# The Geometry of Hamiltonian Mechanics

## Abstract

In this chapter an introduction to Lagrangian and Hamiltonian mechanics is given. An effort is made to present Hamiltonian theory from the analytical mechanics point of view, which unveils the geometrical characteristics of the theory, such as its symplectic symmetry. The relations among the tangent bundle, cotangent bundle and the mixed tangent-cotangent bundle of the configuration manifold are discussed. The Euler–Lagrange equations and Hamilton’s equations of motion are extracted from the principle of least action. The canonical equations are also formulated by the symplectic \(2-\)form and the symplectic transformations are explained. Poisson brackets, which provide the bridge to pass from classical to quantum mechanics are introduced.

## Keywords

Tangent Space Poisson Bracket Tangent Bundle Symplectic Form Symplectic Manifold## References

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