I'm trying to justify the full power of symplectic mechanics yet I keep finding examples of physical systems which are only trivial examples of symplectic mnaifolds, cotangent bundles. What physical systems have phases spaces with are not cotangent bundles?
Since Lagrangian mechanics takes places explicitly on a tangent bundle, I am expected that all examples of non-trivial phases spaces will fail to exhibit the usual duality between Lagrangian and Hamiltonian functions (via Legendre transformation). Is this correct? If so, it would imply that we cannot always use the Lagrangian formalism.