Classical mechanics is the study of second-order systems. The obvious geometric formulation is via semi-sprays, ie second-order vectorfields on the tangent bundle. However, that's not particularly useful as there's no natural way to derive a semi-spray from a function (ie potential).
Lagrangian and Hamiltonian mechanics are two solutions to that problem. While these formalisms are traditionally formulated on the tangent and cotangent bundles (ie velocity and momentum phase space), they were further generalized: Lagrangian mechanics led to the jet-bundle formulation of classical field theory, and Hamiltonian mechanics to the Poisson structure.
The symplectic structure is a stripped-down version of the structure of the cotangent bundle - the part that turned out to be necessary for further results, most prominently probably phase space reduction via symmetries. It doesn't feature prominently in undergraduate mechanics lecture (at least not the ones I attended) because when working in canonical coordinates, it takes a particular simple form - basically the minus in Hamilton's equations, where it's used similarly to the metric tensor in relativity, ie to make a contravariant vector field from the covariant differential of the Hamilton function.
Symplectic geometry also plays its role in thermodynamics: As I understand it, the Gibbs-Duhem relation basically tells us that we're dealing with a Lagrangian submanifold of a symplectic space, which is the reason why the thermodynamical potentials are related via Legendre transformations.
This post imported from StackExchange Physics at 2014-04-01 13:17 (UCT), posted by SE-user Christoph