Revisiting the Lagrange theory for isolated n-particle systems with variable masses connected by an unknown field

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.