We have studied space charge effects in a synchrotron with multi-particle tracking in 2-D and 3-D configuration space (4-D and 6-D phase space, respectively). First, we will describe the modelling of space charge fields in the simulation and a procedure of tracking. Several ways of presenting tracking results will be also mentioned. Secondly, it is discussed as a demonstration of the simulation study that coherent modes of a beam play a major role in beam stability and intensity limit. The incoherent tune in a resonance condition should be replaced by the coherent tune. Finally, we consider the coherent motion of a beam core as a driving force of halo formation. The mechanism is familiar in linac, and we apply it in a synchrotron. ©1998 American Institute of Physics.