The coherent inelastic processes of the type $a \rightarrow b$,
which may take place in the interaction of hadrons and $\gamma$ quanta with
nuclei at very high energies ( the nucleus remains the same ), are
theoretically investigated. For taking into account the influence of
matter inside the nucleus, the optical model based on the concept of
refraction index is applied .
Analytical formulas for the effective
cross section $\sigma_{{\rm coh}} (a \rightarrow b )$ are obtained,
taking into account that at ultrarelativistic energies the
main contribution into
$\sigma_{{\rm coh}} (a \rightarrow b )$
is provided by very small transferred momenta in the vicinity of the
minimum longitudinal momentum transferred to the nucleus. It is shown
that the cross section $\sigma_{{\rm coh}} (a \rightarrow b )$ may be
expressed through the "forward" amplitudes of inelastic scattering
$f_{a + N \rightarrow b + N}(0)$ and elastic scattering
$f_{a + N \rightarrow a + N}(0)$, $f_{b + N \rightarrow b + N}(0)$
on a separate nucleon, and it depends on the ratios $L_a/R$ and
$L_b/R$, where $L_a$,
$L_b$ are the respective mean free paths in the nucleus matter for the
particles $a$, $b$ and $R$ is the nuclear radius.

In doing so,
several characteristic cases with different relations of the magnitudes
$L_a, L_b, R$ are considered in detail. When
$L_a/R \gg 1$, but $L_b/R \ll 1$ ( or, on the contrary,
$L_a/R \ll 1$ but $L_b/R \gg 1$ ), then
the cross section $\sigma_{{\rm coh}} (a \rightarrow b )$ is equal to
the ratio of the "forward" cross sections of inelastic scattering
$a + N \rightarrow b + N$ and elastic
scattering of the particle $b$ (or, respectively, $a$) on a nucleon,
multiplied by
the cross section of scattering on the "black" nucleus $\pi R^2$.
The cases $L_a/R \gg 1$, $L_b/R \lesssim 1$ and
$L_a/R \gg 1$, $L_b/R \ll 1$ (for heavy nuclei) correspond,
in particular,
to the coherent production of vector mesons
$\rho^0, \omega, \phi $ at the interaction of very
high energy photons with nuclei, and some numerical estimates are
made for the coherent process $\gamma \rightarrow \rho^0$ on
the $Pb$ nucleus.
Meantime, when both the conditions $L_a/R \gg 1$ and $L_b/R\gg 1$ are
satisfied,
then $\sigma_{{\rm coh}} (a \rightarrow b )$ is proportional to the factor
$R^4/k^2$, where $k$ is the initial energy of the particle $a$
in the laboratory frame.

The formalism described above is generalized also for the case of coherent
inelastic multiparticle processes on a nucleus of the type
$ a \rightarrow \{ \,b_1, b_2, b_3 ....b_i \,\}$ and for the case
of coherent processes in collisions of two ultrarelativistic nuclei.