calculus on lie groups When looking at the definition of a Lie group it is often stated that is a group which is also a differentiable manifold such that the group operations obey the smooth structure. So, a group on which we can do calculus.
One of the simplest examples is the symmetries of the circle since you can rotate it by any infinitesimal amount. Now, what is the connection of calculus to this?
I see examples mentioning the hydrogen atom etc, but when groups of this nature are mentioned I don't immediately see the connection for why being able to do calculus is important or even really what doing calculus on this type of group means.
Can anyone provide an example?
 A: You're right, there's a disconnect, and it's because Lie theory is an old subject that has undergone reconceptualization.
The classical Lie groups are things like the symmetry groups associated to various physical systems. These usually have special names that look like $SO(2)$, $SL(n,R)$, etc. The way that physicists and applied mathematicians think about these groups are as a family of matrices which form a group under matrix multiplication, and have a distinguished action on a vector space, also by matrix multiplication. 
If you go and look at a modern Lie theory book this perspective is suppressed, as you have observed. Instead, Lie groups are defined to be "groups that are also manifolds", or if you get a really fancy book, they'll say something like "a Lie group is an internal group in the category of smooth manifolds with diffeomorphisms" or something. And then maybe in a couple of exercises they'll get you to put a smooth manifold structure on one of the classical Lie groups. 
(Obviously this is my personal experience, from having tried to learn Lie theory a year ago.) 
Because the problems of interest are so different, I am skeptical that it's actually useful to think of these two versions of Lie theory as being particularly related (even though they are). If you really want to see them fit together in your mind as two legs of the same elephant, I recommend reading Lie groups, Lie algebras, and some of their applications by Gilmore, which is more calculation-and-physics-applications bent but also gives the manifold characterization in full. (However, the way that Gilmore talks about manifolds is also somewhat different from the standard pure math treatment these days! He uses explicit local coordinates all the time and very few commutative diagrams.)
