Colloquium

Colloquium
Hamiltonian monodromy and lattice defects. From elastic pendulum to sunflower patterns. Boris Zhilinskii (Universite du Littoral) 10月8日(金) 15時00分 Hamiltonian monodromy is a simplest obstruction to existence of global action variables in completely integrable classical dynamical problems. Extending monodromy notion to quantum problems with discrete spectrum we represent monodromy as a specific defect of a regular Z^N lattice formed by common eigenvalues of mutually commuting quantum operators. These classical and quantum constructions will be illustrated on an example of "swing spring" or elastic pendulum, the three degree of freedom Hamiltonian dynamical system with 1 : 1 : 2 resonance. The quantum analog of classical swing spring is the vibrational motion of triatomic linear molecule with Fermi resonance between bending and stretching modes. Along with "elementary monodromy defect" more complicated defects of lattices are described as a result of multiple elementary defects. An attempt to find among these defects some universal generic ones leads to a interpretation of the spiral phyllotaxis patterns (widely observed in sunflowers, pineappeles, etc) in terms of a universal specific monodromy defect. How to construct a model dynamical system possessing such a defect remains an open problem. I`ll try to present some arguments supporitng the possible relevance of K3 surfaces to this problem. References 1. R.H. Cushman, H. R. Dullin, A. Giacobbe, D.D. Holm, M. Joyeux, P. Lynch, D.A. Sadovskii, B. I. Zhilinski, CO2 molecule as a quantum realization of the 1:1:2 resonant swing-spring with monodromy. Phys. Rev. Lett. 93, 024302-1-4 (2004) 2.A. Giacobbe, R. Cushman, D. Sadovskii, B. I. Zhilinski, Monodromy of the quantum 1:1:2 resonant swing spring. J. Math. Phys., 45, 5076-5100 (2004) 3. B. I. Zhilinski, Hamiltonian monodromy as lattice defect. in: Topology in Condensed Matter, (Springer Series in Solid-State Sciences, Vol. 150), 2006, pp. 165-186 4. D.A. Sadovskii B. I. Zhilinski, Hamiltonian systems with detuned 1:1:2 resonance. Manifestation of bidromy Ann.Phys. (N.Y) 322, 164-00 (2007) 5. B. Zhilinskii, Monodromy and Complexity of Quantum Systems. in: The Complexity of Dynamical Systems. Ed. J.Dubbeldam, K. Green, and D. Lenstra, WILEY, Weinheim, 2011, pp. 159-181

Hamiltonian monodromy and lattice defects. From elastic pendulum to sunflower patterns.

Boris Zhilinskii
(Universite du Littoral)

10月8日(金) 15時00分

Hamiltonian monodromy is a simplest obstruction to existence of global action variables in completely integrable classical dynamical problems. Extending monodromy notion to quantum problems with discrete spectrum we represent monodromy as a specific defect of a regular Z^N lattice formed by common eigenvalues of mutually commuting quantum operators.
These classical and quantum constructions will be illustrated on an example of "swing spring" or elastic pendulum, the three degree of freedom Hamiltonian dynamical system with 1 : 1 : 2 resonance. The quantum analog of classical swing spring is the vibrational motion of triatomic linear molecule with Fermi resonance between bending and stretching modes.
Along with "elementary monodromy defect" more complicated defects of lattices are described as a result of multiple elementary defects. An attempt to find among these defects some universal generic ones leads to a interpretation of the spiral phyllotaxis patterns (widely observed in sunflowers, pineappeles, etc) in terms of a universal specific monodromy defect. How to construct a model dynamical system possessing such a defect remains an open problem. I`ll try to present some arguments supporitng the possible relevance of K3 surfaces to this problem.

References
1. R.H. Cushman, H. R. Dullin, A. Giacobbe, D.D. Holm, M. Joyeux, P. Lynch, D.A. Sadovskii, B. I. Zhilinski, CO2 molecule as a quantum realization of the 1:1:2 resonant swing-spring with monodromy. Phys. Rev. Lett. 93, 024302-1-4 (2004)
2.A. Giacobbe, R. Cushman, D. Sadovskii, B. I. Zhilinski, Monodromy of the quantum 1:1:2 resonant swing spring. J. Math. Phys., 45, 5076-5100 (2004)
3. B. I. Zhilinski, Hamiltonian monodromy as lattice defect. in: Topology in Condensed Matter, (Springer Series in Solid-State Sciences, Vol. 150), 2006, pp. 165-186
4. D.A. Sadovskii B. I. Zhilinski, Hamiltonian systems with detuned 1:1:2 resonance. Manifestation of bidromy Ann.Phys. (N.Y) 322, 164-00 (2007)
5. B. Zhilinskii, Monodromy and Complexity of Quantum Systems. in: The Complexity of Dynamical Systems. Ed. J.Dubbeldam, K. Green, and D. Lenstra, WILEY, Weinheim, 2011, pp. 159-181