"The origin of symplectic calculus in the work of Lagrange"
Abstract: Between 1808 and 1811, Joseph-Louis Lagrange developed the theory of the "variations of constants" applied to the problem of a planet's stability. He introduced this technique some years before, for the resolution of non-homogeneous differential systems. But, applied to the case of mechanical systems, that construction revealed a strange anti-symmetric structure he used to improve previous results from Laplace and Poisson. That work can be considered as the birth of symplectic calculus. On the way, he discovered the equations attributed later to Hamilton, which describe the evolution of a real perturbed system in the space of unperturbated motions.
The work of Lagrange was, more or less, forgotten for a long time and only rediscovered in the middle of the 20th century when problems of globality in classical mechanics became crucial, in particular with the coming of quantum geometry. Then Lagrange's "symplectic calculus" began to become "symplectic geometry".