I was studying about spherical and cylindrical coordinates but I still cannot understand what its importance in physics and math, that is, what was the motivation for its discovery? Solve a problem in physics or is its importance different?
$\begingroup$
$\endgroup$
3
-
$\begingroup$ Have you tried solving e.g. the heat or wave equation for a spherical domain in Cartesian xyz coordinates? $\endgroup$– Anders SandbergCommented Dec 6, 2020 at 8:06
-
$\begingroup$ it's just a choice of coordinate system to make use of a symmetry of the problem youre looking at $\endgroup$– ctsmdCommented Dec 6, 2020 at 8:48
-
$\begingroup$ It seems you don't have a very good teacher if they couldn't come up with physical examples. $\endgroup$– ProfRobCommented Dec 6, 2020 at 16:30
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
Let's consider an example: The gravity of the earth. Suppose the earth is a sphere (its center of the mass defines the origin of our coordinate system) and that we are interested in the gravity at a hight $h$ above the ground. Newtons law tells us that the gravitational force at a position $\vec r = \vec r_{probe}= (r,\varphi, \theta)$ is given by $$ \tag{1} \vec F_g = \vec F_g(r,\varphi, \theta) = - \gamma \frac{m_{probe} M_{earth}}{r^2} \; \vec e_r $$ where $r = R_{earth} + h$ is the distance of the probe to the earth's center, and $\vec e_r$ is the vector connecting these two points. Note that $\vec e_r=(1,0,0)_{r,\varphi, \theta}$ in spherical coordinates.
Eq. (1) is a simple formula, because it consists only of a single variable, $r$. The two angles $(\varphi, \theta)$ do not appear in the formula. Hence, the 3D problem is reduced to a 1D problem, which is much simpler to solve and understand: From the formula it is evident that the force scales with the distance squared and that it is directed towards the earth's center.
Now, let's express this formula in cartesian coordinates $(x,y,z)$, $$ \vec F_g = \vec F_g(x,y,z) = - \gamma \frac{m_{probe} M_{earth}}{(x^2+y^2 + z^2)^{3/2}} \; \left( \begin{matrix} x\\ y\\ z \end{matrix} \right) $$ Since the cartesian coordinates do not capture the symmetry of the problem, we have to use all three coordinates $(x,y,z)$. Hence, the 3D problem is not reduced to 1D. Note that both descriptions are equivalent: What is true/false for one is also true/false for the other.
The importance of other coordinate systems in physics is that they allow us to understand the problem on a much deeper level. Also, since the math is much simpler they allow us to solve problems which were otherwise beyond our ability.
