What is the role of tensors in classical mechanics?

[aeromrk]

When I took classical mechanics we were given definitions for an
object's moment of inertia, which I understand to be a scalar quantity that
describes that objects tendency to resist rotation about a fixed axis either
about, or some distance from its center of mass.

I was recently reading about how an object's moment of inertia can also be described as a tensor quantity when the axis of rotation is not fixed, but arbitrary.

I have not had much experience with tensors, the extent of my knowledge is mostly conceptual at this point, so I will not be able to decipher the math, however, I was hoping somebody could give me a more conceptual description of what this means, specifically rotation about an arbitrary axis, I'm have a hard time wrapping my head around that. Anybody have an example?

Cheers

 

[chiro]

When I took classical mechanics we were given definitions for an
object's moment of inertia, which I understand to be a scalar quantity that
describes that objects tendency to resist rotation about a fixed axis either
about, or some distance from its center of mass.

I was recently reading about how an object's moment of inertia can also be described as a tensor quantity when the axis of rotation is not fixed, but arbitrary.

I have not had much experience with tensors, the extent of my knowledge is mostly conceptual at this point, so I will not be able to decipher the math, however, I was hoping somebody could give me a more conceptual description of what this means, specifically rotation about an arbitrary axis, I'm have a hard time wrapping my head around that. Anybody have an example?

Cheers

The best way to think about a tensor in this context is to think about both a change in the co-ordinate system from one system to another and then to think about what happens you apply a transformation of one sort (think transformation = a kind of function).

The easiest example of co-ordinate transformations is from Cartesian/Euclidean (the standard basis and the normal intuitive right-angle co-ordinate system) to the polar system. The cartesian system is (x,y,z) and the polar is (r,theta,gamma) or whatever you want to call your angles.

Now tensor theory allows you to figure out how to go from coordinate system A to coordinate system B and also to figure out how to go from the geometry of A to the geometry of B by using the fact that the metric tensor describes the inner products at each element of the bases as well as the metric.

So let's say you have some kind of transformation A(x) where x is a vector in an initial co-ordinate system. Then what you can do using tensor theory is to figure out A(X) where X is in a new co-ordinate system.

The other thing is that tensor theory generalizes to tensors of more than a 2nd order rank. Think of a vector as 1st order and a normal matrix of 2nd order. The tensors are the generalization of a multilinear object.

This means that we can deal with multilinear systems algebraically that are too hard to think about visually in a matrix form (although we could if we wanted find the multilinear system in the form of a reduced matrix), but still be able to analyze the effects of the tensors algebraically.

Think about say when we want to deal with situations where we have del x F, del(F), del . F and so on where we have an arbitrary number of dimensions. Also think about rotations when we have large number of dimensions or when we are working in a non-euclidean co-ordinate system. We can write down rotations about an arbitrary axis in tensor form very easily and if we need to combine this with other tensor formulations, then again using the tensor framework it's a lot easier.

So to sum up, think about tensors as going from on geometry to another and also that tensors generalize the way to go from one geometry to another even if one or another geometry is not flat (non-euclidean) and we can also get the associated inner products and metric information for our curved geometries which means we can do all the geometric calculations that involve distance and angle and relate these quantities between the different geometries.