1. R IEMANNIAN GEOMETRY CRITERION FOR CHAOS IN COLLECTIVE DYNAMICS OF NUCLEI Pavel Stránský, Pavel Cejnar XXI Nuclear Physics Workshop, Kazimierz Dolny, Poland 26 th September 2014 Institute of Particle and Nuclear Phycics Faculty of Mathematics and Physics Charles University in Prague, Czech Republic www.pavelstransky.cz

2. 1.Curvature-based Method - flat X curved space, embedding of a Hamiltonian system into a curved space - estimating the onset of chaos based on overall geometric properties 2.Model - classical dynamics of the Geometric Collective Model (GCM) of atomic nuclei 3.Results and discussion - full map of classical chaos in the GCM - instability predicted by the curvature-based criterion - relation between the curvature-based method and the onset of chaos determined numerically P. Stránský, P. Cejnar, Study of curvature-based criterion for chaos in Hamiltonian systems with two degrees of freedom submitted to Journal of Physics A: Mathematical and Theoretical R IEMANNIAN GEOMETRY CRITERION FOR CHAOS IN COLLECTIVE DYNAMICS OF NUCLEI

3. 1.Curvature-based criterion for chaos in Hamiltonian systems

4. Geometrical embedding Hamiltonian of the motion in the flat Euclidean space with a potential: Why embedding: Riemannian geometry brings in the notion of curvature that could help clarify the sources of instability, and in the same time quantify the amount of chaos in non-ergodic systems Bridge: The equations of motion (Hamilton, Newton) correspond with the geodesic equation L. Casetti, M. Pettini, E.D.G. Cohen, Phys. Rep. 337, 237 (2000) M. Pettini, Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics (Springer New York, 2007) Potential Trajectory x y Hamiltonian of the free motion in a curved space: Geodesic curvature effects topological effects

5. Geodesics & Maps -Generalization of a straight line -Describe a ”free motion” in a curved space -“Shortest path” between two points Visualisation of a curved space - mapping onto the flat space Paris -> Mexico Prague

6. Flat space (dynamics) Curved space (geometry) Potential energy Time Forces Curvature of the potential Metric Arc-length Christoffel’s symbols Riemannian tensor Ricci tensor Scalar curvature Trajectories Hamiltonian equations of motion Geodesics Geodesic equation Tangent dynamics equation Equation of the geodesic deviation (Jacobi equation) Lyapunov exponent

7. Various ways of the geometric embedding 1. Jacobi metric - conformal metric - arc-length - nonzero scalar curvature L. Casetti, M. Pettini, E.D.G. Cohen, Phys. Rep. 337, 237 (2000) (negative only when  V < 0 ) 2. Eisenhart metric - space with 2 extra dimensions - arc-length equivalent with time - only one nonzero Christoffel’s symbol and 3. Israeli method (Horwitz et al.) vanishing scalar curvature L. Horwitz et al., Phys. Rev. Lett. 98, 234301 (2007) - conformal “metric” - metric incompatible connection form metric compatible connection - arc-length proportional to time

8. Curvature and instability Besides solving the equation for the geodesic deviation, can one deduce something about the instability only from the curvature? 1. Riemannian tensor Difficult, the number of components grows with the 4 th power of dimension 2. Scalar curvature R = const Equation of the geodetic deviation Equation of motion for harmonic oscillator with frequency exponential divergence with Lyapunov exponent (isotropic manifold) R < 0 Unstable motion with estimated Lyapunov exponent dim = 2 stable R > 0 unstable R > 0 Equation of motion of a harmonic oscillator with its length (stiffness) modulated in time Unstable if the frequency is in resonance with any of the frequency of the Fourier expansion, even if R(s) > 0 on the whole manifold: Parametric instability The metric-compatible connection is required!

9. Curvature and instability Besides solving the equation for the geodesic deviation, can one deduce something about the instability only from the curvature? 3. Israeli method Using the Israeli connection form, the equation of the geodesic deviation reads as - projector into a direction orthogonal to the velocity Stability matrix Conjecture: A negative eigenvalue of the Stability matrix inside the kinematically accessible area induces instability of the motion. L. Horwitz et al., Phys. Rev. Lett. 98, 234301 (2007) Example of unstable configuration Kinematically accessible area Negative lower eigenvalue of Negative higher eigenvalue of y x

