Non-Lorentzian expansions of the Lorentz force and kinematical algebras

We consider non-Lorentzian expansions, Galilean and Carrollian, of the Lorentz force equation in which both the particle position and the electro-magnetic field are expanded. There are two well-known limits in the case of a constant field, called electric and magnetic, that are studied separately. We show that the resulting equations of motion follow equivalently from considering a non-linear realisation of a certain infinite-dimensional algebras.