Generalized coordinates as components

Question

Why we cannot express Generalized coordinates as a vector like we do with Cartesian coordinates $x$ , $y$ ,$z$ ?

Answer 1

You can put like a vector, for example, if we change the coordinate system $(x,y,z)\Rightarrow (r,\theta,\phi)$ you can put a vector $v=a\hat r+b\hat \theta+c\hat \phi$. I think you are refaring why lagrangian formalism does not appear vectors

Answer 2

Are you familiar with the 1st and 2nd geodetic problems? (https://en.wikipedia.org/wiki/Geodesy#Geodetic_problems)

1. "Given a point (in terms of its coordinates) and the direction (azimuth) and distance from that point to a second point, determine (the coordinates of) that second point."

2. "Given two points, determine the azimuth and length of the line (straight line, arc or geodesic) that connects them."

The reason I mention them is because a point, as defined by coordinates, is not a vector. A point lives in an affine space, which assigns coordinates to every point, and has some arbitrary origin where, usually, all the coordinates are zero. (As opposed to a vector space, where $\vec 0$ has special significance).

The difference between two points is a vector (2nd problem), and a point plus a vector is another point (1st problem).

Now when we use Cartesian coordinates to define a point in 3D Euclidean space

$$ p_i = (x_i, y_i, z_i) $$

we easily find the difference between that point and the origin:

$$ p_i - 0 = (x_i, y_i, z_i) - (0,0,0) = x_i\hat{\bf x} + y_i\hat{\bf y} + y_i\hat{\bf y} \equiv \vec{\bf p}_i = [x_i, y_i, z_i]$$

Notice the striking similarity in the representation of the point, $(x_i, y_i, z_i)$, and the representation of the vector, $[x_i, y_i, z_i]$; they are both three numbers (the same three numbers) in some kind of bracket.

That can lead one to conclude that the point, $p_i$, and vector, $\vec{\bf p}_i$, are the same thing. They are not.

Note that if I represent the point in spherical coordinates:

$$ p_i = (r_i\sin{\theta_i}\cos{\phi_i}, r_i\sin{\theta_i}\sin{\phi_i}, r_i\cos{\theta_i}) $$

then it is still the same point, but it looks nothing like the vector separating it from the origin:

$$\vec{\bf p}_i = p_i - 0 = x_i\hat{\bf x} + y_i\hat{\bf y} + y_i\hat{\bf y} $$

even if I use the local basis $(\hat{\bf r}_i, \hat{\bf \theta}_i, \hat{\bf \phi}_i)$:

$$\vec{\bf p}_i = r_i\hat{\bf r}_i = (r_i, 0, 0)_i$$