Symplectic geometryLecture 15Semi-direct products, their co-adjoint orbits andtheir moment maps.The semi-direct product of a group with a vector space.The “matrix” mnemonic.The co-adjoint representation.Describing the fibration.Describing the fibration, 2.Describing the fibration, 3.“Euclidean” groups.Some identifications.The co-adjoint action.The orbits of SO(V) acting on V are “spheres”.The case of E(3).The zero and two dimensional orbits of E(3).The four dimensional co-adjoint orbits of E(3).The invariants k and s .The moment map for E(3) acting on six-dimensional phase space.Explicit form of the moment map.The inverse image of the moment map.Intrinsic angular momentum.Units.Historical science fiction.The classical meaning of Planck’s constant.Co-adjoint orbits of the Poincare group.The six types of orbits of the Lorentz group:The rest mass.The positive mass co-adjoint orbits.Mass and spin.Notice, in contrast to what is told in physics courses - that spin is a purely quantum effect - we see that spin arises naturally in the classification of the co-adjoint orbits of the Poincare group.Quadrupole moments.*A matrix version.Relation to the symplectic group.The Lie algebra.The moment map.The case of H=Sl(n).The liquid