Why did Charlemagne's empire divide?

The Carolingian Empire divided because the tradition at that time was that a kingdom was divided among the sons of the king. The imperial title could be divided or not, so there could be co-emperors, but there were a number of kingdoms within the empire. These kingdoms went to war with one another for various reasons, and after a lot of changes, the empire was divided with the largest pieces being the Kingdom of the Franks and the Holy Roman Empire. Both of these changed their methods of monarchic succession.