Section 1 defines the geometrical notion of a harmonic map between two generalized Lagrange spaces. Section 2 analyzes the particular case of the harmonic maps between two Lagrange spaces of electrodynamics. Section 3 proves that the smooth solutions of certain important first order DE- or PDE-systems are harmonic maps between convenient generalized Lagrange spaces. Section 4 describes the common main geometrical properties of the generalized Lagrange structures which convert the solutions of the initial first order DE- or PDE-systems into harmonic maps.