Do non-orthogonal coordinate systems mean dependent coordinates?

[gmmaro]

In non-orthogonal coordinate systems, can we say that changing a coordinate could result in changing another coordinate? That is, the coordinates are dependent on each other.

As I understood, non-orthogonal systems will have unit vectors (which are defined to point in the direction of increasing corresponding coordinates) not orthogonal to each other. Doesn’t that mean they could have components along each other’s direction? Thus if we move in the direction of one unit vector, we may “accidently” also move in the direction of another unit vector, yielding a change in another coordinate.

 

[DaveE]

Yes.

 

[Ibix]

Doesn’t that mean they could have components along each other’s direction?

It means that at least one does have a component parallel to at least one other, yes.

Thus if we move in the direction of one unit vector, we may “accidently” also move in the direction of another unit vector, yielding a change in another coordinate.

No, because moving along the direction of one coordinate basis vector is the definition of not changing the other coordinates. A simple example is to draw a non-orthogonal grid on paper. Label one intersection (0,0). All the intersections on one of the lines passing through that must then be (i,0) and those on the other line are (0,j). So if you start at the origin and move in the direction of one coordinate basis you will always have one or other coordinate being zero.

 

[PeroK]

In non-orthogonal coordinate systems, can we say that changing a coordinate could result in changing another coordinate? That is, the coordinates are dependent on each other.

As I understood, non-orthogonal systems will have unit vectors (which are defined to point in the direction of increasing corresponding coordinates) not orthogonal to each other. Doesn’t that mean they could have components along each other’s direction? Thus if we move in the direction of one unit vector, we may “accidently” also move in the direction of another unit vector, yielding a change in another coordinate.

Coordinate unit vectors are defined locally - as are vectors based on them. Take plane polar coordinates as an example, and consider the unit vector $\hat \theta$. The direction of $\hat \theta$ changes at every point. If you fix $r$ and follow $\hat \theta$, then you are not following the same unit vector at every point.

At each point, except the origin, the pair of unit vectors $\hat r, \hat \theta$ gives an orthogonal basis for the tangent space at that point. Note that as the Cartesian unit vectors do not change with position, you can ignore the concept of tangent space and identity the unit vectors as being the same at every point.

In non-orthogonal coordinates, the unit vectors form a non-orthogonal basis for the tangent space at each point. If two unit vectors coincided at a given point, then you would have some sort of coordinate singularity there. The origin is a coordinate singularity in plane polar coordinates, as $\theta$ is not defined at the origin.

Your question, I believe, doesn't properly recognise the local nature of vectors in general, curvilinear (non-Cartesian) coordinate systems.

PS more generally, you may be dealing with a non-Euclidean manifold, where the local nature of vectors becomes even more important.

 

[Svein]

What is "orthogonality"? You need to add some structure to your space in order to define it. The usual version is a Hilbert space, where you define a "scalar product" of two vectors. If that scalar product is 0, the vectors are said to be ortogonal.

 

[PeroK]

To give an example. Suppose two particles at different points both have the velocity $v_x \hat x + v_y \hat y$. Then, the two particles have the same velocity.

But, if they have the same components in plane polar coordinates $v_r \hat r + v_\theta \hat \theta$, then they do not have the same velocity. In other words, the same components represent a different ector at different points - and we see that these components apply technically to a vector in the local tangent space.

 

[Matterwave]

In non-orthogonal coordinate systems, can we say that changing a coordinate could result in changing another coordinate? That is, the coordinates are dependent on each other.

As I understood, non-orthogonal systems will have unit vectors (which are defined to point in the direction of increasing corresponding coordinates) not orthogonal to each other. Doesn’t that mean they could have components along each other’s direction? Thus if we move in the direction of one unit vector, we may “accidently” also move in the direction of another unit vector, yielding a change in another coordinate.

This is one of those things that's way easier to see visually than to describe in words. Geometry is nice like that!

It is easy to draw a non-orthogonal coordinate system: just draw a standard x-y coordinate grid but with your y-axis tilted. Keep your x-axis the same horizontal lines as before, but now your y-axis doesn't go straight up, but at an angle. From there, draw your grid lines parallel to your axes. You find that instead of a grid of squares, you get a grid of rhombus's. When you draw a little arrow at (any of) the intersection following the sides of the rhombus, the angle between the 2 basis vectors is not 90 degrees.

But I think it is not hard to convince yourself that it is certainly possible to only change one coordinate while keeping the other constant, and indeed this IS the case if you move along the basis vectors in this example. Why is that so? Because the grid lines *define* lines of constant coordinate.

If you move horizontally, you are moving along the horizontal grid line and that *by definition* is a line of constant y. Moving along the now tilted y-axis, or any of the infinite parallel displacements of it, similarly *defines* moving in constant x-value.

