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I Generalised momentum

  1. Nov 22, 2016 #1
    I have two books that define generalised momentum differently. Either
    ##p_i = \frac{\partial L}{\partial \dot q_i}##
    or
    ##p_i = \frac{\partial T}{\partial \dot q_i}##.
    Is this since defining generalised momentum only make sense when the potential energy is independent of a coordinate ##q## and hence the above definitions are equal? Or is one of these more general than the other?
     
    Last edited: Nov 22, 2016
  2. jcsd
  3. Nov 22, 2016 #2

    hilbert2

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    The differentiations should be done with respect to the time derivative of ##q##, not ##q## itself. Is ##T## the kinetic energy? In most simple mechanical systems the potential energy is not dependent on velocity.
     
  4. Nov 22, 2016 #3
    Thanks, missed the dots. Added them now. Yes ##T## is the kinetic energy. Is it possible that the potential energy ##V## depends on ##\dot q##? Is one of these definitions correct in that case?
     
  5. Nov 22, 2016 #4

    hilbert2

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    If there are magnetic fields and electric charges in the system, the potential energy depends on velocities. Then you can't use the definition where you differentiate ##T##.
     
  6. Nov 22, 2016 #5
    Cheers! That was a good example!
     
  7. Nov 23, 2016 #6

    vanhees71

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    The momentum canonically conjugated to the generalized coordinate ##q^i## is defined by
    $$p_i=\frac{\partial L}{\partial \dot{q}^i},$$
    and nothing else!
     
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