I Non-orthogonal bases

[geordief]

TL;DR Summary

What are they supposed to model

Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes.

I have seen that this is an important subject in maths

My question is what physical applications does such a model apply to?

I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual space and have come away with little understanding.

So,and pardon the pun I am going back to basics and picking up on this particular question (hopefully others will follow if I feel comfortable as I get my answer)

 

[FactChecker]

Unless two variables, X and Y, are not related, they will not be orthogonal. The vast majority of variables related to the same subject of interest will also be somewhat related to each other. More often than not, two variables must be analyzed if one wants to determine components that are not related. The Graham-Schmidt orthogonalization process helps there.
Example: X=arm length, Y=leg length. You can expect them to be related, but not completely. You can find a component of Y=leg length that is independent of X=arm length. That would be an interesting component to study.

 

[Ibix]

I suspect you are asking this in the context of relativistic physics, in which case the answer is that basis vectors don't model anything. They're just a tool and a matter of choice, and you occasionally find it useful to use non-orthogonal basis vectors. An example is a rotating reference frame, where using orthogonal coordinates leads to simultaneity planes that don't close (so some parts of "now" lie in the future of other parts, which is problematic). It can be easier to define non-orthogonal coordinates (and basis vectors) so that "now" isn't multiply-valued. But if you only need to consider a small part of an orbit it can be easier to stick with an orthogonal basis because the problems with it are outside the region you care about. Your choice; some scenarios need more care depending which one you pick.

 

[DaveE]

My question is what physical applications does such a model apply to?

This appears quite often in engineering. We use eigenvalues and eigenvectors to model dynamic systems to make the analysis easier do and understand. Read a little bit about what these are (you'll study them for real later, I think). These vectors are usually not orthogonal, but the way we want to describe the system behavior at the end is often wrt an orthogonal basis. So we will use linear transforms to switch from one basis to another. One is what we want to measure or control, the other is the way the system naturally responds.

Here's a simpler example that I worked with IRL. Suppose you have to tilt a mirror in a laser system in a desired direction which is describe with orthogonal coordinate basis. The mirror has a fixed pivot point and two actuators located away from that point that can cause it to tilt. However for practical mechanical reasons the triangle formed by these three points that define the mirror plane can't be at right angles. I need to know how much each of the actuators need to be moved to get the desired tilt wrt the orthogonal basis. So, for example, tilt the mirror in the $\bar x$ direction with $\bar y$ unchanged, but my actuators aren't on the $\bar x$ axis, they each change the $\bar x$ and $\bar y$ tilt in different ways.

Anyway, it is a very important tool to understand IMO, you are likely to use it later for real.

 

[DaveE]

There are lots of materials you might run across that naturally have non-orthogonal basis to describe anisotropic behavior like strength or optical properties. Crystals (Quartz, Calcite, etc.) very often have a crystal structure that isn't orthogonal.

The mathematical solutions to many real world problems are much simpler if you arrange your equations to match the natural basis of the system, which is often non-orthogonal. So, it is common to do a simple transform to the natural basis, solve the problem, and do the inverse transform back to the basis you started with. In this way you can solve three simple problems instead of one difficult problem.

A classic example is the evaluation of $e^{\mathbf A t} \equiv 1 + \frac{\mathbf At}{1!} + \frac{ { ({\mathbf A} t)}^2 }{2!} + \frac{ { ({\mathbf A} t)}^3 }{3!} + ...$, where $\mathbf A$ is a matrix. This is really hard unless you can transform your basis so that $\mathbf A$ is diagonal, then it's trivial. This isn't pedantic, it's the basis for modern control systems solutions in applications like fighter jets and chemical factories. You can search for the terms "modern control systems" or "state transition matrix" if you want to know more.

https://en.wikipedia.org/wiki/Matrix_exponential#Computing_the_matrix_exponential

 

[geordief]

Thanks very much for those answers So a tool it is but non orthogonal axes do occur in real applications.(the crystals seem obvious now that it has been pointed out)

Yes I am interested in relativistic physics but lack any expertise. Very much a talentless amateur

 

[FactChecker]

The mathematical solutions to many real world problems are much simpler if you arrange your equations to match the natural basis of the system, which is often non-orthogonal. So, it is common to do a simple transform to the natural basis, solve the problem, and do the inverse transform back to the basis you started with. In this way you can solve three simple problems instead of one difficult problem.

A classic example is the evaluation of $e^{\mathbf A t} \equiv 1 + \frac{\mathbf At}{1!} + \frac{ { ({\mathbf A} t)}^2 }{2!} + \frac{ { ({\mathbf A} t)}^3 }{3!} + ...$, where $\mathbf A$ is a matrix. This is really hard unless you can transform your basis so that $\mathbf A$ is diagonal, then it's trivial.

I would say this a little differently:
There are things (like crystal structure) that are natural to represent using a non-orthogonal basis. For the actual mathematical calculation, transforming to an orthogonal basis can make the calculation easier. Then the solution can be transformed back to the non-orthogonal basis for a natural physical interpretation.