Advantages of Polar Coordinate System & Rotating Unit Vectors

[torito_verdejo]

What is the advantage of using a polar coordinate system with rotating unit vectors? Kleppner's and Kolenkow's An Introduction to Mechanics states that base vectors $\mathbf{ \hat{r}}$ and $\mathbf{\hat{\theta}}$ have a variable direction, such that for a Cartesian coordinates system's base vectors $\mathbf{ \hat{i}}$ and $\mathbf{ \hat{j}}$ we have
$$\mathbf{\hat{r}} = \cos \theta\ \mathbf{\hat{i}} + \sin \theta\ \mathbf{\hat{j}}$$
$$\mathbf{\hat{\theta}} = -\sin \theta\ \mathbf{\hat{i}} + \cos \theta\ \mathbf{\hat{j}}$$
Now, isn't counter-productive to define a coordinate system in terms of another? Why, at least in this book, we choose to use such a dependent coordinate system, instead of using a polar coordinate system employing a radius and the angle that this one forms with a polar axis, that are therefore independent of another coordinate system?

Thank you in advance.

 

[anorlunda]

Summary: What is the advantage of using a polar coordinate system with rotating, not constant unit vectors?

Now, isn't counter-productive to define a coordinate system in terms of another?

Things like coordinate systems are usually chosen to make things simpler.

Simpler or not simpler is by definition a matter of opinion, not fact.

 

[torito_verdejo]

Things like coordinate systems are usually chosen to make things simpler.

Simpler or not simpler is by definition a matter of opinion, not fact.

Well, people can generally agree on a "simpler" way of doing or modeling things, and in that sense my question is not merely about individual opinion.

 

[Chestermiller]

Well, complexity and hence simplicity are pretty non-subjective, and when it comes to the way we model something, people can generally agree on a "simpler" way of doing things. In any case, my question is clearly answerable: what are the advantages of defining a polar coordinate system that relies on the Cartesian coordinate system? How and why is such a system more appropriate than another polar coordinate system that only makes use of a radius, a polar axis and the angle both of these form?

The polar coordinate system does not "rely" on the Cartesian system. The equations you gave merely shows the relationship between the unit vectors in polar coordinates and the unit vectors in Cartesian coordinates. The polar coordinate unit vectors are not constant, but change direction as a function of the polar angle $\theta$.

As to the question about the value of using polar coordinates rather than Cartesian coordinates, this comes into play when we find that it is much simpler to solve certain problems using polar coordinates. This applies strongly when solving distributed parameter problems with boundary conditions applied on cylindrical surfaces.

 

[torito_verdejo]

The polar coordinate system does not "rely" on the Cartesian system. The equations you gave merely shows the relationship between the unit vectors in polar coordinates and the unit vectors in Cartesian coordinates. The polar coordinate unit vectors are not constant, but change direction as a function of the polar angle $\theta$.

As to the question about the value of using polar coordinates rather than Cartesian coordinates, this comes into play when we find that it is much simpler to solve certain problems using polar coordinates. This applies strongly when solving distributed parameter problems with boundary conditions applied on cylindrical surfaces.

Just to clarify, I was not asking about the advantages of a polar coordinate system over a Cartesian one, but about the advantages of a polar coordinate system based on rotating base vectors over another polar coordinate system with simply a radius, a polar axis and the angle between both. However, you made me realize I was getting confused by this "translation" to the Cartesian coordinate system. Thank you. :)

 

[Nugatory]

Thank you for your answer. Just to clarify, I was not asking about the advantage of a polar coordinate system over a Cartesian one, but about the advantage of a polar coordinate system based on rotating base vectors over another polar coordinate system with simply a radius, a polar axis and the angle between both.

Consider an object moving in a circle under the influence of a radially directed force: a rock on a string, or a planet in a circular orbit, or a bicyclist on a banked circular track. In standard polar coordinates its velocity vector is $\frac{V}{R}\frac{d\theta}{dt}\hat{\theta}$ with $V/R$ a constant.

What does this vector look like using the coordinates that you suggest?

 

[torito_verdejo]

Consider an object moving in a circle under the influence of a radially directed force: a rock on a string, or a planet in a circular orbit, or a bicyclist on a banked circular track. In standard polar coordinates its velocity vector is $\frac{V}{R}\frac{d\theta}{dt}\hat{\theta}$ with $V/R$ a constant.

What does this vector look like using the coordinates that you suggest?

Hit and sunk. You just made me understand how useful is to be able, through base vectors, to represent vectors as sums of products. Thank you very much. Excuse me for being so noob.

 

[Chestermiller]

Hit and sunk. You just made me understand how useful is to be able, through base vectors, to represent vectors as sums of products. Thank you very much. Excuse me for being so noob.

You don't need to excuse yourself for anything. We all went through the same issues, so now we're here to help you. Welcome to Physics Forums.