Motion of Particles in External Gravitational-Electromagnetic Field

Given a system of n particles with inertial masses \(m_1,\ldots ,m_n\) , charges \(\sigma _1,\ldots ,\sigma _n\) , places \(\textbf{r}_1(t),\ldots ,\textbf{r}_n(t)\) and velocities \(\textbf{r}'_1(t),\ldots , \textbf{r}'_n(t)\) in the outer gravitational field with vectorial potential \(\textbf{v}(\textbf{x},t)\) , the outer electromagnetical fields with potentials \(\textbf{A}(\textbf{x},t)\) and \(\mathbf {\Psi }(\textbf{x},t)\) and additional conservative field with the classical scalar potential \(V(\textbf{y}_1,\ldots ,\textbf{y}_n,t)\) , consider a Lagrangian