Hydrodynamics and Electromagnetism: Differential–Geometric Aspects and Analogies

The well-known evolution equations of a solenoidal vector field with integral curves frozen into a continuous medium are presented in an invariant form in the four-dimensional spacetime. A fundamental $1$-form ($4$-potential) is introduced, and the problem of variation of the action (integral of the $4$-potential along smooth curves) is considered. The extremals of the action in the class of curves with fixed endpoints are described, and the conservation laws generated by symmetry groups are found. Under the assumption that the electric and magnetic fields are orthogonal, Maxwell's equations are represented as evolution equations of a solenoidal vector field. The role of the velocity field is played by the normalized Poynting vector field.