Apparently disconnected concepts of Differential Geometry on basic Mechanics

Question

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?

Answer 1

The equations (3) should take a general parametrization for the trajectory (we note it $s$), i.e. to each point of the curve we assign an orthonormal basis, so in a sense it is more general than the non-linear change of basis (switching from cartesian to polar coordinates) of the embedding space of the trajectory in which the parameter is necessarily time, see (1) and (2)

If you want, for (1) or (2) we exploit that the the trajectory is fully embedded into $\mathbb{R}^n$ and that the parametrization of it is necessary the time, while for general interpretation of (3) you use a more intrinsic approach (i.e. the embedding of the trajectory is no longer explicit), by using a generic parameter $s$ and a local orthogonal basis of the tangent space. You can find a diffeomorphism between these two manifolds (the embedding space of the motion as $\mathbb{R}^n$ which is called the configurations space and the tangent space of a particular point of the configurations space).

(4) is simply Newton's second law on a generic (curved) manifold, so in a sense it is closer to your (3) but uses time as the generic curve parameter $s$.