We next examine the total energy, E = K + V.

We examine the time derivative of K, and on line four we express the sum over all n particles as being over the half-grid i > k, adding a second function to take care of the top half of the grip where i < k.

We now consider the time derivative of the following function, for reasons that will become clear immediately.

Substituting this into our relation for the time derivative of K,

This shows us that the net change in the total energy E is zero.

The total energy is thus a conserved quantity.