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

[tex]J=\int_a^b L(t,x^\mu,\dot x^\mu) dt[/tex]

From this he derives the Euler-Lagrange equations

[tex]\frac{\partial L}{\partial x^\mu} = \frac{d}{dt}\frac{\partial L}{\partial \dot x^\mu}[/tex]

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

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

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.