Modeling the dynamics of optical manipulation experiments relies upon a precise mathematical representation of electromagnetic fields and the interpretation of optical momentum and stresses in materials. However, the momentum of light within media has been an issue of debate over the past century. Multiple energy-momentum models have been advanced, each, under certain conditions, agreeing with experimental observation and mathematically consistent with classical electromagnetism. The modern view is that the various formulations of electrodynamics represent different divisions of the total energy-momentum tensor, with the separation of field and matter being ambiguous. Recently, a proposed view of photon momentum identified two leading forms as the kinetic and canonical momenta. The Abraham momentum is responsible for the overall center-of-mass translation of a material, while the Minkowski momentum is responsible for translations with respect to the surrounding medium. However, the Abraham momentum corresponds to multiple, unique electromagnetic energy-momentum tensors that attempt to separate field from material responses (e.g. Abraham, Chu, and Einstein-Laub). However, only the form of the kinetic momentum density has been revealed, while the formulation that uniquely separates the kinetic stress tensor has remained ambiguous. In this correspondence, multiple formulations are considered within the framework of relativistic electrodynamics. We apply various mathematical techniques to identify the kinetic subsystem of electrodynamics. While optical manipulation is usually modeled using a stationary medium approximation, the lessons from relativistic electrodynamics reveal a specific distribution of electromagnetic stress in media. The physics of optical and static manipulation of dielectric particles are described within this framework.