Coordinate systems - finding optimal? simple conceptual question

In summary, the conversation discusses the use of Jacobians in transforming coordinate systems in a physics course. This leads to a discussion about deriving an optimal coordinate system for a given problem, with "optimal" meaning the most simplified equation of a surface or bounds of a constraint. The speaker suggests that using Lagrange's method in mechanics, which involves generalized coordinates, may be useful in solving for these coordinates. However, it is not always possible to find a "most simplified" coordinate system for a complex problem. The examples of representing a straight line and a sphere in different coordinate systems are given as illustrations.
  • #1
elegysix
406
15
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?
 
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  • #2
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.
 
  • #3
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.
 

1. What is a coordinate system?

A coordinate system is a mathematical concept that is used to represent the position of a point in space. It is a set of rules that define how points are located and measured in a given space.

2. What are the different types of coordinate systems?

There are several types of coordinate systems, including Cartesian, polar, cylindrical, and spherical. These coordinate systems have different methods for measuring and describing the location of a point in space.

3. How do you find the optimal coordinate system?

The optimal coordinate system depends on the specific problem or situation. It is important to consider the relationships between the variables and the desired outcome in order to determine the best coordinate system for the given problem.

4. What is the difference between an optimal and a simple coordinate system?

An optimal coordinate system is one that is best suited for a specific problem or situation. It takes into account the variables and desired outcome to provide the most accurate and efficient representation. A simple coordinate system, on the other hand, may not take these factors into consideration and may be less accurate.

5. How do you conceptualize a coordinate system?

To conceptualize a coordinate system, it is important to understand the underlying principles and rules that govern it. This includes understanding the different types of coordinate systems, how points are located and measured, and how it can be applied to solve problems in different fields of science and mathematics.

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