I am going to do my degree's dissertation, my advisor suggested I should start reading: "Introduction to symplectic topology" by McDuff and Salamon.
Actually, I find this book very interesting, but I still wonder what's the motivation for symplectic topology. It seems very restrictive and ungeneral so it is hard for me to understand what is the motivation for symplectic topology?
The book talks about how symplectomorphism preserve the solution of hamiltonian system(that if $z$ satisfies $\mathcal{J}_0\dot{z}=\nabla_tH(z)$ and $\psi$ is a symplectomorphism then if $z(t)=\phi(\eta(t))$, it follows that $\mathcal{J}_0\dot{\eta}=\nabla_t(H\psi(z))$)
However, then the book goes into detail into seemly unrelated topics, like masolv index, symplectic balls and non-squeezing, symplectic manifolds and what-not.
So my question is, what is the relation between this and by is symplectic geometry/topology so important and what is it's motivation and its raison-d'être? What do symplectic manifolds have to do with hamiltonian systems?
Thanks to everyone