 #1
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Hi there,
I'm writing a research paper and have hit a roadblock, (wikipedia did not help) and one of my collaborators sent me an email that I do not understand.
I am attempting to find when the following functional is stationary:
[tex]
T = \int\limits_{\lambda_{1}}^{\lambda_{2}}
\sqrt{\sum_{I=1}^{n} \sum_{i=1}^{d}
\left(\frac{d}{d\lambda}\left(\sum_{j=1}^{d} s(\lambda)R_{j}^{i}(\lambda)(q_{I}^{j}( \lambda) + a^{j}(\lambda))\right)\right)^2} d\lambda
[/tex]
This is not necessarily important for anyone who can help answer this question to understand but it helps gives context:
T is the length of all possible curves in configuration space between two particular systems of particles. I represents the number of particles, while i and j are two indices defining the dimension. s is a function describing scale, R is a function describing rotation, q describes the system of particles, and a describes translation. So, the similarity group has 3 elements: the subgroups T^{d} and SO(d) plus the element k, defined by a, R, and s, respectively.
Now, math wise I'm attempting to use the eulerlagrange equation to solve for constraints on s, R, q, and a. Lambda is just a parameter defining where on the trial curve the system of particles is. So, given the eulerlagrange:
[tex]
\frac{\partial{f}}{\partial{x}}  \frac{d}{d \lambda}
\left(\frac{\partial{f}}{\partial{\dot{x}}}\right) = 0
[/tex]
Where f is the integrand of T and x is the variable defining g(\lambda) such that if g(\lambda) is taken to be s(\lambda) then x would be the scaling constant k. In order to avoid a long and drawn out derivation of the conditions on T, the following substitution must be observed, where [tex]\dot{g}(\lambda) = \frac{d}{d \lambda}(g(\lambda))[/tex]:
[tex]
\varepsilon_{I}^{i} = \frac{d}{d \lambda} \left( \sum\limits_{j=1}^{d} s(\lambda) R_{j}^{i}(\lambda)(q_{I}^{j}(\lambda) + a^{j}(\lambda)) \right) [/tex]
Now, I essentially just need the rules for using partial differentiation with respect to Einstein notation.
I got to the point where using regular derivative rules I need to compute:
[tex]
\frac{\partial}{\partial{a^{j}}} \left(\sum\limits_{j} (\dot{s} R_{j}^{i} q_{I}^{j} + s \dot{R}_{j}^{i} q_{I}^{j} + s {R}_{j}^{i} \dot{q}_{I}^{j} + \dot{s} R_{j}^{i} a^{j} + s \dot{R}_{j}^{i} a^{j} + s R_{j}^{i} \dot{a}^{j}) \right)
[/tex]
Now, I am unsure what to do. My collaborator said this:

"You are summing over $j$ in the bracketed expression but, in (6), you want to take derivatives with respect to $a_j$. These two $j$ are not the same!! Call one of them $k$ (say the one that is summed over). Then you get,
[tex]
\frac {\partial a_j} {\partial a_k} = \delta^j_k
[/tex]
where [tex]\delta^j_k[/tex] is 1 if $j=k$ and zero otherwise. This will give a vector with index $j$ when you perform the sum over $k$, which will now be trivial. You must have a vector with these components because you are differentiating with respect to a vector.
In the case of [tex]q_I^i[/tex], you will get two delta functions [tex]\delta^I_K[/tex] and [tex]\delta^i_k[/tex]. The sums over $I$ and $k$ can then be trivially taken. You will see that this leads to a very different result."

