  In the framework of the concept of time correlation functions, we develop a
self-consistent relaxation theory of the transverse collective particle
dynamics in liquids. The theory agrees with well-known results in both the
short-wave (free particle dynamics) and the long-wave (hydrodynamic) limits. We
obtain a general expression for the spectral density~$C_T(k,\omega)$ of
transverse particle current realized in the range of wave numbers $k$. In
domain of microscopic spatial scales comparable to action scale of effective
forces of interparticle interaction, the theory reproduces a transition from a
regime with typical equilibrium liquid dynamics to a regime with collective
particle dynamics where properties similar to solid-state properties appear:
effective shear stiffness and transverse (shear) acoustic waves. In the
framework of the corresponding approximations, we obtain expressions for the
spectral density of transverse particle current for all characteristic regimes
in equilibrium collective dynamics. We obtain expressions for dispersion law
for transverse (shear) acoustic waves and also relations for the kinematic
shear viscosity $\nu$, the transverse speed of sound $v^{(T)}$, and the
corresponding sound damping coefficient $\Gamma^{(T)}$. We compare the
theoretical results with the results of atomic dynamics simulations of liquid
lithium near the melting point.
