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The relationship between the Hamiltonian and Lagrangean functions inanalytical mechanics is a type of duality. The two functions, while distinct,are both descriptive functions encoding the behavior of the same dynamicalsystem. One difference is that the Lagrangean naturally appears as oneinvestigates the fundamental equation of classical dynamics. It is not that wayfor the Hamiltonian. The Hamiltonian comes after Lagrange's equations have beenfully formed, most commonly through a Legendre transform of the Lagrangeanfunction. We revisit the Legendre transform approach and offer a more refinedgeometrical interpretation than what is commonly shown.

Understanding the Hamiltonian Function through the Geometry of Partial Legendre Transforms