 From integrating both sides of the equation with respect to the time, the constrained coordinates of particles at the time, formula_8, are given,

where formula_10 is the unconstrained (or uncorrected) position of the "i"th particle after integrating the unconstrained equations of motion.To satisfy the constraints formula_11 in the next timestep, the Lagrange multipliers should be determined as the following equation,

This implies solving a system of formula_13 non-linear equations

simultaneously for the formula_13 unknown Lagrange multipliers formula_16.This system of formula_13 non-linear equations in formula_13 unknowns is commonly solved using Newton–Raphson method where the solution vector formula_19 is updated using

where formula_21 is the Jacobian of the equations σ"k"