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 ,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 "Φ-parity" duality. Our results are illustrated with many examples.