Geometry of Hamiltonian Systems
A Hamiltonian system may be understood as a fiber bundle E over time R, with the fibers Et, t ∈ R, being the position space. The Lagrangian is thus a function on the jet bundle J over E; taking the fiberwise Legendre transform of the Lagrangian produces a function on the dual bundle over time whose fiber at t is the cotangent space T*Et, which comes equipped with a natural symplectic form, and this latter function is the Hamiltonian.
Read more about this topic: Hamiltonian Mechanics
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