Good resource on covariant/contravariant, magnetic coordinates, and jacobians?

In summary, The speaker is a student in computational plasma physics who is looking for resources to better understand magnetic field aligned coordinates. They have a system of equations that define these coordinates but find them unintuitive. They mention a scalar Jacobian, which is the determinant of the Jacobian matrix, and is used to convert between coordinate systems. The determinant describes how the volume element is affected by the transformation. The speaker also touches on the difference between contravariant and covariant components, which change differently as the coordinate system is changed. The key concept here is the Jacobian, which is used to transform vectors between coordinate systems.
  • #1
chrysaetos13
1
0
First post here in PF, so forgive me if this question is in the wrong place.

I'm a student in computational plasma physics. The code I work with utilizes magnetic field aligned coordinates, and as a necessity, it is sometimes useful to convert between spatially regular coordinates (cartesian, cylindrical, toroidal, etc) to our field aligned coordinates. Although I have a system of equations which defines our magnetic coordinates, and a general idea that "magnetic field lines are straight in these coordinates" I find the coordinate system largely unintuitive.

Does anyone know of a good resource for developing intuition on complicated coordinate systems? I have always felt a little fuzzy even on the relatively simple description of covariant and contravariant coordinates provided in Jackson's E&M. In one sentence, what is the salient benefit of using these coordinates.

I also see descriptions of a scalar Jacobian, which I gather/guess is the determinant of the Jacobian matrix. Is this correct? I believe the matrix can be thought of as a way to convert between coordinate systems. Is there an intuitive, spatial way to think of what this scalar Jacobian value represents?
 
Last edited:
Physics news on Phys.org
  • #2
The determinant of a linear operator describes how the volume element is dilated or shrunk under the action of the operator. Thus it's common to see in volume integrals, as well as easy to work with because unlike the transformations of vectors, this factor is just a number.

I wouldn't describe the distinction between contravariant and covariant as one that is useful so much as necessary. You can decompose a vector in terms of vectors tangent to coordinate axes (these are contravariant components), which is what's usually done, but on a 2d plane, you can use a different basis, one that is perpendicular to all other coordinate axes. These are the covariant components, and as you change coordinate systems, the components change differently, just as a consequence of their differing definitions.

Basically, though, the Jacobian is key, and as you go back and forth between coordinate systems, individual vectors will transform by the Jacobian (or more exactly, but either its inverse or its transpose or both, depending on which coordinate system you start with).
 

1. What is the difference between covariant and contravariant coordinates?

Covariant coordinates are a set of coordinates that transform according to the first-order differential transformation law. In other words, they change when the reference frame changes. Contravariant coordinates, on the other hand, remain unchanged when the reference frame changes.

2. How are magnetic coordinates used in physics?

Magnetic coordinates are a set of coordinates that are used to describe the motion of charged particles in a magnetic field. They are particularly useful in plasma physics and fusion research, where the behavior of charged particles in a magnetic field is of great importance.

3. What is the Jacobian matrix and how is it related to covariant and contravariant coordinates?

The Jacobian matrix is a mathematical tool used to describe the relationship between two sets of coordinates. In the context of covariant and contravariant coordinates, the Jacobian matrix is used to transform between these two coordinate systems.

4. Can you explain the concept of the metric tensor in relation to covariant and contravariant coordinates?

The metric tensor is a mathematical object that describes the relationship between infinitesimal distances in a given coordinate system. In the context of covariant and contravariant coordinates, the metric tensor is used to convert between the two coordinate systems.

5. How are covariant and contravariant coordinates used in general relativity?

In general relativity, covariant and contravariant coordinates play a crucial role in describing the curvature of spacetime. They are used to define the metric tensor, which is in turn used to calculate the geodesic equations that describe the motion of objects in curved spacetime.

Similar threads

  • Advanced Physics Homework Help
Replies
5
Views
2K
Replies
5
Views
805
  • Advanced Physics Homework Help
Replies
13
Views
3K
Replies
11
Views
4K
  • Calculus and Beyond Homework Help
Replies
2
Views
2K
  • Classical Physics
Replies
1
Views
1K
  • STEM Academic Advising
Replies
4
Views
2K
  • Differential Geometry
Replies
23
Views
10K
  • Calculus and Beyond Homework Help
Replies
4
Views
2K
Replies
5
Views
4K
Back
Top