The Existence and Uniqueness of Solutions to N-Body Problem of Electrodynamics

Abstract

Given $n$ charges interacting with each other according to Feynman's law. Let $(r_j(t),v_j(t))$ denote the position and velocity of the charge $q_j.$ The list $y(t)$ of all such vectors is called a trajectory. A Lipschitzian trajectory $x(t), (t\le0),$ with continuous derivative, on which the velocities do not exceed some limiting velocity $v<c,$ where $c$ denotes the speed of light, is called an initial trajectory. A locally Lipschitzian trajectory $y(t)$ is called relativistically admissible if the velocities on it stay below the speed of light $c.$ The author constructs operators $\Phi_{j}$ of a trajectory whose values $\Phi_j(y)(t)$ are linear transformations of $R^3$ into $R^3.$ A point $t=t_1$ on a trajectory $y$ is called singular if either some of the charges collide at the time $t_1$ or the determinant is zero for at least one of the transformations $\Phi_j(y)(t_1).$ The main result is the following: If $x(t) (t\le0)$ is an initial trajectory with nonsingular point $t=0,$ then there exists a unique relativistically admissible trajectory $y(t),$ defined for $t$ in an interval $I\subset < 0,\infty),$ extending the initial trajectory $x(t)$ and having the following properties. (1) No point $t$ on the trajectory $y$ is singular. (2) The trajectory represents a unique solution of the Newton-Einstein momentum-force system of equations under Lorentz forces induced by electromagnetic field in accord to Feynman's law for moving point charges. (3) The trajectory $y$ represents the maximal global solution of the system.