Integrable System - List of Some Well-known Classical Integrable Systems

List of Some Well-known Classical Integrable Systems

1. Classical mechanical systems (finite-dimensional phase space):

  • Harmonic oscillators in n dimensions
  • Central force motion
  • Two center Newtonian gravitational motion
  • Geodesic motion on ellipsoids
  • Neumann oscillator
  • Lagrange, Euler and Kovalevskaya tops
  • Integrable Clebsch and Steklov systems in fluids
  • Calogero–Moser–Sutherland models
  • Swinging Atwood's Machine with certain choices of parameters

2. Integrable lattice models

  • Toda lattice
  • Ablowitz–Ladik lattice
  • Volterra lattice

3. Integrable systems of PDEs in 1 + 1 dimension

  • Korteweg–de Vries equation
  • Sine–Gordon equation
  • Nonlinear Schrödinger equation
  • Boussinesq equation (water waves)
  • Nonlinear sigma models
  • Classical Heisenberg ferromagnet model (spin chain)
  • Classical Gaudin spin system (Garnier system)
  • Landau–Lifshitz equation (continuous spin field)
  • Benjamin–Ono equation
  • Dym equation
  • Three wave equation

4. Integrable PDEs in 2 + 1 dimensions

  • Kadomtsev–Petviashvili equation
  • Davey–Stewartson equation
  • Ishimori equation

5. Other integrable systems of PDEs in higher dimensions

  • Self-dual Yang–Mills equations

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