Trying to Understand Generalized Coordinates

[Vorde]

I am trying to understand what generalized coordinates are but I'm having some trouble. After reading up on them a bit my best understanding of the idea of generalized coordinates is the following:

Because choice of coordinate system is arbitrary when solving physical systems (or anything for that matter), choosing generalized coordinates is the practice of using the simplest possible coordinate system available for your problem, because even though your chosen coordinates might be silly and useless in practice, they result in the cleanest equations.

Is this in any way correct? And if I'm totally off base could someone help me or point to where I can be helped? I've found a lot of pages which mathematically explain them to me but nowhere has yet done a good job of explaining why.

 

[Bacle2]

I'm sorry, I don't know what you mean by generalized coordinates.

In my understanding, coordinates have to see with manifolds.

A related result ,often used in R^n is that of the inverse function theorem,

which tells you when local changes of coordinates are possible.

Edit: I just thought you may have been thinking about tensors and how they

transform under coordinate changes.Is that it? You're right in that sometimes

a coordinate change does simplify calculations.

 

[Vorde]

http://en.wikipedia.org/wiki/Generalized_coordinates

The wikipedia article explains to a level where I understand generalized coordinates but does not get me comfortable with 'why', which is what I was hoping PF could help me with.

 

[chiro]

Hey Vorde.

Probably the best way to think about this is to take a n-dimensional piece of paper and bend it it any way you want and that is your co-ordinate system on a general curved space.

If the object hasn't been changed it's R^n. If it's transformed to something whacky then it's whatever it is.

The co-ordinate systems that are usually considered have a few properties: the first one is that you can take the co-ordinate system and "un-bend it" so that it becomes flat again and this is known as bijectivity between the system and R^n for some positive integer of n.

The other one is that it is "smooth" and this means it can be differentiated so much with respect to the variables that make up the point definitions.

The reason why we do this is because we have cases where each basis vector is not independent.

In R^n, every basis vector does not depend on the other ones but in curved geometries when considering how they are "embedded" in some orthogonal space, this is not true.

To understand this think about the following function examples: x = 2, y = 3 and and y = 2x. In the first example x and y are completely independent but in the second example they are not and the second example happens when things are "curved".

So whenever you have dependencies on the basis vectors in some sense, the geometry is curved.

This happens in the theory of relativity since space and time depend on each other and are not independent like you would have in R^n.

Similarly when you are modelling bodies that deform that retain their mass but can have their boundary change, then you can treat this in the context of a general co-ordinate system.

 

[Stephen Tashi]

http://en.wikipedia.org/wiki/Generalized_coordinates

The wikipedia article explains to a level where I understand generalized coordinates but does not get me comfortable with 'why', which is what I was hoping PF could help me with.

You might get a better intuitive answer in the classical physics section than here among the mathematicians.

A crude way to look at is to notice how people talking about (non-generalized) coordinates are usually thinking about a point in space. The point's coordinates may represent some physical measurements, but that's "background information", so to speak.

The requirement for generalized coordinates is that they give a complete description of "configuration" of "the system", whatever the system happens to be. For example you could represent the first 3 numbers on a lottery ticket as a point (x,y,z) in Cartesian space, but if lottery tickets have 6 numbers on them, this doesn't give complete configuration information. The focus of generalized coordinates is to give a physical description. It will turn out that you can view the coordinates in a context of space and geometry, but that's "background information".

If you have "complete" information about a deterministic physical system at time t = 0, you can predict what it will do in the future. Complete information is more information that the "configuration" of a system. For example, in the Wikipedia, it shows a double pendulum. The two angles are sufficient to determine a possible "configuration" of the pendulum - think of it as a still picture But you can't predict the future movement of the pendulum from a still picture. You'd have to know the current velocities and accelerations of it's parts. Physics problems usually involve not only the generalized coordinates, but also their time derivatives, which give generalized velocities and accelerations.

People in the math section (including me) may be inclined to wonder why a velocity or acceleration can't be part of the "configuration" of a system. Maybe in some kinds of physics it is. You'll have to ask the physicists. As I have seen generalized coordinates used, they represent a "still picture" of a system.

 

[Vorde]

Thank you to both of you. I think I get the gist of what they are, and luckily I won't have to use them in physics for quite a while :)

Thanks again.