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A Question About Partial Derivatives

  1. Mar 14, 2016 #1
    1. The problem statement, all variables and given/known data
    [tex]v_{i}=\dot{x}_{i}=\dot{x}_{i}\left(q_{1},q_{2},..,q_{n},t\right)[/tex]
    [tex]T \equiv \frac{1}{2}\cdot{\sum}m_{i}v_{i}^{2}[/tex]
    [tex] \frac{\partial T}{\partial\dot{q}_{k}}={\sum}m_{i}v_{i}\frac{\partial v_{i}}{\partial\dot{q}_{k}}={\sum}m_{i}v_{i}\frac{\partial x_{i}}{\partial q_{k}}[/tex]


    2. Relevant equations
    Why is it ok to assume:
    [tex]\frac{\partial v_{i}}{\partial\dot{q}_{k}} = \frac{\partial x_{i}}{\partial q_{k}}[/tex]

    3. The attempt at a solution
    I can say that:
    [tex]\frac{\partial x_{i}}{\partial q_{k}}=\frac{\partial x_{i}}{\partial t}\frac{\partial t}{\partial q_{k}}=\frac{v_{i}}{\dot{q_{i}}}[/tex] but it's not the same as written.

    and the expression [tex]\frac{\partial v_{i}}{\partial\dot{q}_{k}}[/tex] says to differentiate the velocity according to change of the q quardinate in time.
     
    Last edited: Mar 14, 2016
  2. jcsd
  3. Mar 14, 2016 #2

    SammyS

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    Maybe try:
    ##\displaystyle \frac{\partial x_i}{\partial t}=\frac{\partial x_i}{\partial q_k} \frac{\partial q_k}{\partial t} ##​
     
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