I'm currently reading "Vector and Tensor Analysis with Applications" by A.I. Borisenko and I.E. Tarapov, and I'm having trouble following a particular mathematical step in where the author projects the moment of inertia tensor onto a set of axes, K. This occurs on page 68 in section 2.4.3. Below is an excerpt:
$$ L = \sum_{j=1}^{n}m_{j}[\mathbf{r}_{j} \times ( \boldsymbol{\omega} \times \mathbf{r}_{j} )] = \sum_{j=1}^{n}m_{j}[\boldsymbol{\omega}(\mathbf{r}_{j}\cdot \mathbf{r}_{j})-\mathbf{r}_{j}(\boldsymbol{\omega} \cdot \mathbf{r}_{j} )] $$
Where L is angular momentum in a system composed of n particles, where the j'th particle has mass $$ m_{j} $$ and $$ \omega $$ is the instantaneous angular velocity of the system.
The they say the project L onto the axes of K to obtain:
$$ L_{i} = \sum_{j=1}^{n}m_{j}(\omega_{i}x_{l}^{(j)}x_{l}^{(j)}-x_{i}^{(j)}\omega_{k}x_{k}^{(j)})\; \; \; (summation \, over \, k \, and \, l) $$
My question then pertains to the indices k and l. Why do we introduce them, What values do they span, and why? ( Assuming R^3 ) I've been stumped on this one, and can't make sense out of it enough do a calculation with it to clarify further. I feel like there's something intrinsic about projection that I'm missing, and it's making the rest of what follows nearly incomprehensible for me unfortunately.
They're just writing a couple dot products in index notation. There's no real meaning or content in doing so, except that some people feel that if you're already using indices elsewhere, everything should be written in terms of indices, even something as simple as a dot product.
I assume you're familiar with the idea that
$$a \cdot b = \sum_i a_i b_i $$
The expression given suppresses the summation symbols and just uses a different letter index than $i$.