Requirement of Holonomic Constraints for Deriving Lagrange Equations

[deuteron]

TL;DR

Why is it required for the constraints to be holonomic to derive the Euler-Lagrange equations of motion?

While deriving the Lagrange equations from d'Alembert's principle, we get from $$\displaystyle\sum_i(m\ddot x_i-F_i)\delta x_i=0\tag{1}$$ to $$\displaystyle\sum_k (\frac {\partial\mathcal L}{\partial\ q_k}-(\frac d {dt}\frac {\partial\mathcal L}{\partial\dot q_k}))\delta q_k=0\tag{2}$$

However, from the above step, we get to the below step only after assuming holonomic constraints:
$$(\frac {\partial\mathcal L}{\partial\ q_k}-(\frac d {dt}\frac {\partial\mathcal L}{\partial\dot q_k})=0.\tag{3}$$

Why is it that we have to assume holonomic constraints for that transition? My guess is that it has something to do with that if the constraints are not holonomic, then the virtual displacement are not always perpendicular to the trajectory of the body, but I can't see the mathematical connection between these.

 

[vanhees71]

There is no restriction to holonomic constraints. You can also treat non-holonomic constraints. If done right, i.e., as a constraint on the "allowed" variations of the trajectories in configuration space, you get the same equations as from d'Alembert's principle. We have a lot of discussions on this in this forum. Just search for "vakonomic dynamics" ;-).