10. Properties of the stability matrix 1. When is big enough, becomes the Hessian matrix for the tangent dynamics 2. Eigenvalues can only decrease within the kinematically accessible domain The size of the negative eigenvalue region can only grow with energy, or remain the same 3. The lower eigenvalue is continuous on the boundary of the accessible domain f = 2 condition for inflexion points of the curve 4. The lower eigenvalue is zero on the boundary when concave convex The curvature-based criterion for the onset of chaos can be partly translated into the language of the shape of the equipotential contours. concave potential surface - dispersing convex potential surface - focusing

11. Instability threshold Scenario A - Penetration - region of negative, which exists outside the accessible region, starts overlapping with it at some energy E - equipotential contours undergoes the convex-concave transition

12. Instability threshold Scenario A - Penetration - region of negative, which exists outside the accessible region, starts overlapping with it at some energy E - equipotential contours undergoes the convex-concave transition

13. Instability threshold Scenario B - CreationScenario A - Penetration - region of negative, which exists outside the accessible region, starts overlapping with it at some energy E - region of negative eventually appears somewhere inside the accessible region at some energy E - all the equipotential contours convex - equipotential contours undergoes the convex-concave transition - necessary condition

14. 2. Model (Geometric collective model of nuclei)

15. Surface of homogeneous nuclear matter: Monopole deformations = 0 (even-even nuclei – collective character of the lowest excitations) - Does not contribute due to the incompressibility of the nuclear matter Dipole deformations = 1 - Related to the motion of the center of mass - Zero due to momentum conservation - “breathing” mode Geometric collective model

16. T…Kinetic term V…Potential Neglect higher order terms Quadrupole tensor of collective coordinates (2 shape parameters, 3 Euler angles) Corresponding tensor of momenta Quadrupole deformations = 2 G. Gneuss, U. Mosel, W. Greiner, Phys. Lett. 30B, 397 (1969) 4 external parameters neglect Geometric collective model Surface of homogeneous nuclear matter:

17. T…Kinetic term V…Potential Neglect higher order terms Quadrupole tensor of collective coordinates (2 shape parameters, 3 Euler angles) Corresponding tensor of momenta Scaling properties 4 external parameters Adjusting 3 independent scales energy (Hamiltonian) 1 “shape” parameter size (deformation) time 1 “classicality” parameter sets absolute density of quantum spectrum (irrelevant in classical case) P. Stránský, M. Kurian, P. Cejnar, Phys. Rev. C 74, 014306 (2006) neglect (order parameter) Geometric collective model Surface of homogeneous nuclear matter: Quadrupole deformations = 2

18. Principal Axes System (PAS) Shape variables: Shape-phase structure Spherical ground-state shape V  V  B A C=1 Phase separatrix   We focus only on the nonrotating case J = 0 ! grey lines – equivalent dynamical configurations covers all inequivalent configurations of the GCM Deformed ground-state shape control parameter

19. Hamiltonian It describes: Motion of a star around a galactic centre, assuming the motion is cylindrically symmetric (Hénon-Heiles model) Collective motion of an atomic nucleus (Bohr model) … but also (for example):

20. 3.Results and discussion (Israeli geometry method applied to GCM)

21. Complete map of classical chaos in the GCM Integrable limit Veins of regularity chaotic regular control parameter Shape-phase transition P. Stránský, P. Hruška, P. Cejnar, Phys. Rev. E 79 (2009), 046202 “Arc of regularity” Integrable limit deformed shape spherical shape Saddle point / local maximum and minimum convex-concave border transition (stability / instability) regular (mexican hat potential)

22. Complete map of classical chaos in the GCM Integrable limit Veins of regularity chaotic regular control parameter Shape-phase transition P. Stránský, P. Hruška, P. Cejnar, Phys. Rev. E 79 (2009), 046202 Integrable limit deformed shape spherical shape Saddle point / local maximum and minimum convex-concave border transition (stability / instability) regular “Arc of regularity” (mexican hat potential)

23. Integrable limit Veins of regularity chaotic regular control parameter Shape-phase transition P. Stránský, P. Hruška, P. Cejnar, Phys. Rev. E 79 (2009), 046202 “Arc of regularity” Integrable limit deformed shape spherical shape regular Saddle point / local maximum and minimum convex-concave border transition (stability / instability) Stability (Application of the curvature-based method)

24. Stability matrix S Kinematically accessible area Negative lower eigenvalue of Negative higher eigenvalue of Stability-instability transition, as predicted by the Israeli method Low-energy region where the regular harmonic approximation is valid Stable-unstable transition according to the geometric criterion saddle point of the potential Local maximum of the potential – sharp minimum of regularity concave-convex transition of the equipotential contour “Regular vein” – strongly pronounced local maximum of regularity (a) (b) (c) (e) (g) (h)

