classical chaos in geometric collective model pavel stránský, pavel cejnar, matúš kurian...
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Basics of classical GCM Lagrangian 5 coordinates 5 velocitiesTRANSCRIPT
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Classical Chaos in
Geometric Collective Model
Pavel Stránský, Pavel Cejnar, Matúš Kurian
Institute of Particle and Nuclear PhycicsFaculty of Mathematics and PhysicsCharles University in Prague, Czech Republic
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1. Classical GCM and its dynamics2. Scaling properties3. Angular momentum and equations of
motion4. Poincaré sections and measure of chaos5. Numerical results for 6. Numerical results for
0J
Outline
0zJ
J
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Basics of classical GCMLagrangian VTL
5 coordinates5 velocities
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Scaling propertiesof Lagrangian
General Lagrangian:
transformation of 3 fundamental physical units:
size (deformation)energy (Lagrangian)time
Important example:
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Introduction of angular momentum
Spherical tensor of rank 1:
Spherical symmetry of the Lagrangian – angular momentum is conserved.
2 special cases:0zJ
In Cartesian frame (Jx, Jy, Jz) we choose rotational axis paralel with z +
Nonrotating case
0J
Nonzero variables:
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New coordinatesWell-known Bohr coordinates:
Generalization:
In this new coordinates kinetic and potential terms in Lagrangian reads asand angular momentum
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Solution of the Lagrangeequation of motion
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How to use these trajectories
to clasify the system?
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Measures of Chaos1. Lyapunov exponent (for a trajectory in the phase space):• positive for chaotic trajectories• slow convergence
Deviation of two neighbouring trajectories in phase space
2. Poincaré sections, surface of the sections3. SALI (Smaller Alignment Index)
• reach zero for chaotic trajectories• fast convergence
Ch. Skokos, J. Phys. A: Math. Gen 34 (2001), 10029; 37 (2004), 6269
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Poincaré sections
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Poincaré sections
- surface
For this example (GCM with A = -5.05,
E = 0, J = 0)
freg=0.611
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Poincaré sectionsFor systems with trajectories laying on 4- or higher-
dimensional manifolds (practically systems with more than 2 degrees of freedom)IT IS NOT POSSIBLEto use surface of sections to measure quantity of chaos
Fishgraph A = -2.6, E = 24.4
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Results for J = 0(using Poincaré
sections)
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A = -0.84
Dependence of freg on energy
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Dependence of freg on energy
• full regularity for E near global minimum of potential • complex behaviour in the intermediate domain• sharp peak for E = 23 if A > -0.8• logaritmic fading of chaos for large E
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• for B = 0 system is integrable -> fully regular• for small B chaos increases linearly, but the increase stops earlier than freg = 0• for very large B system becomes regular
Dependence of freg on B (on A) for E = 0
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Results for 0zJ(using Lyapunov exponents)
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Noncrossing ruleQuadrupole deformation tensor in Cartesian (x, y, z) components
Difference of the eigenvalues
It can be zero only if Jz = 0.
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Increasing j
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Summary
1. There is only 1 essential external parameter in our truncated form of GCM
2. GCM exhibits complex interplay between regular and chaotic types of motions depending on the control parameter A and energy E
3. Poincaré sections are good tools to quantify regularity of classical 2D system
4. The effect of spin cannot be treated in a perturbative way5. With increasing J the system overall tend to suppress the
chaos for small B and to enhance it for large B6. SALI method could be succesfuly used to analyse efects of
general spin
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Thank you for your attention
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