# Requirement of Holonomic Constraints for Deriving Lagrange Equations

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.

• Is this from some reference? Which page? Is it online? Sep 1 at 18:02
• @Qmechanic it is from the lecture notes published by my professor... Sep 1 at 18:23
• Judging by your follow-up questions: I get the impression that what you are looking for is a conversational forum. However, the stackexchange environment is not a conversational environment. The stackexchange concept is that you take the time to formulate a focused question, such as to allow a focused answer. The concept is that if there is a new question then that new question should be submitted as a separate question. If you prefer a conversational forum: an example of that is: physicsforums Sep 1 at 19:19