EulerLagrange Equations with constraint depend on 2nd derivative?by birulami Tags: constraint, depend, derivative, equations, eulerlagrange 

#1
Nov412, 09:38 AM

P: 148

I am reading the book of Neuenschwander about Noether's Theorem. He explains the EulerLagrange equations by starting with
[tex]J=\int_a^b L(t,x^\mu,\dot x^\mu) dt[/tex] From this he derives the EulerLagrange 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 EulerLagrange equations? If the latter is true, how would we introduce constraints on the [itex]\dot x^\mu[/itex]? 



#2
Nov412, 10:44 AM

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 nonintegrable equation of both the generalized coordinates and velocities.



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