I am a programmer trying to simulate the motion of particles involved in $\beta^-$ decay (in 2D). While trying to conserve energy and momentum throughout the process, I end up with a system of equations for the trajectories of the electron ($_e$) and neutrino ($_n$) after $\beta^-$ decay, $$ |v_n|^2 = \mathbf{v_{xn}}^2 + \mathbf{v_{yn}}^2 $$ $$ |v_e|^2 = \mathbf{v_{xe}}^2 + \mathbf{v_{ye}}^2 $$ $$ p_x = m_e \mathbf{v_{xe}} + m_n \mathbf{v_{xn}} $$ $$ p_y = m_e \mathbf{v_{ye}} + m_n \mathbf{v_{yn}} $$ Everything not bolded is a known constant and everything bolded are unknowns (4 total). Assuming the constants are given such that this system has infinite solutions (i.e. there is enough energy for $\beta^-$ decay to occur in the first place), how can I choose a solution for the unknowns randomly?
EDIT: I realized the equations for momentum included incorrect squares, so I removed them. Now I'm not sure if this system even has infinite solutions, but intuitively I think it should.
With reference to the diagram below, Equations 1 and 2 tell you that for the electron the vector $ (m_e \mathbf{v_{xe}},m_e \mathbf{v_{ye}}) $ has length $ r_e = v_e $, and similarly for the neutrino, $ r_n = v_n $. So in momentum space the vector momenta of the electron and neutrino lie somewhere on two circles about the origin (see sketch on the left). Equations 3 and 4 tell you that the vector sum of the momenta of the electron and neutrino lies at a point $ (p_x, p_y) $, and as you can see from the sketch on the right, there will generally only be two solutions for the momenta of the electron and neutrino. So two solutions, not an infinite number.