Classical Chaos in Geometric Collective Model Pavel Stránský, Pavel Cejnar, Matúš Kurian Institute of Particle and Nuclear Phycics Faculty of Mathematics and Physics Charles University in Prague, Czech Republic

1.Classical GCM and its dynamics 2.Scaling properties 3.Angular momentum and equations of motion 4.Poincaré sections and measure of chaos 5.Numerical results for 6.Numerical results for Outline

Basics of classical GCM Lagrangian 5 coordinates 5 velocities

Scaling properties of Lagrangian General Lagrangian: transformation of 3 fundamental physical units: size (deformation) energy (Lagrangian) time Important example:

Introduction of angular momentum Spherical tensor of rank 1: Spherical symmetry of the Lagrangian – angular momentum is conserved. 2 special cases: In Cartesian frame (J x, J y, J z ) we choose rotational axis paralel with z + Nonrotating case Nonzero variables: 

New coordinates Well-known Bohr coordinates: Generalization: In this new coordinates kinetic and potential terms in Lagrangian reads as and angular momentum

Solution of the Lagrange equation of motion

How to use these trajectories to clasify the system?

Measures of Chaos 1. 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 sections 3. SALI (Smaller Alignment Index) reach zero for chaotic trajectories fast convergence Ch. Skokos, J. Phys. A: Math. Gen 34 (2001), 10029; 37 (2004), 6269

Poincaré sections

- surface For this example (GCM with A = -5.05, E = 0, J = 0) f reg =0.611

Poincaré sections For systems with trajectories laying on 4- or higher-dimensional manifolds (practically systems with more than 2 degrees of freedom) IT IS NOT POSSIBLE to use surface of sections to measure quantity of chaos Fishgraph A = -2.6, E = 24.4

Results for J = 0 (using Poincaré sections)

A = Dependence of f reg 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

for B = 0 system is integrable -> fully regular for small B chaos increases linearly, but the increase stops earlier than f reg = 0 for very large B system becomes regular Dependence of f reg on B (on A) for E = 0

Results for (using Lyapunov exponents)

Noncrossing rule Quadrupole deformation tensor  in Cartesian (x, y, z) components Difference of the eigenvalues It can be zero only if J z = 0.

Increasing j

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 way 5.With increasing J the system overall tend to suppress the chaos for small B and to enhance it for large B 6.SALI method could be succesfuly used to analyse efects of general spin

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