Multidimensional center-of-mass and relative coordinates

Question

For a two-dimensional cartesian coordinate system $(x,y)$ describing two particles of unit mass, one frequently encounters a transformation into center-of-mass and relative coordinates $(R,r)$ defined by

$$ \begin{pmatrix}R\\r\end{pmatrix} = \begin{pmatrix}1&1\\1&-1\end{pmatrix} \begin{pmatrix}x\\y\end{pmatrix} $$

How can this concept reasonably be extended to higher dimensions?

Here are my thoughts. In principle, the problem can be approached quite directly as the following: Find a unitary transformation matrix $V$ with a row of ones on top, $$ V = \begin{pmatrix}1&1&\cdots&1&1\\\\ & &U\\ &\end{pmatrix} $$ This can be accomplished, for example, by starting from the identity matrix, plugging in the row of ones on top, and applying Gram-Schmidt orthonormalization to obtain the matrix U. This, however, is not going to produce a "balanced" set of coordinates (whatever that means). Thus, I'm looking for approaches where $U$ is reasonably chosen, say as compact or sparse as possible, or with other suitable properties. This is also where I'd be interested what references in the literature did.

Answer 1

How can this concept reasonably be extended to higher dimensions?

first create a vector $\vec{x}$ with n components , for example $n=5$

$$\vec{x}= \left[ \begin {array}{c} x_{{1}}\\ x_{{2}} \\ x_{{3}}\\ x_{{4}} \\ x_{{5}}\end {array} \right] $$

then create the vector $\vec{r}$ with $n-1$ relative equations

$$\vec{r}=\left[ \begin {array}{c} x_{{1}}-x_{{2}}\\ x_{{2}}- x_{{3}}\\ x_{{3}}-x_{{4}}\\ x_{{4} }-x_{{5}}\end {array} \right] $$

the matrix

$U=\frac{\partial \vec{r}}{\partial \vec{x}}\quad,(n-1\times n)$ the Jacobi-Matrix

$$U=\left[ \begin {array}{ccccc} 1&-1&0&0&0\\ 0&1&-1&0&0 \\ 0&0&1&-1&0\\0&0&0&1&-1 \end {array} \right] $$

you can write a small program to build the matrix V

$$V=\left[ \begin {array}{ccccc} 1&1&1&1&1\\ 1&-1&0&0&0 \\ 0&1&-1&0&0\\ 0&0&1&-1&0 \\ 0&0&0&1&-1\end {array} \right] $$