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Clearly, the number of new coordinates cannot be smaller than the number of degrees of freedom within the system.

But otherwise, there seem to be little restriction on maps between coordinate system. In particular, in the context of the Lagrangian formalism, authors regularly argue that essentially "any set of generalized coordinates" can be used. This, of course, cannot be correct because we can't map all coordinates to, say, $0$.

So what are the precise conditions suitable maps between coordinate systems need to fulfil?

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    $\begingroup$ I think nonsingular and suitably differentiable? $\endgroup$ – tfb Feb 22 at 10:43
  • $\begingroup$ This depends on context. $\endgroup$ – Qmechanic Feb 22 at 11:55
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What are the conditions coordinate transformations need to fulfill in general?

if you go from a set of generalized coordinates to a new set of generalized coordinates, i see three criteria that must fulfill.

I) the number of the new set of the generalized coordinates must be the same as the old one

II) The metric of the new set of the generalized coordinates must be invertible ($\det(G)\ne 0$)

III) for a system with constrained conditions: the new set of the generalized coordinates must fulfilled the constrained equations

Example I:

Position vector:

$\vec{R}=\begin{bmatrix} x\\ y\\ \end{bmatrix}=\begin{bmatrix} q_1\\ q_2\\ \end{bmatrix}$

we have two degrees of freedom with $x=q_1\,,y=q_2$ the generalized coordinates

we can write the position vector $\vec{R}$ in polar coordinates

$\vec{R}=\begin{bmatrix} r\,\cos(\theta)\\ r\,\sin(\theta)\\ \end{bmatrix}=\begin{bmatrix} w_1\,\cos(w_2)\\ w_1\,\sin(w_2)\\ \end{bmatrix}$

with $r=w_1\,,\theta=w_2$ the "new" generalized coordinates

The metric is:

$G= \left[ \begin {array}{cc} 1&0\\ 0&{r}^{2} \end {array} \right]\quad,\det(G)=r^2\ne 0$

Fazit: criteria I) and II) are fulfilled

Example II: pendulum

the constrained equation is:

$F_c=x^2+y^2=l^2\quad \Rightarrow\quad y=\sqrt{l^2-x^2}$

The position vector:

$\vec{R}=\begin{bmatrix} x\\ y\\ \end{bmatrix}\mapsto \begin{bmatrix} x\\ \sqrt{l^2-x^2}\\ \end{bmatrix}=\begin{bmatrix} q\\ \sqrt{l^2-q^2}\\ \end{bmatrix}$

we have one degree of freedom with the generalized coordinate $q$

if we go to polar coordinate $x=l\cos(\theta)\,,y=l\sin(\theta)$ we can see that this transformation satisfied the constrained equation $F_c$ and the position vector is now :

$\vec{R}=\begin{bmatrix} x\\ y\\ \end{bmatrix}\mapsto \begin{bmatrix} l\cos(\theta)\\ l\sin(\theta)\\ \end{bmatrix}= \begin{bmatrix} l\cos(w)\\ l\sin(w) \end{bmatrix}\quad \Rightarrow\quad G=l^2\,,\det(G)\ne 0$

with $w$ the new generalized coordinate.

Fazit: criteria I) II) and III) are fulfilled.

how to calculate the metric $G$:

$G=J^T\,J$

where $J$ the Jacobi matrix $J=\frac{\partial \vec{R}}{\partial \vec{q}} $ and $\vec{q}$ is the vector of the generalized coordinates $(q_1\,,q_2\,\ldots\,,q_n)$

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