Exploring Uncommon Coordinate Systems in Physics

[Falgun]

I have come across Cartesian, Polar, Cylindrical & Spherical Coordinate Systems so far and was wondering if someone could tell me which are the uncommon systems used in physics which everyone says that they exist but no one explicitly mentions. Is there a "standard reference" or are they just passed down as word of mouth? Sorry if the question is a bit misguided because I have no idea about this topic.

 

[anuttarasammyak]

All of them are common each making use of symmetry of the system you are tackling. That's the reason why you learn them in your curriculum.

 

[Falgun]

All of them are common each making use of symmetry of the system you are tackling. That's the reason why you learn them in your curriculum.

I understand that. I am asking if you could name and explain some uncommon ones?

 

[anuttarasammyak]

As a short example skew coordinates. In GR metric tensor $g_{ij}$ defines what coordinate system the system has.

 

[jasonRF]

There is a table of them here:

https://en.m.wikipedia.org/wiki/Orthogonal_coordinates

 

[Falgun]

There is a table of them here:

https://en.m.wikipedia.org/wiki/Orthogonal_coordinates

This is what I was looking for

 

[alan123hk]

There are many other orthogonal coordinate systems, such as parabolic cylindrical coordinates, elliptic cylindrical coordinates, and oblate spherical coordinates, which seem to be not very commonly used. I don't know under what circumstances they will be used to study physics.

https://en.wikipedia.org/wiki/Parabolic_cylindrical_coordinates
https://en.wikipedia.org/wiki/Elliptic_cylindrical_coordinates
https://en.wikipedia.org/wiki/Oblate_spheroidal_coordinates

 

[vanhees71]

Parabolic coordinates are of some use in the Coulomb-scattering problem in quantum mechanics and for the bound state problem when treating the Stark effect. They are always a good choice if you have a spherically symmetric problem in a situation, where some direction is preferred by the physical situation. In the scattering problem that's the direction of the incoming-particle momentum; in the case of the Stark effect it's the direction of the external electric field.

These 3D orthogonal coordinates are the ones for which the 3D Laplace operator and thus the Helmholtzoperator separates. For a thorough treatment, see the great books by Morse and Feshbach, Methods of Theoretical Physics (2 vols.).

 

[Dale]

Most coordinate systems have no name. They are described by their mathematical properties instead. These can be equations to transform from named systems to the unnamed system or the metric written in the new coordinates.

 

[Andy Resnick]

I have come across Cartesian, Polar, Cylindrical & Spherical Coordinate Systems so far and was wondering if someone could tell me which are the uncommon systems used in physics which everyone says that they exist but no one explicitly mentions. Is there a "standard reference" or are they just passed down as word of mouth? Sorry if the question is a bit misguided because I have no idea about this topic.

Moon and Spencer's "Field Theory Handbook" is a compendium of 40 coordinate systems, providing separation equations and solutions for the Laplace and Helmholtz equations in each.

 

[mpresic3]

If you study geodesy using Heiskanen and Moritz, you will run into elliptic coordinates pretty extensively

 

[alan123hk]

Moon and Spencer's "Field Theory Handbook" is a compendium of 40 coordinate systems, providing separation equations and solutions for the Laplace and Helmholtz equations in each.

There are many different coordinate systems to choose from. If the shape of the object is deliberately designed to correspond to these special coordinate systems in the engineering design, this may simplify the analysis and speed up the efficiency of the simulation calculation.

 

[DrDu]

The $\mathrm{H}_2^+$ molecular hamiltonian is separable in elliptical coordinates.

 

[DrDu]

How about Rindler coordinates in SR?

 

[vanhees71]

Milne coordinates are useful to express the Bjorken solution of relativistic hydro in an elegant way.

https://www.physik.uni-bielefeld.de/~borghini/Teaching/Hydrodynamics15/06_09-slides.pdf

 

[rude man]

In mechanical dynamics the 'tangential & normal' component system is often used since in curvilinear motion acceleration is thereby relatively simply represented.

 

[Meir Achuz]

There is a table of them here:

https://en.m.wikipedia.org/wiki/Orthogonal_coordinates

That is great reference. I bookmarked it