left(x^{i},u^{i},u_{j}^{i},u_{jk}^{i}\cdots\right)=0$$

Where $$u^{i}$$ is a section of some principal $$G_{E}$$ bundle $$\pi:E\rightarrow M$$, and $$u_{j}^{i}=\partial u^{i}/\partial x^{j}$$ and so on up to order $$k$$. We know that $$\mathcal{R}$$ describes a submanifold of the kth order jet bundle over $$E$$:

$$\mathcal{R}\subset J^{k}E$$

Then $$\mathcal{R}$$ can be viewed as a fiber bundle over $$E$$ and consequently $$M$$. I am interested in finding the $$\mathit{internal}$$ symmetries of such a differential equation, which map solutions $$u^{i}$$ to solutions. If $$J^{k}E$$ is a $$G_{J^{k}E}$$ bundle over $$M$$ then the paper “internal external and generalized symmeties” (by Anderson, Kamran, and Olver) describes the group of internal symmeties $$G_{I}$$ of $$\mathcal{R}$$ as a reduction of the $$G_{J^{k}E}$$ bundle $$J^{k}E$$ to the bundle $$\mathcal{R}$$, such that: $$G_{I}\subset G_{J^{k}E}$$

In my case however we have that $$u^{i}$$ is subject to nonholonomic constraints, then $$\mathcal{R}$$ is a submanifold of the $$\mathit{nonholonomic}$$ jet bundle $$\tilde{J}^{k}E$$ described by the iteration:

$$\tilde{J}^{k}E=J^{1}J^{1}\cdots J^{1}E=\left(\prod_{i=1}^{k}J_{i}^{1}\right)E$$

In order to construct $$\mathcal{R}$$ and see the structure group, I've been building $$\tilde{J}^{k}E$$ up one iterative jet bundle at a time starting with $$J^{1}E$$ and proceeding to $$J^{1}J^{1}E$$ and so on..

This is extremely tedious and the fiber space in general become extremely complicated very quickly after a few iterations.

I much easier way I've found is to construct $$J^{1}E$$ and apply all 0th and 1st order constraints to reduce the space of $$J^{1}E$$ and then proceed to $$J^{1}J^{1}E$$ and apply all constraints up to second order constraints. If I continue this to kth order the approach is much easier than applying all constraints ex post facto.

However, I'm not sure if this is a valid approach, do applying my mth order constraints on the mth iteration lead to the same manifold as applying them after construction? i.e does the reduction of my jet bundle commute with the operation of building it up? In what cases is this true?

Apologies in advance, I study physics so hopefully my terminology is correct, please ask if something is unclear.

EXAMPLE: To give an example of constraints I'm talking about is suppose my $$u^{i}$$ is a section of the oriented orthonormal frame bundle of $$M$$, and we make the choice of Levi-Cevita connection on $$M$$.

The connection on $$M$$ induces a Levi-Cevita connection on $$FM$$ and each jet space (applying this condition reduces the space significantly via a splitting of the tangent space of each bundle). As each bundle has a Riemannian structure, I've been applying a conditions of orthonormal frames or basis on every jet bundle also. This also reduces my space quite a bit and makes it more managable (but is that valid?) I'm not applying the explicit relations in $$\mathcal{R}$$ until the total bundle is constructed.

For me this is analogous to a reduction Of the frame bundle, it changes the form of equations but not their validity.