The review of the development of modified betatron - the circular accelerators and charge particle accumulators with focusing by longitudinal magnetic field is presented. Such accelerators may have numerous applications like, generation of powerful electron beams with current about several kA and energy 20-100 MeV, radiological and electron therapy, electron cooling of ions with circulating electron beam, generation antihydrogen and positronium. The methods of the computation particle dynamics in modified betatron are considered. The results of experimental studies of beam dynamics in modified betatron LEPTA, JINR are described.

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