  A four-dimensional differential Euler-Lagrange equation for continuously
distributed materials is derived based on the principle of least action, and
instead of Lagrangian, this equation contains the Lagrangian density. This
makes it possible to determine the density of generalized four-momentum in
covariant form as derivative of the Lagrangian density with respect to
four-velocity of typical particles of a system taken with opposite sign, and
then calculate the generalized four-momentum itself. It is shown that the
generalized four-momentum of all typical particles of a system is an integral
four-vector and therefore should be considered as a special type of
four-vectors. The presented expression for generalized four-momentum exactly
corresponds to the Legendre transformation connecting the Lagrangian and
Hamiltonian. The obtained formulas are used to calculate generalized
four-momentum of stationary and moving relativistic uniform systems for the
Lagrangian with particles and vector fields, including electromagnetic and
gravitational fields, acceleration field and pressure field. It turns out that
the generalized four-momentum of a moving system depends on the total mass of
particles, on the Lorentz factor and on the velocity of the systems center of
momentum. Besides, an additional contribution is made by the scalar potentials
of the acceleration field and the pressure field at the center of system. The
direction of the generalized four-momentum coincides with the direction of
four-velocity of the system under consideration, while the generalized
four-momentum is part of the relativistic four-momentum of the system.
