Force-free magnetic fields (f.f.f) are considered as the first approximation of magnetic hydrodynamic equations in the case when the energy of the field exceeds the thermal energy of the medium. Such a relation of energies takes place in the upper atmosphere of the Sun in active regions.
The consequence of the virial theorem obtained shows that for any solution of the corresponding non-linear system of equations only two cases are possible: either the total energy of the field is given by a divergent integral, or in some regions the force-free character of field is destroyed. This permits the conclusion that it is impossible to build f.f. current systems everywhere, and therefore 'boundary' problems for this type of fields are of the same importance as for harmonic fields.
Integral relations are obtained which are the necessary conditions for the solution of boundary problems. According to the classical principle of Thompson the harmonic fields are always stable, while f.f.f, may be stable or unstable.
It is shown that: (1) arbitrary f.f.f, are stable to small changes of boundary conditions; (2) among f.f.f, the hydrodynamically stable configurations exist.
The hydrodynamic stability condition restricts the size of force-free currents in such configurations.