Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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?

    Answer:

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

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: In proving that the Lagrange equations hold in any coordinate system
  1. Lagrange's equations (Replies: 7)

  2. Lagrange's Equations (Replies: 7)

Loading...