Are Cylindrical Polar Coordinates Generalized Coordinates? A Discussion

[neelakash]

meaningless question(?)

I have faced a weird question: Examine if the cylindrical polar co-ordinates represent a set of genralized co-ordinates?

To me it looks a meaningless question. Unless we are given a configuration to describe,how can we be able to predict?For a point particle it may reresent---(if rho, theta and phi describe fully the configuration,each being independent of other), but for a rigid body, it certainly does not. For, we are to consider the Eulerian angles which give the space orientation of the body.

Please comment on it,so that the thing gets clarified.

(Please Note that this is not a homework problem---I am asking this to clarify my idea).

 

[Defennder]

I don't know if this helps, but theta of the polar coordinates is defined with reference to the x-axis, so without having the x-axis you can't locate a specific point (x0, y0) which would be able to do so with Cartesian axes. So all (r,theta) tells us is a circular locus of points.

 

[neelakash]

I did not get your point.

 

[Defennder]

Never mind then, I don't think what I wrote really answered your question, which I have yet to figure out.

 

[Lojzek]

If I understand the question, you must find out whether the function that transforms
coordinates into points in space is bijective. The answer is no: there are several sets of coordinates that represent the same point (r=0, fi=arbitrary).

 

[neelakash]

But what prevents cylindrical polar co-ordinates from being a set of generalized co-ordinates?May be this is not the only set, but how do we know if it is a valid set?

One possibility comes in my mind.We have S=(rho)(phi);so, for a fixed rotation distance, rho and phi are related by this equation.But a set of generalized co-ordinates must be independent of one another.

 

[Defennder]

If I understand the question, you must find out whether the function that transforms coordinates into points in space is bijective. The answer is no: there are several sets of coordinates that represent the same point (r=0, fi=arbitrary).

What's "fi" here? Is it theta for polar coordinates? And if you were to set "fi" to be arbitrary, isn't it apparent that you'll definitely have more than one more point that satifies that condition? The same things occurs in Cartesian coordinates with x=0 y=arbitrary.

@OP: What exactly do you mean by "generalised coordinates"? Defining it would help clear things up a lot.

 

[Andy Resnick]

I have faced a weird question: Examine if the cylindrical polar co-ordinates represent a set of genralized co-ordinates?

To me it looks a meaningless question. Unless we are given a configuration to describe,how can we be able to predict?For a point particle it may reresent---(if rho, theta and phi describe fully the configuration,each being independent of other), but for a rigid body, it certainly does not. For, we are to consider the Eulerian angles which give the space orientation of the body.

Please comment on it,so that the thing gets clarified.

(Please Note that this is not a homework problem---I am asking this to clarify my idea).

I don't understand your question, exactly... but using the term "generalized co-ordinates", in my mind, means either doing mechanics in a coordinate-free representation (e.g. $$\nabla\bullet\textbf{V}$$) or using index notation without specifying which coordinate system is used (here, $$\partial_iV^i$$).

Perhaps the question relates to how cylindrical coordinates involve scale factors when performing various differential operations becasue the unit vectors vary in space?

 

[Lojzek]

What's "fi" here? Is it theta for polar coordinates? And if you were to set "fi" to be arbitrary, isn't it apparent that you'll definitely have more than one more point that satifies that condition? The same things occurs in Cartesian coordinates with x=0 y=arbitrary.

@OP: What exactly do you mean by "generalised coordinates"? Defining it would help clear things up a lot.

My fi is your theta: the angle between the origin-point vector and x-axis (it is called fi in our schools).
It is not true that you get different points with (r=0, fi=arbitrary)! You always get the origin (0,0) in cartesian coordinates.

 

[dipstik]

i owuld say no, because generalized coordinates refer to unspecified coordiante systems, but in your case, the coordinates system has been specified.