Integrability of a differential condition

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Main Question or Discussion Point

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.
 

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  • #2
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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].
 
  • #3
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Thanks, Shyan! :D
 

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