As a run-up to the "Introduction to Integrability" series (see my pinned tweet), I decided to share some interesting bits from the history of integrable systems.
Let's start at the beginning, shall we? So Newton...
#1 It all started with Newton solving the gravitational 2-body problem and deriving Kepler's laws of planetary motion as a result. I would argue that this was possible, because the Kepler problem is (super)integrable. [1/2]
This roughly means that there are many conversed physical quantities like energy, angular momentum, and the Laplace-Runge-Lenz vector. These conservation laws restrict the motion and allow for explicit analytic solutions of otherwise difficult equations. [2/2]
The 19th century was the golden age of integrable systems. The greatest mathematicians made their marks:
• Euler (1758)
• Lagrange (1888)
• Kovalevskaya (1889)
studied spinning tops
• Jacobi (1839) geodesics on ellipsoids
• C. Neumann (1859) oscillator on spheres
but then...
In 1890 Poincaré showed that the gravitational 3-body problem is not an integrable system and that integrability is a very fragile property of Hamiltonian systems.
This development pretty much killed the interest of researchers and the field laid dormant for several decades...
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