I am interested in how Lagrangian and Hamiltonian mechanics and then symplectic geometry was developed starting from Newtonian mechanics. We can start by assuming that Newtonian mechanics tells us that motion of a particle under an applied force is a second order differential equation in the configuration space. I know that first Lagrangian mechanics was developed and then Hamiltonian mechanics.
I guess in formulating Lagrangian and Hamiltonian mechanics, they were motivated by optics and variational ideas within. Infact in Arnold's book he gives a sketch of the idea showing that every concept in Huygen's Optics has an analogy in Hamiltonian mechanics. Yet I do not understand the physical or mathematical motivation why people wanted to formulate mechanics like optics?
After this point I guess it is not so hard to pass to symplectic geometry and cotangent bundles. Indeed one can see that a set of canonical transformations for Hamiltonian systems on $\mathbb{R}^n$ is the one which is the transition functions for the space of differential 1-forms (i.e cotangent space). From here I think it is evident why one should formulate Hamiltonian mechanics on the cotangent space.