We consider three types of electron accelerators that can be used for various applications, such as industrial, medical, cargo inspection, and isotope production applications, and that require small- and medium-sized machines, namely classical microtron (CM), race-track microtron (RTM), and multisection linac. We review the principles of their operation, the specific features of the beam dynamics in these machines, discuss their advantages and weak points, and compare their technical characteristics. In particular, we emphasize the intrinsic symmetry of the stability region of microtrons. We argue that RTMs can be a preferable choice for medium energies (up to 100 MeV) and that the range of their potential applications can be widened, provided that the beam current losses are significantly reduced. In the article, we analyze two possible solutions in detail, namely increasing the longitudinal acceptance of an RTM using a higher-order harmonic accelerating structure and improving beam matching at the injection.

We model the phase oscillations of electrons in race-track microtrons by means of an area preserving map with a fixed point at the origin, which represents the synchronous trajectory of a reference particle in the beam. We study the nonlinear stability of the origin in terms of the synchronous phase -- the phase of the synchronous particle at the injection. We estimate the size and shape of the stability domain around the origin, whose main connected component is enclosed by the last rotational invariant curve. We describe the evolution of the stability domain as the synchronous phase varies. Besides, we approximate some rotational invariant curves by level sets of certain Hamiltonians. Finally, we clarify the role of the stable and unstable invariant curves of some hyperbolic (fixed or periodic) points.

We consider a cosmological model with a variable gravitational constant, G, based on a scalar-tensor theory. Using the recent observational data for the Hubble diagram of type Ia supernovae (SNeIa) we find a phenomenological expression describing the variation of G. The corresponding variation of the fine structure constant \alpha within multidimensional theories is also computed and is shown not to support known constraints on \Delta \alpha / \alpha.