How Do You Derive a Tensor Matrix from a Potential Energy Function?

[KleZMeR]

Homework Statement

I am looking at Goldstein, Classical Mechanics. I am on page 254, and trying to reference page 190 for my confusion.

I don't understand how they got from equation 6.49 to 6.50, potential energy function to tensor matrix. I really want to know how to calculate a tensor from a function of this type (any type), but somehow the Goldstein text is not clear to me.

Homework Equations

$V = \frac{k}{2} (\eta_{1}^2+2\eta_{2}^2 +\eta_{3}^2-2\eta_{1}\eta_{2}-2\eta_{2}\eta_{3})$

\begin{array}{ccc} k & -k & 0 \\ -k & 2k & -k \\ 0 & -k & k \end{array}

The Attempt at a Solution

The solution is given. I think this is done by means of equation 5.14, but again, I am not too clear on this.

 

[ShayanJ]

$\mathcal V=\frac 1 2 \vec \eta^T V \vec\eta=\frac 1 2 (\eta_1 \ \ \ \eta_2 \ \ \ \eta_3) \left(\begin{array}{ccc} k \ \ \ \ -k \ \ \ \ 0 \\ -k \ \ \ \ 2k \ \ \ \ -k \\ 0 \ \ \ \ -k \ \ \ \ k \end{array} \right)\ \left( \begin{array}{c} \eta_1 \\ \eta_2 \\ \eta_3 \end{array} \right)$

 

[KleZMeR]

Thanks Shyan, but how do I decompose the potential function to arrive at this? Or, rather, how do I represent my function in Einstein's summation notation? I believe from what you are showing that my potential function itself can be written as a matrix and be decomposed by two multiplications using $\eta^T , \eta$?

 

[ShayanJ]

The potential function is a scalar so you can't write it as a matrix. And the thing I wrote, that's the simplest way of getting a scalar from a vector and a tensor. So people consider this and define the potential tensor which may be useful in some ways.
In component notation and using Einstein summation convention, its written as:
$\mathcal V=\frac 1 2 \eta_i V^i_j\eta^j$
But the potential function itself, is just $\mathcal V$ in component notation because its a scalar and has only one component!

 

[KleZMeR]

Thank you! That did help a LOT. Somehow I keep resorting back to the Goldstein book because it is the same notation we use in lecture and tests, but it does lack some wording in my opinion. I guess the explanation you gave would be better found in a math-methods book.