  The overlap between the spectroscopic Galactic Archaeology with HERMES
(GALAH) survey & $Gaia$ provides a high-dimensional chemodynamical space of
unprecedented size. We present a first analysis of a subset of this overlap, of
7066 dwarf, turn-off, & sub-giant stars. [...] We investigate correlations
between chemical compositions, ages, & kinematics for this sample. Stellar
parameters & elemental abundances are derived from the GALAH spectra with the
spectral synthesis code SME. [...] We report Li, C, O, Na, Mg, Al, Si, K, Ca,
Sc, Ti, V, Cr, Mn, Co, Ni, Cu, Zn, Y, as well as Ba & we note that we employ
non-LTE calculations for Li, O, Al, & Fe. We show that the use of astrometric &
photometric data improves the accuracy of the derived spectroscopic parameters,
especially $\log g$. [...] we recover the result that stars of the
high-$\alpha$ sequence are typically older than stars in the low-$\alpha$
sequence, the latter spanning $-0.7<$[Fe/H]$<+0.5$. While these two sequences
become indistinguishable in [$\alpha$/Fe] vs. [Fe/H] at the metal-rich regime,
we find that age can be used to separate stars from the extended high-$\alpha$
& the low-$\alpha$ sequence even in this regime. [...] we find that the old
stars ($>8$ Gyr have lower angular momenta $L_z$ than the Sun, which implies
that they are on eccentric orbits & originate from the inner disk. Contrary to
some previous smaller scale studies we find a continuous evolution in the
high-$\alpha$-sequence up to super-solar [Fe/H] rather than a gap, which has
been interpreted as a separate "high-$\alpha$ metal-rich" population. Stars in
our sample that are younger than 10 Gyr, are mainly found on the low
$\alpha$-sequence & show a gradient in $L_z$ from low [Fe/H]
($L_z>L_{z,\odot}$) towards higher [Fe/H] ($L_z<L_{z,\odot}$), which implies
that the stars at the ends of this sequence are likely not originating from the
close solar vicinity.
