  The relationship between the Hamiltonian and Lagrangean functions in
analytical mechanics is a type of duality. The two functions, while distinct,
are both descriptive functions encoding the behavior of the same dynamical
system. One difference is that the Lagrangean naturally appears as one
investigates the fundamental equation of classical dynamics. It is not that way
for the Hamiltonian. The Hamiltonian comes after Lagrange's equations have been
fully formed, most commonly through a Legendre transform of the Lagrangean
function. We revisit the Legendre transform approach and offer a more refined
geometrical interpretation than what is commonly shown.
