The origins of symplectic calculus in Lagrange's work (Q1594936)

Since Kepler ephemerides of a planet were no question but there was no general approach to the global problem of real movements within the whole planetary system. To that end Lagrange has invented in 1808-1811 a method of variations of constants. In order to describe the movement of one planet around the Sun (solving a second order differential equation) one needs 6 constants but an instantaneous shock (e.g., impact of an asteroid) yields another 6 constants which describe the movement of that planet along another ellipse. Lagrange considered interactions of other planets within the system as a series of infinitesimal and continuous shocks. In that way he came to a differentiable description of a perturbated planet as a curve in the space of all Keplerian movements. And it was here that a symplectic structure appeared for the first time (the term ``symplectic'' has been, however, introduced by Hermann Weyl only in 1946). The paper is a fine reconstruction of Lagrange's procedures.