Proving Euler-Lagrange for constrained system

railblue

1. The problem statement, all variables and given/known data

Given two Euler-Lagrange systems with generalized coordinates ##r_1## and ##r_2,## and input ##u_1## and ##u_2##. Suppose now that a constraint is placed on them such that ##r_1 = f_1(q)## and ##r_2 = f_2(q)##.

Propose a Lagrangian for the constrained system and show that is is also Euler-Lagrange


2. Relevant equations
Where should I even be starting on this type of proof?


3. The attempt at a solution
I do know that the generalized coordinate themselves are constrained now, but do they contribute at all to reducing the degree of freedom?