Infinite sequence of Poincare group extensions: structure and dynamics

We study the structure and dynamics of the infinite sequence of extensions of the Poincare algebra whose method of construction was described in a previous paper (Bonanos and Gomis J. Phys. A: Math. Theor. 42 (2009) 145206 (arXiv: hep-th/0808.2243)). We give explicitly the Maurer-Cartan (MC) 1-forms of the extended Lie algebras up to level 3. Using these forms and introducing a corresponding set of new dynamical couplings, we construct an invariant Lagrangian, which describes the dynamics of a distribution of charged particles in an external electromagnetic field. At each extension, the distribution is approximated by a set of moments about the world line of its center of mass and the field by its Taylor series expansion about the same line. The equations of motion after the second extensions contain back-reaction terms of the moments on the world line.