# Moment of inertia in n dimensions.

1. Dec 10, 2011

### tarnhelm

I've just been thinking about moments of inertia in n dimensions and I just want to establish if this makes any sense:

I'm considering doing a Monte Carlo evaluation of the moment of inertia of any n-ball - a solid sphere in n dimensions. Now I think you can say that the moment of inertia of a sphere in 2d space - a circle - is (1/2)mr^2, this being about an axis through the centre, which in 2d space is merely the point in the centre. Now it is pretty run of the mill to integrate a circle like this in 3d space to get the moment of inertia of a sphere (or at least you're integrating an infinitesimally thin cynlinder). My question is - are the notions of mass and density in 3d the same as in 2d (and presumably by extension in all dimensions)? What is the exact relationship between a 2d "mass" or "density" and a 3d one? For one thing, density in 2d would have to have different units than in 3d. What exactly would a 2d mass be?

Of course it's possible that none of this makes any sense. After all, you can't really have mass in two dimensions, can you?

Last edited: Dec 10, 2011
2. Dec 10, 2011

### robphy

Possibly useful:
http://books.google.com/books?id=8vlGhlxqZjsC Tensor Calculus by Synge and Schild
Search for moment of inertia and get to page 161 to read about the fourth-order moment of inertia tensor in N-dimensions.