25. The Riemannian geometry indicator gives a good estimate on the stability. However, it does not capture the full richness of the dynamics of a Hamiltonian system. Conclusions: 1.The curvature-based criterion for the onset of chaos gives a fast indicator of stability of a Hamiltonian system without the need of solving equations of motion. In 2D systems it exactly corresponds to the convex-concave change in equipotential contours (Scenario A). 2.This criterion, although only approximate, works well in many physical systems: L. Horwitz et al., Phys. Rev. Lett. 98, 234301 (2007) [2D Toda lattice] Y.B. Zion and L. Horwitz, Phys. Rev. E 76, 046220 (2007) [3D Yang-Mills system] Y.B. Zion and L. Horwitz, Phys. Rev. E 78, 036209 (2008) [2D coupled HO, 2D quartic oscillator] J. Li and S. Zhang, J. Phys. A: Math. Theor. 43, 125105 (2010) [Dicke model] A list of counterexamples (unbound systems) is given in X. Wu, J. Geom. Phys. 59, 1357 (2009). A detailed discussion of the GCM and the Creagh-Whelan model is given in our paper submitted recently to J. Phys. A: Math. Theor. 3.The complete study of the dynamics in the GCM shows rough coincidence of the criterion and the numerically determined inset of chaos, although some deviations are observed (chaotic dynamics penetration into stable region, completely regular dynamics appearing in unstable region, instability predicted for the integrable configuration).

26. The Riemannian geometry indicator gives a good estimate on the stability. However, it does not capture the full richness of the dynamics of a Hamiltonian system. T HANK YOU FOR YOUR ATTENTION And special thanks to all organizers of this inspiring Nuclear Physics Workshop. Conclusions: 1.The curvature-based criterion for the onset of chaos gives a fast indicator of stability of a Hamiltonian system without the need of solving equations of motion. In 2D systems it exactly corresponds to the convex-concave change in equipotential contours (Scenario A). 2.This criterion, although only approximate, works well in many physical systems: L. Horwitz et al., Phys. Rev. Lett. 98, 234301 (2007) [2D Toda lattice] Y.B. Zion and L. Horwitz, Phys. Rev. E 76, 046220 (2007) [3D Yang-Mills system] Y.B. Zion and L. Horwitz, Phys. Rev. E 78, 036209 (2008) [2D coupled HO, 2D quartic oscillator] J. Li and S. Zhang, J. Phys. A: Math. Theor. 43, 125105 (2010) [Dicke model] A list of counterexamples (unbound systems) is given in X. Wu, J. Geom. Phys. 59, 1357 (2009). A detailed discussion of the GCM and the Creagh-Whelan model is given in our paper submitted recently to J. Phys. A: Math. Theor. 3.The complete study of the dynamics in the GCM shows rough coincidence of the criterion and the numerically determined inset of chaos, although some deviations are observed (chaotic dynamics penetration into stable region, completely regular dynamics appearing in unstable region, instability predicted for the integrable configuration).

27. The Riemannian geometry indicator gives a good estimate on the stability. However, it does not capture the full richness of the dynamics of a Hamiltonian system. T HANK YOU FOR YOUR ATTENTION And special thanks to all organizers of this inspiring Nuclear Physics Workshop. Conclusions: 1.The curvature-based criterion for the onset of chaos gives a fast indicator of stability of a Hamiltonian system without the need of solving equations of motion. In 2D systems it exactly corresponds to the convex-concave change in equipotential contours (Scenario A). 2.This criterion, although only approximate, works well in many physical systems: L. Horwitz et al., Phys. Rev. Lett. 98, 234301 (2007) [2D Toda lattice] Y.B. Zion and L. Horwitz, Phys. Rev. E 76, 046220 (2007) [3D Yang-Mills system] Y.B. Zion and L. Horwitz, Phys. Rev. E 78, 036209 (2008) [2D coupled HO, 2D quartic oscillator] J. Li and S. Zhang, J. Phys. A: Math. Theor. 43, 125105 (2010) [Dicke model] A list of counterexamples (unbound systems) is given in X. Wu, J. Geom. Phys. 59, 1357 (2009). A detailed discussion of the GCM and the Creagh-Whelan model is given in our paper submitted recently to J. Phys. A: Math. Theor. 3.The complete study of the dynamics in the GCM shows rough coincidence of the criterion and the numerically determined inset of chaos, although some deviations are observed (chaotic dynamics penetration into stable region, completely regular dynamics appearing in unstable region, instability predicted for the integrable configuration).