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The origins of symplectic calculus in Lagrange’s work. (Les origines du calcul symplectique chez Lagrange.) (French) Zbl 0966.01017
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.

01A55 History of mathematics in the 19th century
37-03 History of dynamical systems and ergodic theory
53-03 History of differential geometry
70H05 Hamilton’s equations
70-03 History of mechanics of particles and systems