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Lagrangian equations - ring which is sliding along a wire

  1. Oct 28, 2017 #1
    1. The problem statement, all variables and given/known data
    Hello. I have this problem:
    I have a ring which is sliding along a wire in the shape of a spiral because of gravity.
    Spiral (helix) is given as the intersection of two surfaces: x = a*cos(kz), y = a*sin(kz). The gravity field has the z axis direction.
    I have to find motion equations and find the wire reaction as a function of time.

    2. Relevant equations

    I have to solve it with this equation:
    equation.png

    3. The attempt at a solution
    This is the first time, when I solve a example with Lagrangian equations, so I am not sure what to do.
    I created this equations:
    solution.jpg

    I know, it is not the end. I have to find λ, motion equations and the wire reaction. But firstly, please, tell me, if I am right or where is mistake and why. Then I will continue.

    Thank you very much.
     
  2. jcsd
  3. Oct 28, 2017 #2

    BvU

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    Hi,
    can you tell me what
    your Lagrangian is ?
    your generalized coordinates are ?
    all variables and given/known data are, i.e. what your relevant equation symbols represent ?
    Then we can continue. :wink:
     
  4. Oct 28, 2017 #3
    Do I really need the Lagrangian? When I read the study text, there was Lagrangian only in Lagrange's equations of the second kind, not in Lagrange's equations of the first kind. And I wrote there Lagrange's equations of the first kind, because in my task was, that I have to use this.

    My study text is in different language, but the type of equations is same like this on 3rd page: https://www.physast.uga.edu/ag/uplo...8011 - HMWK 04 - Lagrange Eqs of 1st Kind.pdf

    Sorry, I am confused, it is new for me.
     
  5. Oct 29, 2017 #4

    BvU

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    Well, I'm grateful because I had to find out about these first kind Lagrange equations (look suspiciously like Newton's) . At least that revealed what ##\Phi## stands for: the constraint equations. And they are supposed to come in the form ##\Phi_\alpha = 0##. Can you make that explicit for me, so we can check your ##\partial\Phi_\alpha\over \partial x_j## ?

    Furthermore: I see only one ##\lambda##. How many constraints do you have ?

    Another tack (problem solving skills): approaching this from the other end: what kind of motion do you expect ? Do your intermediate equations fit that ?
     
    Last edited: Oct 29, 2017
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