Integrable Systems

3Integrable Systems

The aim of this chapter is to study the Arnold–Liouville theorem and its connection with completely integrable systems. Many integrable systems are studied in detail, including: the problem of the rotation of a rigid body about a fixed point – the Euler problem of a rigid body, the Lagrange top, the Kowalewski top and other special cases, such as the Hesse–Appel’rot top, Goryachev–Chaplygin top and Bobylev–Steklov top; the problem of motion of a solid in an ideal fluid – Clebsch’s case and Lyapunov– Steklov’s case; the Yang–Mills field with gauge group SU(2). Some of these problems will be studied in detail in other chapters, using other methods.

3.1. Hamiltonian systems and Arnold–Liouville theorem

DEFINITION 3.1.– A Hamiltonian system is a triple (M, ω, H), where (M, ω) is a 2n-dimensional symplectic manifold (the phase space) and H ∈ 𝒞^∞(M) is a smooth function (Hamiltonian).

Recall section 1.5, where we showed that we have a complete characterization of Hamiltonian vector field

[3.1]

where H : M → ℝ is the Hamiltonian, J = J(x) is a skew-symmetric matrix, possibly depending on x ∈ M, and for which the corresponding Poisson bracket satisfies the Jacobi identity: {{H, F}, G}+ {{F, G}, H} + {{G, H}, F} = 0.

DEFINITION 3.2.– A Hamiltonian system [3.1] is called ...