# Coordinate systems - finding optimal? simple conceptual question

## Main Question or Discussion Point

today in my physics course we were using jacobians to transform coordinate systems.

This made me wonder if there was a way of deriving an optimal coordinate system to use for a given problem.
-optimal meaning most simplified equation of a surface or bounds of a constraint (ex. cylindrical coordinates for modeling a solenoid, polar for pendulum motion)

I know that usually we just look at a problem and use what we think will simplify things the most, but I think it would be useful in complex problems to derive it.

Lagrange's method in mechanics uses generalized coordinates, so maybe I could use it to solve for them somehow?

lurflurf
Homework Helper
Not possible, at least in general. What does most simplified mean? What Lagrangian would you propose for these most simplified coordinates? Sometimes there is a system that will be complicated no matter the coordinates. Other times different parts of a system are simple in different coordinates, but when the interaction of the parts requires a third coordinate system.

As an example of 'most simplified' - representing a straight line is easiest in cartesian, and more complex in cylindrical, even more so in spherical and others.

Another example is a sphere, very simple in spherical coordinates, more complex in cylindrical, and even more in cartesian.

I don't know that using the lagrange's method I'm familiar with - relating energies and potentials to generalized coordinates - would actually be useful. I just know it involves generalized coordinates.