---
tag: N0N7CMRCMH
kind: concept
name: Integrable system
msc: [37J35, 70H06, 37K10]
specializes: [0EDKGA2FAK]
sources:
  - url: https://en.wikipedia.org/wiki/Integrable_system
same_as:
  wikidata: Q1957758
  wikipedia: Integrable system
---

## Summary

An integrable system is a [[Hamiltonian system|0EDKGA2FAK]] with as many independent conserved quantities in involution as it has degrees of freedom.
Such a system can be solved, in principle, by quadratures, and its motions on compact regular level sets are quasi-periodic motions on invariant tori.
The word is also used for infinite-dimensional systems, such as the Korteweg–De Vries equation.
These have infinitely many conserved quantities and can be solved by inverse scattering.
Integrable systems are the starting point of perturbation theory, where the question is what survives when the system is slightly changed.

## Explanations

### L1 human

Most mechanical systems cannot be solved by a formula, but integrable ones can.
They have enough conserved quantities, like energy and angular momentum, to pin down each motion.
Picture each motion as winding around the surface of a doughnut, at fixed speeds in each direction.
Two independent harmonic oscillators are integrable: the energy of each oscillator is conserved, and each one simply turns around a circle.