  It is shown that the formulation of the SU(3) Yang-Mills quantum Hamiltonian
in the "flux-tube gauge" $A_{a1}=0$ for all a=1,2,4,5,6,7 and $A_{a2}=0$ for
all a=5,7 allows for a systematic and practical strong coupling expansion of
the Hamiltonian in $\lambda\equiv g^{-2/3}$, equivalent to an expansion in the
number of spatial derivatives. Introducing an infinite spatial lattice with box
length a, the "free part" is the sum of Hamiltonians of Yang-Mills quantum
mechanics of constant fields for each box, and the "interaction terms" contain
higher and higher number of spatial derivatives connecting different boxes. The
Faddeev-Popov operator, its determinant and inverse, are rather simple, but
show a highly non-trivial periodic structure of six Gribov-horizons separating
six Weyl-chambers. The energy eigensystem of the gauge reduced Hamiltonian of
SU(3) Yang-Mills mechanics of spatially constant fields can be calculated in
principle with arbitrary high precision using the orthonormal basis of all
solutions of the corresponding harmonic oscillator problem, which turn out to
be made of orthogonal polynomials of the 45 components of eight irreducible
symmetric spatial tensors. First results for the low-energy glueball spectrum
are obtained which substantially improve those by Weisz and Ziemann using the
constrained approach. Thus, the gauge reduced approach using the flux-tube
gauge proposed here, is expected to enable one to obtain valuable
non-perturbative information about low-energy glueball dynamics, using
perturbation theory in $\lambda$.
