Dynamical Systems
Integrable Systems
The three-body problem is non-integrable - Poincare proved it in 1890. The two-body problem is integrable and solvable in closed form. A single extra body produces chaos.
- **Optical fibers:** solitons are used in telecommunications to transmit data without dispersion
- **Bose-Einstein condensates:** solitons are observed experimentally in quantum gases
- **Orbital mechanics:** action-angle variables are the foundation of perturbation theory in celestial mechanics
- **Quantum physics:** the inverse scattering method underlies quantum integrability and the Bethe Ansatz
Предварительные знания
- Hamiltonian mechanics: Poisson brackets and conserved quantities
- Phase portraits and invariant manifolds
- Linear operators and spectral theory
Liouville-Arnold Theorem
The two-body problem (Kepler, 1609) is the first example of a fully integrable system: 6 degrees of freedom, 6 conserved quantities (E, L, the Laplace-Runge-Lenz direction) guarantee an analytic solution via elliptic orbits.
How many independent integrals of motion in involution are needed for the integrability of a system with n degrees of freedom?
Liouville-Arnold theorem: n independent integrals in involution for a system with n degrees of freedom guarantee integrability and motion on tori.
Solitons and the Inverse Scattering Method
In 1965 Zabusky and Kruskal discovered, in numerical simulations of the KdV (Korteweg-de Vries) equation, solitons - nonlinear waves that pass through one another without changing shape. This discovery revived interest in integrable systems.
What do the discrete eigenvalues in the inverse scattering problem for the KdV equation correspond to?
In the inverse scattering method discrete levels lambda_n = -kappa_n^2 represent solitons, while the continuous spectrum represents the dispersive part.
Lax Pairs and the Poisson Bracket
A Lax pair (L, B) is an elegant algebraic structure that generates infinite families of integrals of motion. Lax showed in 1968 that the KdV equation is equivalent to the evolution of the operator L by the formula dL/dt = [B, L], where the bracket is the commutator.
The Lax pair formalism unifies integrable systems: the KdV equation, the nonlinear Schroedinger equation, the Toda lattice, and the sine-Gordon equation are all described by Lax pairs with infinite families of integrals.
Why are the traces tr(L^k) integrals of motion in systems with a Lax pair?
The equation dL/dt = [B, L] is an isospectral deformation: the eigenvalues of L are preserved, and the traces tr(L^k), being symmetric functions of the eigenvalues, are also constant.
Connections to other areas
Integrable systems sit at the intersection of differential geometry, Lie algebra, and mathematical physics.
- Symplectic geometry — Liouville tori are Lagrangian submanifolds of symplectic space
- Lie algebra — Lax pairs use algebraic structures: the commutator [B, L] is an operation in a Lie algebra
- Quantum groups — Quantum integrability (Bethe Ansatz) generalizes classical Lax pairs to the operator level
Key results
- Liouville integrability: n integrals in involution, motion on tori T^n
- Action-angle variables: Hamiltonian flow is uniform rotation with frequencies omega_i = dH/dJ_i
- KdV solitons: localized nonlinear waves preserving their shape under collisions
- Inverse scattering method: nonlinear evolution becomes a linear scattering problem
- Lax pairs: dL/dt = [B,L] generates infinite families of integrals via tr(L^k)