How to derive a symmetric tensor?

[Cathr]

Let $Q_ik$ be a symetric tensor, so that $Q_ik= \frac{m}{2} \dot x_i \dot x_j + \frac{k}{2} x_i x_j$ (here k is also a sub, couldn't do it better with LaTeX).
How do we derive such a tensor, with respect to time? And what could such a tensor mean in a physical sense? It really looks like the tensor for the total energy, except that I don't understand the need for adding indices to create a symmetrical form.

 

[Cathr]

I apologize, the formula is actually $Q_{ij}= \frac{m}{2} \dot x_i \dot x_j + \frac{k}{2} x_i x_j$ .