Thanks everybody for the input.
I can see two explanations listed : mental or physical.
I would rather opt for the mental explanation because this never happens when I'm sitting around all day long at home.
I fly for a very large operation with routes all over Europe (just on my type), but for personal, quality of life, reasons I signed up for an option with very few routes that I end up knowing by heart. It's a bit boring to fly always the same route. Even going everywhere in Europe it would just mainly be sit around waiting (at least for the flight).
I never regret my choice of working in the airline because it's still my dream job, the quality of life and the technical aspects suit me very well, but I just don't understand why the boredom has a physical expression.
On a typical day, I will try to be on my phone as much as possible between flights as I feel it helps me unwind, but maybe it's a wrong impression and I should try stopping.
Or, maybe the other explanation is a bit of mental pressure (even if bored, one has to be able to react if anything goes sideways), and/or the pressure (my airplane goes up to 5000 to 8000ft cabin altitude depending on the flight). The real boredom comes from flying 4-leg days. On the last day of a 4-5 day shift, sometimes I don't even remember if I did the departure briefing.
About the physical side, for context I am 28, rather slim but I don't work out or do sport really regularly. I still live a somewhat active lifestyle. For example just this week I spent two entire days out, walking, shopping, sightseeing.. I ran with my sister's dog. I often walk instead of taking the tube, even up to 2-4km depending on the time on my hands. I have a two story apartment and often forget things downstairs or upstairs... My girlfriends keep me very busy as well.
I try to avoid eating bull**** all the time (which is difficult because our colleagues bring us exactly that) but I'm not very keen on it so still end up for example ordering desert at a restaurant and I don't always turn it down at private dinners.
I put on some weight last year (1-2kg) when I decided to eat more and work out at the same time (to gain mass), part of it should be muscle and part of it fat.. I looked very slim and wanted to diminish that (my BMI is between 20 and 21).
My favourite drink at work is tea, I try to avoid hot water as I was told they don't clean the hot water tank, but still take some hot tea sometimes. The rest of the time, I put one tea bag and keep it in my glass with cold mineral water for 4-5 hours, only before 4pm. After that time I will tend to drink cold herbal tea (no caffeine).
I don't like coffee. I will sometimes ask for a coke which has the caffeine, but try to avoid it for sugar reasons, just like apple juice. Tomato juice is a drink of choice because no sugar.
Regarding sports, I could do more, I admit it could have some benefits. Last time I tried to motivate myself to do sports, it lasted for 3-4 months only, then I stopped for different reasons. I usually jog around my home, I have several loops between 1-3km. I bought some very good shoes for skydiving but I don't really intend on doing many more jumps (I did seven, until the first solo flight) because of it taking a lot of time, and the activity not being very rewarding for a beginner. I could use these shoes for jogging, which would help me greatly reduce the risk of any pain (which would come automatically if I jogged to much with bad shoes).
I also wanted to work out but I durably hurt my elbow and stopped immediately (I have a little pain in the elbow when doing pull ups, which I think is preventing me from lifting weights...) I should maybe go see a sports doctor to ask why the pain isn't going away and how to lift weight.
From a health point of view, is it better to work out or jog ?
From a fitness (esthetic) point of view, I think the best is to work out 100% first to gain mass then jog to eliminate the fat, but it's very demanding to do so at a good level. I also bought 4000€ of measure clothing, so it would be a bit stupid to put on a significant amount of weight (be it muscle or fat) and having to throw away the brand new suits and shirts...
But it seems to take a long term dedication that I will never have.
Finally, regarding my sleep patterns, I am a late owl. So when on my own, I will sleep typically from 3 to 12 (9 hours), and maybe from midnight to twelve if very tired from an early wakeup. When I have an early wakeup from home, I have trouble sleeping before midnight, so I typically end up sleeping 4-6 hours before my first day. If there is a second early, I will have much less trouble sleeping early and try to have at least 6 hours if there is no third early, 7 hours if there is. And 8 hours if there is a fourth early (even if it means going to sleep at 8pm without eating)
I don't work for a very demanding carrier, I work airline but no more than 13 days per month. I can't say I'm overbooked. If I worked 18 days (the maximum) I would make 30% more as well which would be very motivating.