How do eagles fly slowly for a long time? Eagles fly slowly for a long time. 
Many other species fly faster and move their wings faster. But eagles keep their wings steady, and move only their tail.


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*How do they move slowly in the air, without falling down?

*Can this eagle flying technique be used in aviation?
 A: Eagles (and most large birds) fly by soaring; it's much more energy efficient than flapping their wings. We do use the technique for our own flights.
The reason eagles and other soaring birds do this rather than flap is that they generally hunt from the air and so spend a lot of time waiting for prey. They don't need to go anywhere specific and don't need to move fast until they spot prey. Until then they want to use as little energy as possible.
A: Every fixed-wing aircraft has a (fairly slow, just above stall) speed at which it minimizes its descent rate while gliding, so that speed is good for loitering. If it can also find an updraft, it can stay up indefinitely without any power.
But if the aircraft needs high speed, that also means high drag, so it needs to expend power. In an aircraft, that means it needs to generate thrust. In a bird, it means flapping wings.
You can try it yourself.
Make a paper airplane with extra-wide wings.
(Put a paper clip near the nose so it flies stably, not in a scallop-pattern.)
Then turn up the back edge so it flies slowly at a high angle of attack.
You should be able to get it to float along lazily while coming down quite slowly.
Then create an updraft by having a fan blow up underneath it, and it should go up.
A: How does they move slowly in air, without falling down?
One possibility is soaring using a ridge lift - typically a situation when the wind is approx. perpendicular to a mountain ridge. The air is lifted at the front side of the ridge and an eagle can soar in the lifting air stream. This can also work without the wind,
Which is a situation of thermal flying. Typically, the ground is heated by the Sun, the air layer just above the ground is heated by conduction and at some moment it forms a kind of bubble that starts to rise. This bubble is usually long, resembling a column and lasts until the warm air is depleted. The situation can repeat (this behavior is called an interval). If a ridge is oriented south, then the Sun can create a thermal wind (intervals) that enables a bird to soar.
Can we use eagle's flying technic for flights? Yes, however, man will never be that good.
soaring: https://www.youtube.com/watch?v=63qJn9HrB7E
thermal flight: https://www.youtube.com/watch?v=KXqTCM0-zXQ
Edit:  Just for completness - there exists also a wave soaring, that is reachable for gliders and maybe for hangliders, probably not possible for birds and paragliders - see pictures here : http://www.ssa.org/GliderLiftSources
