I Variation principle - holonomic constraints

[QuasarBoy543298]

in order to extend hamilton's principle to include holonomic constraints, out lecturer did the following :
when we are under constraints, we cannot consider the variations of the coordinates as independent of each other.
we know that the constraint equations are f_a = 0.
we can multiply each constraint by an arbitrary constant λ_a.
∑f_a*λ_a = 0, so we can just add it to the Lagrangian, as we treat the λ_a
as independent coordinates.

from there we just followed the same thing we did in order to find the original EL equations.my question is - why can we treat the λ_a as independent coordinates.
also, why doing so gives us the option to treat the other coordinates as independent from each other (vary them by δq_i without considering the constraints).

besides, if someone can explain a little the background of this process because to me it feels like we just added something that relates to the constraints into the Lagrangian and got new equations plus the original constraints.thanks in advance !

 

[BvU]

Did you already study Lagrange multipliers in calculus ?
It basically has to do with the two gradients (of the constraint(s) and of the objective function to be minimized): they have to be collinear.
You remove (number of constraints) degrees of freedom from an n$\times$n problem, so you add n $\lambda$'s to come back to an n$\times$n optimization problem again.