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Lagrange equation: when exactly does it apply?

  1. Jan 27, 2015 #1
    Hi! Does the Lagrange equation ONLY apply when the constraints are holonomic? What about the constraining forces acting on the system (i.e. normal force, or other perpendicular forces), do they make a system holonomic?

    What about the Lagrange equation with the general force on the right hand side. I read in Goldstein that it can be, for instance, a non-conservative frictional force. Why? Where did that come from?
     
  2. jcsd
  3. Jan 28, 2015 #2
    BTW, I am talking about the Euler-Lagrange equation. This one, $$ \sum_j \frac{\partial L }{\partial q_j} - \frac{d}{d t} \frac{\partial L }{\partial \dot{q_j}} = 0$$ in case there was any confusion.

    But what is up with the modified equation, ##\frac{\partial L }{\partial q_j} - \frac{d}{d t} \frac{\partial L }{\partial \dot{q_j}} = Q_j## ? When does this apply to a system, and for which generalized forces ##Q_j##s? It was not derived in Goldstein's book, just given.
     
  4. Feb 1, 2015 #3
    Another question, if somebody wants to answer: does ##\frac{\partial T}{\partial q_j}##, where ##T## is the kinetic energy of the system, always equal zero? Or do there exist situations where the kinetic energy has an explicit dependence on position?

    It might seem like a strange question because kinetic energy is defined using total velocity, but I ask because one form of Lagrange's equation is ##\frac{d}{dt} \frac{\partial T}{\partial \dot{q_j}} - \frac{\partial T}{\partial q_j} = Q_j##.
     
  5. Feb 1, 2015 #4
    It certainly can, in spherical coordinates (or polar) you have position dependence in the kinetic term.
     
  6. Feb 1, 2015 #5
    Check http://physics.clarku.edu/courses/201/sreading/AJP73_March2005_265-272.pdf [Broken] paper out. Does that help answer your questions?
     
    Last edited by a moderator: May 7, 2017
  7. Feb 4, 2015 #6
    I hate to answer your question this way, but if you reread Goldstein chapter 1 and 2 enough, you will answer your questions. This was true for me.
     
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