Symplectic geometry and Hamiltonian mechanics

A symplectic structure is some at first strange sounding extra structure you can put on a manifold. By a miracle, a lot of the shapes arising in geometric representation theory have this extra structure, and this structure can be exploited to prove very useful things. In this blogpost, my friend and I motivate the definition of symplectic structures from mechanics, and say a little at the end about where they appear in pure math (by the way, the two of us are mathematicians, and we only learned physics to better appreciate the symplectic structures which were showing up in our work!).