- #1
Lajka
- 68
- 0
Hello,
I wasn't quite sure where to make this topic, so I hope I didn't do wrong by putting it here.
The question I'm having is somewhat difficult to describe and I guess it's more of a mathematical question really, but since I'm learning mechanics now and came up with it, I thought it would be best to ask it here.
So what I'm interested in is what really makes generalized coordinates... coordinates?
Let me explain this.
The way I figure coordinates is that they are just a numbers, a coefficients if you will, which multiply the basis vectors in order to produce a needed vector as a superposition of these. So, the first thing I need is a basis, a set of linearly independent vectors which are able to span the whole vector space.
Since I learn mechanics, it's only natural to observe R3 space, and a standard basis for that is, of course, (1,0,0), (0,1,0) and (0,0,1) vectors, or [tex]\vec{i}[/tex], [tex]\vec{j}[/tex] and [tex]\vec{k}[/tex] vectors.
So, if I have my coordinates [tex](x,y,z)[/tex], the only thing I need to do is to multiply my unit vectors with them, add them, and voila, I have my desired vector.
My problem here is because I can't think of any other coordinates that satisfy all these conditions.
For example, I'm well aware of spherical coordinates [tex](r,\varphi,\theta)[/tex] and I'm well aware that these are equivalent to a former triple [tex](x,y,z)[/tex] in a sense that they can fully determine any position, and that they also have three orthogonal unit vectors, but here's the thing...
First of all, you don't even use all three of these unit vectors, you just use one of them to determine the position, and not one of them is even constant! I don't have problem with this from a physical point of view, but I do have from a mathematical point of view. Like I said before, if I have new coordinates, I need a new basis as well, and although these unit vectors are orthogonal, they are not constant, and that just doesn't make sense if they are to constitute a basis.
Further more, if these new coordinates are coordinates on their own right, they should be able to stand alone for themselves, but that's not the case! If I want to determine a position in spherical system, i write [tex]\vec{r}=r*\vec{r_{0}}[/tex], right? But to calculate [tex]\vec{r_{0}}[/tex], I need my [tex]\varphi[/tex], [tex]\theta[/tex] AND [tex]\vec{i}[/tex] and [tex]\vec{j}[/tex] unit vectors from before!
WHY on Earth would I need these OLD unit vectors when I should be able to calculate everything without them? Remember, I have chosen a new basis and new coordinates now! Old coordinates, old basis, begone!
What I'm trying to say is that, for me, the process is simple: I have coordinates, I have constant basis and I calculate a needed vector. That's the definition that I've learned from math when I was learning vector spaces, and as far as I can tell, the only basis I've seen that satisfies these rigorous conditions is the triple [tex](\vec{i},\vec{j},\vec{k})[/tex].
All other coordinates (and their unit vectors consequently) depend on that triple in some way, so how can one look at them as a set of independent coordinates in their own right?
When I see in my book from mechanics something like "generalized coordinates are defined as [tex](q_{1},,q_{2},q_{3})[/tex], and that can be [tex](x,y,z)[/tex] or [tex](r,\varphi,\theta)[/tex] or...", I can't help but wonder if that's really fair, because I only see one example of 'true' coordinates with fully independent, constant basis.
So, what am I missing here?
P.S. I should say that I'm also aware that [tex](x,y,z) \mapsto (r,\varphi,\theta)[/tex] is a map and I know what a Jacobian is and so forth, I've learned all that from a multivariable calculus, but I don't know how and if any of that can be used here for some reasoning. After all, I used that for integrating functions, this something different, these are vector spaces.
I wasn't quite sure where to make this topic, so I hope I didn't do wrong by putting it here.
The question I'm having is somewhat difficult to describe and I guess it's more of a mathematical question really, but since I'm learning mechanics now and came up with it, I thought it would be best to ask it here.
So what I'm interested in is what really makes generalized coordinates... coordinates?
Let me explain this.
The way I figure coordinates is that they are just a numbers, a coefficients if you will, which multiply the basis vectors in order to produce a needed vector as a superposition of these. So, the first thing I need is a basis, a set of linearly independent vectors which are able to span the whole vector space.
Since I learn mechanics, it's only natural to observe R3 space, and a standard basis for that is, of course, (1,0,0), (0,1,0) and (0,0,1) vectors, or [tex]\vec{i}[/tex], [tex]\vec{j}[/tex] and [tex]\vec{k}[/tex] vectors.
So, if I have my coordinates [tex](x,y,z)[/tex], the only thing I need to do is to multiply my unit vectors with them, add them, and voila, I have my desired vector.
My problem here is because I can't think of any other coordinates that satisfy all these conditions.
For example, I'm well aware of spherical coordinates [tex](r,\varphi,\theta)[/tex] and I'm well aware that these are equivalent to a former triple [tex](x,y,z)[/tex] in a sense that they can fully determine any position, and that they also have three orthogonal unit vectors, but here's the thing...
First of all, you don't even use all three of these unit vectors, you just use one of them to determine the position, and not one of them is even constant! I don't have problem with this from a physical point of view, but I do have from a mathematical point of view. Like I said before, if I have new coordinates, I need a new basis as well, and although these unit vectors are orthogonal, they are not constant, and that just doesn't make sense if they are to constitute a basis.
Further more, if these new coordinates are coordinates on their own right, they should be able to stand alone for themselves, but that's not the case! If I want to determine a position in spherical system, i write [tex]\vec{r}=r*\vec{r_{0}}[/tex], right? But to calculate [tex]\vec{r_{0}}[/tex], I need my [tex]\varphi[/tex], [tex]\theta[/tex] AND [tex]\vec{i}[/tex] and [tex]\vec{j}[/tex] unit vectors from before!
WHY on Earth would I need these OLD unit vectors when I should be able to calculate everything without them? Remember, I have chosen a new basis and new coordinates now! Old coordinates, old basis, begone!
What I'm trying to say is that, for me, the process is simple: I have coordinates, I have constant basis and I calculate a needed vector. That's the definition that I've learned from math when I was learning vector spaces, and as far as I can tell, the only basis I've seen that satisfies these rigorous conditions is the triple [tex](\vec{i},\vec{j},\vec{k})[/tex].
All other coordinates (and their unit vectors consequently) depend on that triple in some way, so how can one look at them as a set of independent coordinates in their own right?
When I see in my book from mechanics something like "generalized coordinates are defined as [tex](q_{1},,q_{2},q_{3})[/tex], and that can be [tex](x,y,z)[/tex] or [tex](r,\varphi,\theta)[/tex] or...", I can't help but wonder if that's really fair, because I only see one example of 'true' coordinates with fully independent, constant basis.
So, what am I missing here?
P.S. I should say that I'm also aware that [tex](x,y,z) \mapsto (r,\varphi,\theta)[/tex] is a map and I know what a Jacobian is and so forth, I've learned all that from a multivariable calculus, but I don't know how and if any of that can be used here for some reasoning. After all, I used that for integrating functions, this something different, these are vector spaces.