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


by birulami
Tags: constraint, depend, derivative, equations, eulerlagrange
birulami
birulami is offline
#1
Nov4-12, 09:38 AM
P: 148
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]?
Phys.Org News Partner Physics news on Phys.org
Sensitive detection method may help impede illicit nuclear trafficking
CERN: World-record current in a superconductor
Beam on target: CEBAF accelerator achieves 12 GeV commissioning milestone
vanhees71
vanhees71 is offline
#2
Nov4-12, 10:44 AM
Sci Advisor
Thanks
P: 2,131
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.


Register to reply

Related Discussions
Euler-Lagrange equations Calculus & Beyond Homework 3
Euler-Lagrange Equations Advanced Physics Homework 3
Euler-Lagrange 2nd derivative Advanced Physics Homework 2
Euler-Lagrange equations in QFT?? Quantum Physics 11
Euler-Lagrange equations in QFT?? General Physics 2