1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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)

    [tex]
    \frac{1}{2} \frac{\partial \ddot T}{\partial\ddot q_j} - \frac{3}{2} \frac{\partial T}{\partial q_j} = Q_j
    [/tex]

    where T is kinetic energy and qj generalized coordinates.



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



    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!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Alternative Lagrange equation
  1. Lagrange equations: (Replies: 1)

Loading...