Trying to Sleep is Exhausting

The biggest problem I have when I wake up too early is to try and do what my body wants to do.
I can see I have a good hour and a half before I need to actually get up. I am sleepy. I want to sleep. I have no reason not to sleep.
Except I have to work at trying to sleep. And that is tiring. I have to force my brain to concentrate on sleeping. It is not easy to do, because you almost have to do the opposite of concentration. Probably some Zen hippy could describe it better than me. I usually give myself a mental problem to work on that will hopefully distract my thought process; My current favourite is to list the countries of the world, in order, by where they are on the coastline.
In any case, trying to stay in that mode of trying to sleep becomes too much after awhile. It is easier to just wake up and spend some time reading.
This does not solve the fundamental problem that everything wants to go to sleep.