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Homework Help: Alternative Lagrange equation

  1. Mar 26, 2017 #1
    1. The problem statement, all variables and given/known data
    Show that for an arbitrary ideal holonomic system (n degrees of freedom)

    \frac{1}{2} \frac{\partial \ddot T}{\partial\ddot q_j} - \frac{3}{2} \frac{\partial T}{\partial q_j} = Q_j

    where T is kinetic energy and qj generalized coordinates.

    2. Relevant equations
    Lagrange's equation
    \frac{d}{dt} \frac{\partial T}{\partial\dot q_j} - \frac{\partial T}{\partial q_j} = Q_j

    3. The attempt at a solution
    We know that [tex] T(q_1,...q_n,\dot q_1,...,\dot q_n,t) [/tex]
    The idea is to express [tex] \dot T [/tex] and [tex] \ddot T [/tex] and then plug it into initial equation in order to obtain equivalence with Lagrange's equation.

    So we write

    [tex] \frac {dT}{dt}=\dot T=\frac{\partial T}{\partial \dot q_j} \ddot q_j + \frac{\partial T}{\partial q_j} \dot q_j + \frac{\partial T}{\partial t} [/tex]

    So I figure that I should express [tex] \ddot T [/tex]
    in the same manner, but I'm stuck at doing the chain rule for the first 2 terms.
  2. jcsd
  3. Mar 28, 2017 #2
    As you say, T = T(q1,q2,...,q1d,q2d,...) so the partial of T with respect to qidd is necessarily zero.

    I really don't believe the result that you are trying to prove. I can say with confidence that in almost 60 years of doing dynamics, I have never seen this expression anywhere, and it looks entirely bogus to me. I look forward to whatever light others may bring to this issue. Maybe there is something new under the sun after all!
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