I think I should convince myself that studying Differential Geometry is not just a lot of formal machinery with no purpose. It certainly is a lot of formal machinery, so the purpose of this note is to consider the purpose...

To get a dynamic model of sand behavior, I want to write a Lagrangian in terms of some smallish set of abstract coordinates and make sure that it behaves in the same way, statistically, as the detailed Lagrangian for all the grains of sand.

The detailed Lagrangian for all the grains of sand has O(N) dimensions where N is the number of grains of sand. In this case the big O notation means that the number of dimensions is some small constant times N. If I treat each sand grain as a rigid body, then there are 3 coordinates for its center and 3 coordinates for the rotations so the number of dimensions is 6N. If the sand grain is not rigid, then it can deform and we need to add coordinates to describe the deformation. But the truth is, I'm not going near this O(N) dimensional manifold. To describe say 50 meters of sand bed below a tall building I'm going to want to use maybe 1,000 degrees of freedom, maybe 1,000,000 if I feel like getting detailed information. The number of sand grains in 50 meters by 10 meters by 10 meters with packing fraction of say 55% and average grain volume of $4/3\pi r^3$ with $r = 1{\rm mm}$ is around $10^{11}$ so I need to make some correspondence between a manifold of high dimensions and a much lower dimensional manifold in which I will describe a solution.

The way I want to do that is by blurring the information about grains of sand into a small region in $R^3$ the actual place under the ground in which the sand grains live.

I need a language to describe how a flow in the sand grain configuration manifold relates to a flow of the much lower dimensional sand-grain statistics manifold.

And we also want some kind of distance metric: a flow in the configuration manifold should map to a flow in the statistics manifold, and for every flow in the configuration manifold the error between the implied flow in the statistics manifold gotten by projecting from the "real" solution, and the calculated flow in the statistics manifold from its own set of equations should be bounded by some acceptable error tolerance. Defining that correspondence and the method of mapping between the manifolds is why I think I need to study Differential Geometry.

In particular, in the 3D space we've got a fiber of statistics, and these are controlled by say 1000 or so control points. The control points move according to a Lagrangian through a 3000 dimensional space. Every point in that 3000 dimensional space defines a function on 3 dimensional space. Flows of energy in 3D will correspond to changes in the 3000 dimensional Lagrangian. It's that net flow of energy in 3D that should be similar between the real system and the statistical system.

It would be nice to have a toy problem where some of these things come into play and I could solve a much reduced problem and compare the method to the exact solution. I was looking at the 2D swing spring as a model of a system with "small" degrees of freedom (if the spring is stiff). I think I should probably find something a bit "larger". What kind of system could I get "exact" numerical results on so that I could compare some statistical type abstraction to it?

Finding the toy problem is the key as Paul Newton says. What is the perfect toy?