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

by birulami
Tags: constraint, depend, derivative, equations, eulerlagrange
 P: 148 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 $\dot x^\mu$?
 Sci Advisor Thanks P: 2,149 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.

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