This morning I read a post on Open Culture, by Sheerly Avni , about the Raptor Resource Project in Iowa, (USA) which included this video. It's a 10 minute collage of 24-hours of first egg pipping and hatch. You can see the mother eagle carefully turning the eggs, and at the very end, see the first chick.
You can watch the third egg hatch on this video - There are many more videos on their YouTube channel.
Click here to go to the Resource Project's home page. From there, you can click through to their webcam "nest cams" page, where you can watch what's happening in about a dozen nests. (Remember that there's a time difference between Iowa and Switzerland.) You can watch the nests of Eagles, Ospreys, Falcons, Kestrals and Owls.
Here's what's going on in the eagles' nest now - this is the live webcam stream (you'll have to watch a very short ad first :-( )
Live video by Ustream
I learned from the Project's blog that eagles incubate for approximately 34 - 37 days, that nest-building seems to be part of the courtship ritual of the male and female bird, and that the eggs are laid in a separate "cup" of soft material, built inside the regular nest, which helps keep the eggs warm.
On Learner.org's Journey North webpage, you can read a "list of duties" for an Eagle couple when raising a family, and links to pages with more information about which duties are for the female, and which for the male.
Now that you're really interested, you'll find more links, information, and webcams watching Eagles and Ospreys in Maine at this webpage.