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I am having difficulties on the following question:

The equations of motion for a system of n particles are: $$m \ddot{x}_i = - \dfrac{\partial U(x_1,...,x_n)}{\partial x_i}$$ $$\ddot{x}_i = \dfrac{d^2x_i}{dt^2}$$ where $m$ is mass and $x_i$ is the coordinate of particle i. $U(x_1,...,x_n)$ is the potential energy of the system.
Given $$U(x_1,...,x_n) = \sum^{n-1}_{i=1} \frac{k}{2} (x_{i+1} - x_i)^2 + \sum^{n}_{i,j =1} \frac{\lambda}{4}(x_i - x_j)^4$$ Find the equations of motion for particle i using the Kronecker delta.

Could someone please help me with this as I am not even sure how to start this question.

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    $\begingroup$ Just do the partial derivatives and plug them into the first equation. $\endgroup$
    – Sebastian Riese
    Commented May 11, 2015 at 12:04
  • $\begingroup$ But then how do I use the Kronecker delta? $\endgroup$
    – user
    Commented May 11, 2015 at 12:07
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    $\begingroup$ $\frac{\partial x_i}{\partial x_j} = \delta_{ij}$ $\endgroup$
    – AV23
    Commented May 11, 2015 at 12:10
  • $\begingroup$ Would someone mind showing me a worked answer to this question, so I know what answer I have to aim for. $\endgroup$
    – user
    Commented May 11, 2015 at 12:32
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    $\begingroup$ All you need is in the comments and the task. Try to work with this and wrap your head around these concepts, if more specific questions arise, ask them. Do not expect us to do your homework (especially as this can be found in any book on mathematical methods). $\endgroup$
    – Sebastian Riese
    Commented May 11, 2015 at 12:40

1 Answer 1

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You apply the chain rule basically, so I'll try help you with the first sum and leave the second for you.

https://i.sstatic.net/XNkzF.jpg your final comment is correct but the first sum should be over a dummy variable j, and the second should summed over dummy variables j,k.

FrolovOut

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    $\begingroup$ This site has MathJax enabled so that you don't have to copy notes to imgur. $\endgroup$
    – Kyle Kanos
    Commented May 11, 2015 at 18:33
  • $\begingroup$ Now I understand, you use them just like Levi - Civita tensors. Thanks FrolovOut $\endgroup$
    – user
    Commented May 11, 2015 at 20:35

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