  We propose a new classical approach for describing a system composed of $n$
interacting particles with variable mass connected by a single field with no
predefined form ($n$-VMVF systems). Instead of assuming any particular nature
or analytical function for representing the variation of the masses or field,
we propose them as unknown functions dependent on the particle positions and
velocities. The work presents the Lagrangian theory which incorporates such
variations which are find using only first principles. The consideration of
mass as unknown quantity lead us to modify the D'Alembert's principle to ensure
the compliance of the relativity principle. Also, because the addition of new
variables to the system, we add a new and independent set of Lagrange equations
depending on the $3$-D angular coordinates for the system of equations remain
solvable. The four-dimensional space-time naturally appears in the problem when
the position of the particle is expressed as a function of angular coordinates.
This transformation set the $3-$D space of the angular coordinates as the
stereographic projection of the $4-$D sphere defined by the Lorentz condition
in the space-time. We identify two sets of constraints, each one for every
coordinate system, by forcing the system to satisfy the laws of the
conservation of the linear and angular momentum. The $\ddot{x}$ dependency's of
the constraints functions set the necessity of extending the classical theory
up to the second order of the Lagrange function. The obtained constraints are
added to the initial Lagrangian by the Lagrange multiplier method and obtain
not one, but two Lagrange functions and with them, two set of Lagrange
equations for finding the final solution.
