Symmetries and Integrability

Abstract

This is a survey on finite-dimensional integrable dynamical systems related to Hamiltonian $G$-actions. Within a framework of noncommutative integrability we study integrability of $G$-invariant systems, collective motions and reduced integrability. We also consider reductions of the Hamiltonian flows restricted to their invariant submanifolds generalizing classical Hess--Appel'rot case of a heavy rigid body motion.

36 pages

References in corpus (14)

• Integrable geodesic flow with positive topological entropy
• Nonholonomic LR systems as Generalized Chaplygin systems with an Invariant Measure and Geodesic Flows on Homogeneous Spaces
• Torus Actions and Integrable Systems
• The Poincare'-Lyapounov-Nekhoroshev theorem
• Magnetic Geodesic Flows on Coadjoint Orbits
• Bi-Hamiltonian partially integrable systems
• Systems of Hess-Appel'rot type
• Noncommutative integrability on noncompact invariant manifolds
• Invariant fibration of geodesic flows
• Projections of Jordan bi-Poisson structures that are Kronecker, diagonal actions, and the classical Gaudin systems
• Complete families of commuting functions for coisotropic Hamiltonian actions
• Necessary conditions for partial and super-integrability of Hamiltonian systems with homogeneous potentia
• Partial Reductions of Hamiltonian Flows and Hess-Appel'rot Systems on SO(n)
• Differential Galois obstructions for non-commutative integrability

Cited by in corpus (12)

• Poisson-Lie analogues of spin Sutherland models
• On the Hamiltonian formulation of the trigonometric spin Ruijsenaars-Schneider system
• Extended Hamilton-Jacobi theory, contact manifolds and integrability by quadratures
• The Jacobi-Rosochatius problem on an ellipsoid: the Lax representations and billiards
• Systems of Hess-Appel'rot Type and Zhukovskii Property
• Geodesic flows on Riemannian g.o. spaces
• Integrable multi-Hamiltonian systems from reduction of an extended quasi-Poisson double of $U(n)$
• Poisson reductions of master integrable systems on doubles of compact Lie groups
• Bi-Hamiltonian structure of Sutherland models coupled to two $\mathfrak{u}(n{)}^{\ast }$-valued spins from Poisson reduction
• Poisson-Lie analogues of spin Sutherland models revisited
• Action-angle variables for geodesic motions in Sasaki-Einstein spaces $Y^{p,q}$
• Contact line bundles, foliations, and integrability