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In proving that the Lagrange equations hold in any coordinate system

  1. Oct 10, 2013 #1
    I was checking the proof of this, when things came vague at one point.

    It goes as follows, how to prove that Lagrange's equations hold in any coordinate system?


    Let q[itex]_{a}[/itex] = q[itex]_{a}[/itex](x[itex]_{1}[/itex],..., x[itex]_{3N}[/itex], t)

    here the possibility of using a coordinate system that changes with time is included.

    By chain rule: [itex]\dot{q_{a}}[/itex]= [itex]\partial[/itex]q[itex]_{a}[/itex]/[itex]\partial[/itex]x[itex]^{A}[/itex]([itex]\dot{x^{A}}[/itex]) + [itex]\partial[/itex]q[itex]_{a}[/itex]/[itex]\partial[/itex]t

    To be in a good coordinate system, we should be able to invert this relationship, call this equation (1):

    [itex]\dot{x_{a}}[/itex]= [itex]\partial[/itex]x[itex]_{a}[/itex]/[itex]\partial[/itex]q[itex]^{A}[/itex]([itex]\dot{q^{A}}[/itex]) + [itex]\partial[/itex]x[itex]_{a}[/itex]/[itex]\partial[/itex]t

    Then the book said, now we can examine L(x[itex]^{A}[/itex], [itex]\dot{x^{A}}[/itex]) when we substitute in x[itex]^{A}[/itex](q[itex]_{a}[/itex], t)

    Using (1), we have

    [itex]\partial[/itex]L/[itex]\partial[/itex]q[itex]_{a}[/itex] =

    ([itex]\partial[/itex]L/[itex]\partial[/itex]x[itex]^{A}[/itex])([itex]\partial[/itex]x[itex]^{A}[/itex]/[itex]\partial[/itex]q[itex]_{a}[/itex]) + [itex]\partial[/itex]L/[itex]\partial[/itex][itex]\dot{x^{A}}[/itex]([itex]\partial[/itex][itex]^{2}[/itex]x[itex]^{A}[/itex]/[itex]\partial[/itex]q[itex]_{a}[/itex]q[itex]_{b}[/itex]*[itex]\dot{q_{b}}[/itex] + [itex]\partial[/itex][itex]^{2}[/itex]x[itex]^{A}[/itex]/[itex]\partial[/itex]q[itex]_{t}[/itex]q[itex]_{a}[/itex])

    My question is I don't understand where the q[itex]_{b}[/itex] come from??

    Thank you for your time!
  2. jcsd
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