Dualities of Self-Dual Nonlinear Electrodynamics

For any causal nonlinear electrodynamics theory that is "self-dual" (electromagnetic U (1) -duality invariant), the Legendre-dual pair of Lagrangian and Hamiltonian densities \L, H\ are constructed from functions \, h\ on R^+ related to a particle-mechanics Lagrangian and Hamiltonian. We show how a `duality' relating to h implies that L and H are related by a simple map between appropriate pairs of variables. We also discuss Born's "Legendre self-duality" and implications of a new "-parity" duality. Our results are illustrated with many examples.