Coordinate systems - finding optimal? simple conceptual question

[elegysix]

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?

anyone have any idea about this?

 

[lurflurf]

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.

 

[elegysix]

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.