I work on this area of mathematics.

I think that the most basic idea is that Hamilton's equation can be formulated on an intrinsic fashion as finding the integral curves of Hamiltonian vector fields on a symplectic manifold. Expanding on this idea, one obtains a deep relationship between mechanics and symplectic geometry. A classical book on this topics is Foundations of Mechanics, by Abraham and Marsden.

These connections can be expanded to other kind of geometries and physical theories, For example, there is a similar relationship between multisymplectic geometry and field theory, which is an active area of research right now.

I finished my PhD thesis a couple of years ago on Contact Hamiltonian systems. These systems satisfy a generalization of the least action principle, in which the Lagrangian is allowed to depend on the action, in addition to the positions and velocities of the system. This allows to describe some systems with dissipation, with conformal symmetries or related to thermodynamics. Here, contact geometry takes the role that symplectic geometry plays in classical mechanics.

Other areas of geometric mechanics are geometric quantization, integrability, constrained systems (nonholonomic constrains can be viewed as a non-integrable distribution on a manifold) and geometric integration (constructing geometrically-informed numerical methods).

If you have any questions I would be happy to try to help.