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It is shown that the electric current in a plane superconducting circuit, placed between the undulator poles, is proportional to the transversal component of velocity of a particle moving in the magnetic field of the undulator. If the particle has nonrelativistic transversal velocity, then the properties of undulator radiation are completely defined by the transversal velocity. This gives the possibility of studying the dynamics and the radiation of a relativistic particle, just by measuring the current in the superconducting circuit. Comprehensive magnetic measurements can be used to adjust the magnetic field of an undulator and to detect or correct field imperfections before installation in an accelerator.