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

TL;DR

This paper proves the existence and uniqueness of solutions for the relativistic N-body problem in electrodynamics, ensuring well-defined trajectories under Feynman's law with no singularities, extending initial conditions globally.

Contribution

It establishes the conditions under which unique, maximal, relativistically admissible trajectories exist for the N-body electrodynamics problem, extending initial trajectories without singularities.

Findings

Unique solutions exist for initial trajectories with nonsingular points.

Solutions extend globally and are maximal in the system.

Trajectories avoid singularities and satisfy the Newton-Einstein equations.

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.