  We employ generalized Euler coordinates for the $n$ body system in $d \geq
n-1$ dimensional space, which consists of the centre-of-mass vector, relative
(mutual), mass-independent distances $r_{ij}$ and angles as remaining
coordinates. We prove that the kinetic energy of the quantum $n$-body problem
for $d \geq n-1$ can be written as the sum of three terms: (i) kinetic energy
of centre-of-mass, (ii) the second order differential operator $\Delta_{rad}$
which depends on relative distances alone and (iii) the differential operator
$\Omega$ which annihilates any angle-independent function. The operator
$\Delta_{rad}$ has a large reflection symmetry group $Z_2^{\oplus
\frac{n(n-1)}{2}}$ and in $\rho_{ij}=r_{ij}^2$ variables is an algebraic
operator, which can be written in terms of generators of their {\it hidden}
algebra $sl(\frac{n(n-1)}{2}+1, R)$. Thus, $\Delta_{rad}$ makes sense of the
Hamiltonian of a quantum Euler-Arnold $sl(\frac{n(n-1)}{2}+1, R)$ top in a
constant magnetic field. It is conjectured that for any $n$, the
similarity-transformed $\Delta_{rad}$ is the Laplace-Beltrami operator plus
(effective) potential; thus, it describes a $\frac{n(n-1)}{2}$-dimensional
quantum particle in curved space. This was verified for $n=2,3,4$. After
de-quantization the similarity-transformed $\Delta_{rad}$ becomes the
Hamiltonian of the classical top with variable tensor of inertia in an external
potential.
  This approach allows a reduction of the $dn$-dimensional spectral problem to
a $\frac{n(n-1)}{2}$ -dimensional spectral problem if the eigenfunctions depend
only on relative distances. We prove that the ground state function of the $n$
body problem depends on relative distances alone.
