  Many particle quantum hydrodynamics based on the Darwin Hamiltonian (the
Hamiltonian corresponding to the Darwin Lagrangian) is considered. A force
field appearing in corresponding Euler equation is considered in details.
Contributions from different terms of the Darwin Hamiltonian in the Euler
equation are traced. For example, the relativistic correction to the kinetic
energy of particles leads to several terms in the Euler equation, these terms
have different form. One of them has a form similar to a term appearing from
the Darwin term. Hence, the two different mechanisms give analogous
contributions in wave dispersion. Microscopic analog of the Biot-Savart law,
called the current-current interaction and describing an interaction of moving
charges via the magnetic field, is also included in our description. The
semi-relativistic generalization of the quantum Bohm potential is obtained.
Contribution of the relativistic effects in the spectrum of plasma collective
excitations is considered.
