Natural oscillation of plasma bunches in a resonator

The Lagrange functions of a plasma bunch, a chain of plasma bunches and electromagnetic field in a resonator loaded by a plasma bunch are defined in the paper. Dispersion equations for three cases are derived by the variation method. It is shown that in case of low plasma density natural frequency of the bunch oscillations follows the Langmuir dependence, but at high plasma density it aspires to the frequency of half-wave vibrator. The chain of plasma bunches possesses weak space dispersion. At commensurability of natural frequencies of the bunch and hollow resonator, frequency of the resonator loaded by the bunch never becomes equal to the frequency of any mode of the resonator vortex field