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

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

Basics of classical GCMLagrangian VTL

5 coordinates5 velocities

Scaling propertiesof 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:0zJ

In Cartesian frame (Jx, Jy, Jz) we choose rotational axis paralel with z +

Nonrotating case

0J

Nonzero variables:

New coordinatesWell-known Bohr coordinates:

Generalization:

In this new coordinates kinetic and potential terms in Lagrangian reads asand angular momentum

Solution of the Lagrangeequation of motion

How to use these trajectories

to clasify the system?

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

Poincaré sections

Poincaré sections

- surface

For this example (GCM with A = -5.05,

E = 0, J = 0)

freg=0.611

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

Results for J = 0(using Poincaré

sections)

A = -0.84

Dependence of freg on energy

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

• 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

Results for 0zJ(using Lyapunov exponents)

Noncrossing ruleQuadrupole deformation tensor in Cartesian (x, y, z) components

Difference of the eigenvalues

It can be zero only if Jz = 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 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

Thank you for your attention