I)For this question I will use apparently disconected concepts of vector calculus and differential geometry to ilustrate my doubts. Moreover, I'll refer to [1]FINN.E; Fundamental university physics Volume 1: Mechanics and to [2]ANTON.H; Calculus: Multivariable, in some results.

II) I'll need to be a little bit prolix in this question to explain my point

Consider velocity [1]: \begin{equation} \vec{v} = \vec{r}\frac{dr(t)}{dt}+\vec{\theta}\frac{d\theta (t)}{dt}..........(1) \end{equation} In the equation above we cleary see a particle under general curvilinear motion. Which means that the terms :

$$\vec{r}\frac{dr(t)}{dt}$$ and $$\vec{\theta}\frac{d\theta (t)}{dt}$$

are respectively the radial velocity and tangencial velocity (where $\vec{r}$ and $\vec{\theta}$ are simply the basis of polar coordinates).

If we consider $ \displaystyle \frac{dr(t)}{dt} = 0$ (the condition of time independent radius), then, equation (1) becomes:

$$ \vec{v} =\vec{\theta}\frac{d\theta (t)}{dt}..........(2a) $$

and then the acceleration: $$\vec{a} = -\vec{r}\left (\frac{d\theta(t)}{dt}\right )^2 + \vec{\theta}\frac{d^{2} \theta(t)}{dt^{2}}..........(2b)$$ This analysis is quite right, but [2] gives us another way to see the problem:

For a particle moving along a curve C in 2-space or 3-space,the velocity and acceleration vector can be written as:

$$ \vec{v} = \frac{ds(t)}{dt}\vec{T(t)}..........(3a) \\ \vec{a} = \frac{d^{2}s(t)}{dt^{2}}\vec{T(t)} + \kappa (t)\left (\frac{ds(t)}{dt}\right )^2 \vec{N(t)}..........(3b) $$

Here the vectors $\vec{T(t)}$ and $\vec{N(t)}$ are precisely the vectors of TNB-triad (frenet trihedron), and $\kappa (t)$ is the Curvature.

Now, consider that we know the "general force":

$$ F^a = m\left( \frac{d^2x^a}{dt^2} + \Gamma^{a}_{bc}\frac{dx^b}{dt}\frac{dx^c}{dt} \right)..........(4)$$

So, the Christoffel symbols vanish on cartesian coordinates but in a curvilinear coordinate system, say polar coordinates, we can derive velocities and accelerations in the form of the set of equations 2.

My first question is: the set of equations 2 are the description of motion using a polar coordinate system. So, the set of equations 3 are independent of coordinate choice? And if so, why use polar coodinates then?

My second question is: How can we relationate the equations 2, 3 and 4?

I) For this question I will use apparently disconnected concepts of vector calculus and differential geometry to illustrate my doubts. Moreover, I'll refer to [1]FINN.E; Fundamental university physics Volume 1: Mechanics and to [2]ANTON.H; Calculus: Multivariable, in some results.

II) I'll need to be a little bit prolix in this question to explain my point

Consider velocity [1]: \begin{equation} \vec{v} = \vec{r}\frac{dr(t)}{dt}+\vec{\theta}\frac{d\theta (t)}{dt}\tag{1} \end{equation} In the equation above we cleary see a particle under general curvilinear motion. Which means that the terms :

$$\vec{r}\frac{dr(t)}{dt}$$ and $$\vec{\theta}\frac{d\theta (t)}{dt}$$

are respectively the radial velocity and tangencial velocity (where $\vec{r}$ and $\vec{\theta}$ are simply the basis of polar coordinates).

If we consider $ \displaystyle \frac{dr(t)}{dt} = 0$ (the condition of time independent radius), then, equation (1) becomes:

$$ \vec{v} =\vec{\theta}\frac{d\theta (t)}{dt}\tag{2a} $$

and then the acceleration: $$\vec{a} = -\vec{r}\left (\frac{d\theta(t)}{dt}\right )^2 + \vec{\theta}\frac{d^{2} \theta(t)}{dt^{2}}\tag{2b}$$ This analysis is quite right, but [2] gives us another way to see the problem:

For a particle moving along a curve C in 2-space or 3-space,the velocity and acceleration vector can be written as:

$$ \vec{v} = \frac{ds(t)}{dt}\vec{T(t)}\tag{3a} $$ $$ \vec{a} = \frac{d^{2}s(t)}{dt^{2}}\vec{T(t)} + \kappa (t)\left (\frac{ds(t)}{dt}\right )^2 \vec{N(t)}\tag{3b} $$

Here the vectors $\vec{T(t)}$ and $\vec{N(t)}$ are precisely the vectors of TNB-triad (frenet trihedron), and $\kappa (t)$ is the Curvature.

Now, consider that we know the "general force":

$$ F^a = m\left( \frac{d^2x^a}{dt^2} + \Gamma^{a}_{bc}\frac{dx^b}{dt}\frac{dx^c}{dt} \right)\tag{4}$$

So, the Christoffel symbols vanish on Cartesian coordinates but in a curvilinear coordinate system, say polar coordinates, we can derive velocities and accelerations in the form of the set of equations 2.

My first question is: the set of equations 2 are the description of motion using a polar coordinate system. So, the set of equations 3 are independent of coordinate choice? And if so, why use polar coordinates then?

My second question is: How can we relate the equations 2, 3 and 4?