Euler-Lagrange Equations with constraint depend on 2nd derivative?

[birulami]

I am reading the book of Neuenschwander about Noether's Theorem. He explains the Euler-Lagrange equations by starting with

$$J=\int_a^b L(t,x^\mu,\dot x^\mu) dt$$

From this he derives the Euler-Lagrange equations

$$\frac{\partial L}{\partial x^\mu} = \frac{d}{dt}\frac{\partial L}{\partial \dot x^\mu}$$

which is all well comprehensible. Then he describes how to introduce constraints of the form $h(t,x^\mu)=0$ to form a lagrangian with constraint $L_c = L+\lambda h$.

My question: The constraint does not depend on $\dot x^\mu$. Is this just to simplify the derivation in this case or would a constraint $$h(t,\dot x^\mu)=0$$ invalidate the Euler-Lagrange equations? If the latter is true, how would we introduce constraints on the $\dot x^\mu$?

 

[vanhees71]

This is called a holonomic constraint (but it's rheonomic because it's explicitly time dependent). A constraint is anholonomic if it's a non-integrable equation of both the generalized coordinates and velocities.