It's been a long time since I used Mathematica, but I always used it for symbolic computations (e.g. "derivative of x^2 = 2x"), especially when I was learning calculus and differential equations. As far as I'm aware R doesn't even natively have a way of storing symbolic equations, let alone solving them (at the least, it's not its main use-case).

But I could be wrong; I used mathematica and R for very different things so I may just be unaware of their similarities.

In principle R language is equally potent than Mathematica's one (functional+self-modifying), so Mathematica clone can be done in R (and in fact there is a built-in function "D" which calculates derivatives of native expressions symbolically).

Of course this won't ever happen because of the paradigm -- R is for data crunching and does this way better than Mathematica.

https://code.google.com/p/ryacas/ to name one. It's not native but yet so aren't a lot of libraries in other languages.

Hasn't been updated since 2007... math doesn't really change that often but that does not look promising...

There's "can do" and then there's "is like."

Hats of if you can do http://blog.wolfram.com/2012/10/23/calculating-the-energy-be... in R or, for that matter, about half of the other examples given in that blog.

Mathematica provides symbolic computation, R does not.