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Discrete Lagrangian

  1. May 24, 2016 #1
    1. The problem statement, all variables and given/known data
    In this exercise, we are given a discrete Lagrangian which looks like this: http://imgur.com/TL0P61r. We have to minimize the discrete S with fixed point [tex]r_i[/tex] and [tex]r_f[/tex] and find the the discrete equations of motions.
    In the second part we should derive a discrete trajectory for U(r) = 0 and U(r) = mgz.
    2. Relevant equations
    [tex] j = t_i + j \Delta t[/tex] ; [tex] \Delta t = (t_f - t_i)/N [/tex]
    [tex]r_j = r(t_i + j \Delta t)[/tex]
    [tex]v_j = \frac{r_j - r_{j-1}}{\Delta t}[/tex]

    j = 0, ....., N
    U(r) is the potential energy
    And the discrete action:
    [tex]S(r_j) = \sum_{j=1}^N \Delta t \left(\frac{1}{2}m\left(\frac{\boldsymbol{r_j} - \boldsymbol{r_{j-1}}}{\Delta t}\right)^2 - \frac{1}{2}\left(U(\boldsymbol{r_j})+ U(\boldsymbol{r_{j-1}})\right)\right)[/tex]
    3. The attempt at a solution
    Since S is a function, I thought I could take the gradient and set that to 0 to derive the discrete equations of motions: [tex]
    0 = \frac{\partial}{\partial r_k} S = \sum_{j=1}^N \Delta t \left(\frac{1}{2}m\frac{\partial}{\partial r_k}\left(\frac{\boldsymbol{r_j} - \boldsymbol{r_{j-1}}}{\Delta t}\right)^2 - \frac{1}{2}\left(\frac{\partial}{\partial r_k} U(\boldsymbol{r_j})+ \frac{\partial}{\partial r_k}U(\boldsymbol{r_{j-1}})\right)\right)
    \\
    \rightarrow \sum_{j=1}^N \Delta t \left(m \left(\frac{\boldsymbol{r_j} - \boldsymbol{r_{j-1}}}{\Delta t}\right)\left(\frac{\delta_{jk}-\delta_{j-1,k}}{\Delta t}\right) -\frac{1}{2} \left(\frac{\partial U(\boldsymbol{r_j})}{\partial r_k}\delta_{jk} + \frac{\partial U(\boldsymbol{r_{j-1}})}{\partial r_k}\delta_{j-1,k}\right)\right)
    \\
    \rightarrow 0 = m \left(\frac{-\boldsymbol{r_{k+1}}-\boldsymbol{r_{k-1}}+2\boldsymbol{r_k}}{\Delta t^2}\right) - \frac{1}{2} \left(\frac{\partial U(r_k)}{\partial r_k} + \frac{\partial U(r_{k+1})}{\partial r_k}\right)
    [/tex]

    The first term seems to be right, it looks like F = ma at least. I am not sure about the second part though because of that factor of 1/2 and I am not sure if the derivatives of the potential energy are correct.
    For part b, I think I would have to insert the potential energy in the discrete equations of motions and integrate two times with respect to the time. But I am not sure how an integral would work here because of those indices.
    Anyways, I would be really grateful for any advice! Thank you
     
  2. jcsd
  3. May 25, 2016 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    I think everything is good so far.
    I don't think the very last term on the right is correct. You might reconsider what the expression ##\sum_{j=1}^N \frac{\partial U(\boldsymbol{r_{j-1}})}{\partial r_k}\delta_{j-1,k}## reduces to.
     
  4. May 26, 2016 #3
    Wouldn't it reduce to [tex] \frac{\partial U(r_{k})}{\partial r_k} [/tex] [tex]j = r_{k+1}[/tex] because only for j=k+1 the kroenecker delta would give me a 1? And when you would have [tex] \frac{\partial U(r_{k+1-1})}{\partial r_k} = \frac{\partial U(r_{k})}{\partial r_k} [/tex] and then I would have two [tex]\frac{\partial U(r_{k})}{\partial r_k}[/tex] and I could cancle the 1/2 out?
     
  5. May 26, 2016 #4

    TSny

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    Homework Helper
    Gold Member

    Yes.
     
  6. May 26, 2016 #5
    Hey thank you very much! Now the discrete equation looks like newtons 2nd law just as was asked :')
     
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