Orthogonal coordinates just means "grid lines intersect at right angles". But when you move along a grid line, no matter the kind of coordinates you have, you are by definition moving in a way that keeps *all other coordinates constant*. This is *the* defining property of a coordinate system.

I will make the following statement without proof (but it's true, see e.g. Geometrical Methods of Mathematical Physics by Bernard Schutz):

In a general N-dimensional manifold, given N vector fields which are linearly independent at each point, these vector fields form a coordinate basis if and only if the Lie bracket between all pairs of vector fields is 0. All pairwise Lie brackets being 0 means that moving along the congruences (the grid lines in our example) of one vector field is moving along constant affine parameters for the congruences of all other vector fields. We then can use the affine parameters as our coordinate system.

 

[pasmith]

In non-orthogonal coordinate systems, can we say that changing a coordinate could result in changing another coordinate? That is, the coordinates are dependent on each other.

As I understood, non-orthogonal systems will have unit vectors (which are defined to point in the direction of increasing corresponding coordinates) not orthogonal to each other. Doesn’t that mean they could have components along each other’s direction? Thus if we move in the direction of one unit vector, we may “accidently” also move in the direction of another unit vector, yielding a change in another coordinate.

Coordinates are independent, whether orthogonal or nonorthogonal. Introducing such a dependency restricts you to a curve or a surface rather than the whole space.

Consider the following example. A point on the plane can be written as $$x(\tfrac{1}{\sqrt{2}},\tfrac{1}{\sqrt{2}}) + y(0,1) = xe_x + ye_y.$$ Adding 1 to $x$ is not at all the same as adding 1 to $y$.

To find the components of a vector $v$, we must solve $$
\begin{pmatrix} v \cdot e_x \\ v \cdot e_y \end{pmatrix} =
\begin{pmatrix} e_x \cdot e_x & e_y \cdot e_x \\ e_y \cdot e_x & e_y \cdot e_y \end{pmatrix}
\begin{pmatrix} x \\ y \end{pmatrix} =
\begin{pmatrix} 1 & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$$ and we must invert a non-diagonal matrix. If $e_x$ and $e_y$ had been orthogonal, this matrix would have been the identity.

 

[pervect]

In non-orthogonal coordinate systems, can we say that changing a coordinate could result in changing another coordinate? That is, the coordinates are dependent on each other.

As I understood, non-orthogonal systems will have unit vectors (which are defined to point in the direction of increasing corresponding coordinates) not orthogonal to each other. Doesn’t that mean they could have components along each other’s direction? Thus if we move in the direction of one unit vector, we may “accidently” also move in the direction of another unit vector, yielding a change in another coordinate.

The language here is confusing. I wouldn't say that changing one coordinate result in changing another coordinate, given that changing a coordinate is defined by holding the other coordinates constant.

If you have a 3 dimensional manifold with coordinates p,q, and r, $\partial / \partial p$ is defined as changing p while holding q and r constant, and it's defined a coordinate basis vector. One of the important concepts of a mathematical treatment of differential geometry is the identification of partial derivative operators as vectors. The coordinate basis vector, for example $\partial / \partial p$ vector does NOT necessarily have a unit length however. This coordinate basis is also called a holonomic basis.

While the vector $\partial / \partial p$ is always a coordinate basis vector, it's not always of unit length.

When you start scaling non unit length coordinate basis vectors to different vectors that are unit length, you are probably using a non-holonomic basis, or a non-coordinate basis (unless all the scale factors are unity).

There are some important differences between holonomic and nonholonmic bases. The usual language is to say that basis vectors in a holonomic basis commute, and that in a non-holonomic basis, the basis vectors do not all commute, there is at least one pair of basis vectors that do not commute in a non-holonomic basis.

The idea of "commuting" may seem strange, I find it easier to think about commuting in terms of the partial derivative operators rather than the vectors, but differential geometry strongly identifies the two.

So, I would say that it is not right to say that changing one coordinate changes another. But if you are worried that things might not work exactly the way you are used to when you move to a general basis, this is definitely a valid concern. It's difficult to address specifics, though.

Coordinate basis vectors are not necessarily orthogonal, and are not necessarily unit length either. For example, polar coordinates (r,$\phi$) are orthogonal, but are not unit length, because $\partial / \partial \phi$ is not unit length.

Using an orthonormal basis is generally a lot more intuitive than a coordinate basis. Sometimes, though, the math is presented in a way that specifically requires the use of a coordinate basis rather than a non-holonomic basis. For example, in General Relativity, computing the Christoffel symbols is usually done specifically in a coordinate basis.