- #1
Bashyboy
- 1,421
- 5
Hello,
I am currently reading about the topic alluded to in the topic of this thread. In Taylor's Classical Mechanics, the author appears to be making a requirement about any arbitrary coordinate system you employ in solving some particular problem. He says,
"Instead of the Cartesian coordinates r = (x,y,z), suppose that we wish to use some other coordinates. These could be spherical polar coordinates [itex](r, \theta, \phi)[/itex], r cylindrical coordinates [itex]\rho , \phi , z)[/itex], or any set of "generalized coordinates" q1, q2, q3, i the property that each position specifies a unique value of (q1, q2, q3); that is,
[itex]q_i = q_i(\vec{r})[/itex] for [itex]i = 1,2~and~3[/itex],
[itex]\vec{r}=\vec{r}(q_1,q_2,q_3)[/itex]
I understand that [itex]\vec{r}[/itex] is our position vector function, and that requiring coordinates (q1, q2, 3) in our arbitrary coordinate system to be able to be represented by a position vector makes sense, although I wouldn't mind someone telling me the exact reason we require this. Again, it also makes sense that we would require (q1, q2, q3) and the position vector function representing that point to specify one unique point--if this weren't true, then a particle could occupy to positions at the same time, although I wouldn't mind someone telling me the exact reason we require this. But I don't understand the requirement [itex]q_i = q_i(\vec{r})[/itex], what does this mathematical statement mean?
I am currently reading about the topic alluded to in the topic of this thread. In Taylor's Classical Mechanics, the author appears to be making a requirement about any arbitrary coordinate system you employ in solving some particular problem. He says,
"Instead of the Cartesian coordinates r = (x,y,z), suppose that we wish to use some other coordinates. These could be spherical polar coordinates [itex](r, \theta, \phi)[/itex], r cylindrical coordinates [itex]\rho , \phi , z)[/itex], or any set of "generalized coordinates" q1, q2, q3, i the property that each position specifies a unique value of (q1, q2, q3); that is,
[itex]q_i = q_i(\vec{r})[/itex] for [itex]i = 1,2~and~3[/itex],
[itex]\vec{r}=\vec{r}(q_1,q_2,q_3)[/itex]
I understand that [itex]\vec{r}[/itex] is our position vector function, and that requiring coordinates (q1, q2, 3) in our arbitrary coordinate system to be able to be represented by a position vector makes sense, although I wouldn't mind someone telling me the exact reason we require this. Again, it also makes sense that we would require (q1, q2, q3) and the position vector function representing that point to specify one unique point--if this weren't true, then a particle could occupy to positions at the same time, although I wouldn't mind someone telling me the exact reason we require this. But I don't understand the requirement [itex]q_i = q_i(\vec{r})[/itex], what does this mathematical statement mean?