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Integrability of a differential condition

  1. Dec 25, 2014 #1
    I'm reading "The variational principles of mechanics", written by C. Lanczos and he said that, if one have the condition [itex]dq_3 = B_1 dq_1 + B_2 dq_2 [/itex] and one want to know if there is a finite relation between the [itex]q_i[/itex], on account the given condition, one must have the condition [tex] \frac{\partial B_1}{\partial q_2} = \frac{\partial B_2}{\partial q_1}[/tex].

    What I don't understand is how the condition on the differentials imply the equality between the derivatives above.
     
  2. jcsd
  3. Dec 25, 2014 #2

    ShayanJ

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    Gold Member

    The point is, under certain conditions(which are always satisfied in physics), we have [itex] \frac{\partial^2 f}{\partial q_1 \partial q_2}=\frac{\partial^2 f}{\partial q_2 \partial q_1} [/itex]. Now for [itex] dq_3 [/itex] to be integrable, there should be some f such that [itex] B_1=\frac{\partial f}{\partial q_1} [/itex] and [itex] B_2=\frac{\partial f}{\partial q_2} [/itex]. But how can we check that? We can use the equation I mentioned:
    [itex] \frac{\partial^2 f}{\partial q_1 \partial q_2}=\frac{\partial^2 f}{\partial q_2 \partial q_1} \Rightarrow \frac{\partial B_2}{\partial q_1}=\frac{\partial B_1}{\partial q_2} [/itex].
     
  4. Dec 26, 2014 #3
    Thanks, Shyan! :D
     
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