Symplectic and Contact Geometries. Conservation Laws

Abstract

Allan Weinstein is a leading expert in symplectic geometry. The discovery of momentum map by him and Marsden led them to a very important theorem, called now the Marsden—Weinstein symplectic reduction theorem proved in 1974. It was a great shock for Weinstein when, while studying the thick three volumes of forgotten monograph of Lie Theorie der Transformationgruppen, called earlier shortly ‘Lie-Engel’, and has discovered that a number of famous theorems proved in nineteen sixties — nineteen seventies was known to Lie already 90 years earlier. ‘Lie-Engel’ knew also the coadjoint map Ad_G*: g* → g* and the momentum map µ: ℝ²ⁿ →g* and its Ad*-equivariance. He knew that the momentum map µ gives the first integrals of Hamiltonian action (for a G-invariant Hamiltonian.) In a sense Lie knew the famous Emmy Noether theorem on conservation laws (in hamiltonian formalism) already before her birth. And it is known how impressive and influential this theorem of Noether of 1918 was, and still is.

To Sophus Lie, the father of symplectic and contact geometries