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This paper aims to present some selected fundamentals of the theory of a broad class of gyro-devices in a systematic and consistent manner and with sufficient detail necessary for understanding the underlying physical principles of their operation. The focus of this work is on the derivation and analysis of important invariants (constants of motion), as well as on comments concerning their analytical power and the physical insights they provide.

Gyrotrons are used as high-power sources of coherent radiation operating in pulsed and CW regimes in many scientific and technological fields. In this paper, we discuss two of their numerous applications. The first one is in gyrotron-powered electromagnetic wigglers and undulators. The second one is for driving high-gradient accelerating structures in compact particle accelerators. The comparison, between the requirements imposed by these two concepts on the radiation sources on one hand and the output parameters of the currently available high-performance gyrotrons on the other hand, show that they match each other to a high degree. We consider this as a manifestation of the feasibility and potential of these concepts. It is believed that after the first successful proof-of-principle experiments they will find more wide usage in the advanced FEL and particle accelerators.