Symplectic Geometry in Radar: 2-Form of Maxwell Equations and Wigner/Woodward Ambiguity Function

We will introduce the symplectic geometry structures that appear naturally in two radar application domains. We will start, in a first part, by introducing the closed 2-form for the Maxwell equations characterizing the symplectic manifold structure associated with the electromagnetic wave equations. In a second part, we will consider the Woodward Radar self-ambiguity function through Wigner function, in the case of the Heisenberg-Weyl symmetry group. The Wigner function is then defined on the coadjoint orbit (action of the Lie group on the dual of its Lie algebra), with a Liouville invariant measure (invariant measures under the group action) and a symplectic structure that make them geometrically similar to classical phase spaces. This ensures its interpretation as a quasi-distribution on a phase space. We will recall in introduction a synthetical history of Symplectic Geometry and its introduction by Jean-Marie Souriau in a seminal paper in 1953 to geometrize classical Mechanics based on Lagrange bracket.