On derivation of equations of electrodynamics and gravitation from the principle of least action, the Hamilton–Jacobi method, and cosmological solutions

In classical texts, equations for fields are proposed without derivation of right-hand sides. Below, the right-hand sides of the Maxwell and Einstein equations are derived within the framework of the Vlasov–Maxwell–Einstein equations from a classical, but slightly more general principle of least action and Hamilton–Jacobi equations are used to obtain cosmological solutions.