2011/06/30 by R. M. Angelo, R M Angelo, E. I. Duzzioni +3
Mathematics · Physics and Astronomy · #Algebraic number #Chaotic #Dynamical systems theory #Hamiltonian (control theory) #Hamiltonian system #Integrable system #Invariant (physics) #Lie algebra #Nonlinear Waves and Solitons #Quantum #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #quant-ph
paper · pdf · doi:10.1088/1751-8113/45/5/055101
published as J. Phys. A: Math. Theor. 45 (2012) 055101 · extended version; title changed; 10 pages; 4 figures; accepted in J. Phys. A
arxiv created 2011/12/16 · openalex publication_date 2012/01/16 · arxiv updated 2012/01/20 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
The notion of integrability is discussed for classical nonautonomous systems with one degree of freedom. The analysis is focused on models which are linearly spanned by finite Lie algebras. By constructing the autonomous extension of the time-dependent Hamiltonian, we prove the existence of two invariants in involution which are shown to obey the criterion of functional independence. The implication of this result is that chaotic motion cannot exist in these systems. In addition, if the invariant manifold is compact, then the system is Liouville integrable. As an application, we discuss regimes of integrability in models of dynamical tunneling and parametric resonance, and in the dynamics of two-level systems under generic classical fields. A corresponding quantum algebraic structure is shown to exist which satisfies analog conditions of Liouville integrability and reproduces the classical dynamics in an appropriate limit within the Weyl–Wigner formalism. The quantum analog is then conjectured to be integrable as well.