Can anyone help me interpret this or help me out? If you have time I would love to just send you the paper on what I've done so far, it's about 17 pages and 4 of them are incorrect because they lead off of a mistake I made right at this point. If you need any clarification whatsoever I can email you the paper in a .pdf or .tex if you post your email here or send it to me in a pm.
If anyone can help me that would be so unbelievably helpful, I've asked a professor at my university and he was not familiar with Einstein notation so he could not help.
Thanks,
Sam Reid
I'm writing a research paper and have hit a roadblock, (wikipedia did not help) and one of my collaborators sent me an email that I do not understand.
I am attempting to find when the following functional is stationary:
[tex]
T = \int\limits_{\lambda_{1}}^{\lambda_{2}}
\sqrt{\sum_{I=1}^{n} \sum_{i=1}^{d}
\left(\frac{d}{d\lambda}\left(\sum_{j=1}^{d} s(\lambda)R_{j}^{i}(\lambda)(q_{I}^{j}( \lambda) + a^{j}(\lambda))\right)\right)^2} d\lambda
[/tex]
This is not necessarily important for anyone who can help answer this question to understand but it helps gives context:
T is the length of all possible curves in configuration space between two particular systems of particles. I represents the number of particles, while i and j are two indices defining the dimension. s is a function describing scale, R is a function describing rotation, q describes the system of particles, and a describes translation. So, the similarity group has 3 elements: the subgroups T^{d} and SO(d) plus the element k, defined by a, R, and s, respectively.
Now, math wise I'm attempting to use the eulerlagrange equation to solve for constraints on s, R, q, and a. Lambda is just a parameter defining where on the trial curve the system of particles is. So, given the eulerlagrange:
[tex]
\frac{\partial{f}}{\partial{x}}  \frac{d}{d \lambda}
\left(\frac{\partial{f}}{\partial{\dot{x}}}\right) = 0
[/tex]
Where f is the integrand of T and x is the variable defining g(\lambda) such that if g(\lambda) is taken to be s(\lambda) then x would be the scaling constant k. In order to avoid a long and drawn out derivation of the conditions on T, the following substitution must be observed, where [tex]\dot{g}(\lambda) = \frac{d}{d \lambda}(g(\lambda))[/tex]:
[tex]
\varepsilon_{I}^{i} = \frac{d}{d \lambda} \left( \sum\limits_{j=1}^{d} s(\lambda) R_{j}^{i}(\lambda)(q_{I}^{j}(\lambda) + a^{j}(\lambda)) \right) [/tex]
Now, I essentially just need the rules for using partial differentiation with respect to Einstein notation.
I got to the point where using regular derivative rules I need to compute:
[tex]
\frac{\partial}{\partial{a^{j}}} \left(\sum\limits_{j} (\dot{s} R_{j}^{i} q_{I}^{j} + s \dot{R}_{j}^{i} q_{I}^{j} + s {R}_{j}^{i} \dot{q}_{I}^{j} + \dot{s} R_{j}^{i} a^{j} + s \dot{R}_{j}^{i} a^{j} + s R_{j}^{i} \dot{a}^{j}) \right)
[/tex]
Now, I am unsure what to do. My collaborator said this:

"You are summing over $j$ in the bracketed expression but, in (6), you want to take derivatives with respect to $a_j$. These two $j$ are not the same!! Call one of them $k$ (say the one that is summed over). Then you get,
[tex]
\frac {\partial a_j} {\partial a_k} = \delta^j_k
[/tex]
where [tex]\delta^j_k[/tex] is 1 if $j=k$ and zero otherwise. This will give a vector with index $j$ when you perform the sum over $k$, which will now be trivial. You must have a vector with these components because you are differentiating with respect to a vector.
In the case of [tex]q_I^i[/tex], you will get two delta functions [tex]\delta^I_K[/tex] and [tex]\delta^i_k[/tex]. The sums over $I$ and $k$ can then be trivially taken. You will see that this leads to a very different result."

Can anyone help me interpret this or help me out? If you have time I would love to just send you the paper on what I've done so far, it's about 17 pages and 4 of them are incorrect because they lead off of a mistake I made right at this point. If you need any clarification whatsoever I can email you the paper in a .pdf or .tex if you post your email here or send it to me in a pm.
If anyone can help me that would be so unbelievably helpful, I've asked a professor at my university and he was not familiar with Einstein notation so he could not help.
Thanks,
Sam Reid