Extended relativistic dynamics of charged spinning particle in quaternionic formulation

The central idea in the present paper is that corresponding to some increment of the particle’s energy there should correspond an extension of the degrees of freedom in the description. We then suggest to extend the formulations of Newtonian and relativistic mechanics. We start from the relativistic Lorentz-force equations, explore an algorithm of extension and use the latter to build an extension of the Newtonian equations of motion. The mapping between momenta of the Extended Newtonian and relativistic mechanics is built on the basis of Vieta’s formulae of a quadratic polynomial. The equations of motion in the external e.m. fields are presented in the basis of the quaternion algebra. Further, the algorithm of extension is used a second time and leads to Doubly Extended Newtonian Mechanics (DENM). On making use of Vieta’s formulae on the cubic polynomial from DENM equations we derive equations of the Extended Relativistic Mechanics (ERM). In the polar representation the dynamic equations of DENM are given by Jacobi equations for elliptic functions, whereas equations of motion of ERM are represented by Weierstrass equations for elliptic functions. The equations of particle motion under e.m. fields are given in the basis of the algebra of quaternions. The algorithm of extension is repeated n-times to obtain Extended Newtonian mechanics of (n+1)-order and the corresponding mapping onto the hyper-relativistic dynamics is constructed. The mapping is given by Vieta’s formulae between roots and coefficients of (n+1)-degree polynomial. The equations of motion in the external e.m. fields are given within the algebra of quaternions.