Electromagnetic field theory without divergence problems: 1. The Born Legacy

A fully consistent classical relativistic electrodynamics with spinless point charges is constructed. The classical evolution of the electromagnetic fields is governed by the nonlinear Maxwell--Born--Infeld field equations, the classical evolution of the point charges by a many-body Hamilton--Jacobi law of motion. The Pauli principle for bosons can be incorporated in the classical Hamilton--Jacobi formalism. The Cauchy problem is explained and illustrated with examples. The question of charge-free field solitons is addressed also and it is shown that if they exist, their peak field strengths must be enormous. The value The value of Born's constant is shown to be a subtle open issue.