In the book it does mention that Gandalf has a staff and thus summons the eagle. But this wouldn't make sense in the movie because the last we see of Gandalf before this scene was him spinning up to the rooftop without his staff. So for me it makes sense because why would they give back his staff, especially since PJ decides to make the staff a symbol of power. (This we see in TTT when Gandalf uses it to get rid of Saruman influencing Theoden)
I think another reason that there is no staff is because PJ said in one of the commentaries that he didn't like to have the wizards shooting lighteningbolts from their staffs and in the book this happens.
Quote:
|
The figure(Gandalf) lifted his arms and a light flashed from the staff that he wielded.
|
Found in the chapter In the house of Tom Bombadil. for my version it is page 167.
I don't really care about this change because I think the alternative is pretty cool. But I must admit that there is a lot of staff losing and finding going on throughout these movies. You never know when Gandalf will lose his staff. the only one who seems to be consistent is Saruman.