The inception of Symplectic Geometry: the works of Lagrange and Poisson during the years 1808-1810

TL;DR

This paper explores the early development of symplectic geometry through the foundational works of Lagrange and Poisson between 1808 and 1810, highlighting their contributions to the concept's origins.

Contribution

It provides a historical analysis of the initial ideas and mathematical formulations that led to symplectic geometry, emphasizing Lagrange's and Poisson's roles.

Findings

Lagrange's work introduced the variation of constants method.

Poisson defined the composition law now known as the Poisson bracket.

The paper reinterprets historical works using modern mathematical notation.

Abstract

The concept of a symplectic structure first appeared in the works of Lagrange on the so-called "method of variation of the constants". These works are presented, together with those of Poisson, who first defined the composition law called today the "Poisson bracket". The method of variation of the constants is presented using today's mathematical concepts and notations.