R IEMANNIAN GEOMETRY CRITERION FOR CLASSICAL CHAOS

1. RIEMANNIAN GEOMETRY CRITERION FOR CLASSICAL CHAOS Pavel Stránský www.pavelstransky.cz Institutode Ciencias Nucleares, Universidad Nacional Autónoma de México In collaboration with: Pavel Cejnar Institute of Particle and Nuclear Phycics, Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic Michal Macek Racah Institute of Physics, The Hebrew University, Jerusalem, Israel 30thJuly 2013 XLIIIEscuela Latino Americana de Física, 1a ”Marcos Moshinsky“, Colegio Nacional, México

2. 1. Geometrical Method - flat X curved space (embedding of a Hamiltonian system with a potential in a flat space into a curved space with a Riemannian tensor) 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 Geometrical method - relation between the Geometrical method and the shape of equipotential surfaces

3. 1. Geometrical Method for Hamiltonian systems

4. Geometrical embedding Hamiltonian of a free particle in a curved space: Hamiltonian inthe flat Eucleidian space with a potential: A suitable metric gij y Geodesic x Trajectory Potential • Bridge: • The equations of motion (Hamilton, Newton) correspond with the geodesic equation • 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 L. Casetti, M. Pettini, E.D.G. Cohen, Phys. Rep. 337, 237 (2000)

5. Generalization of a straight line • Describe a ”free motion” in a curved space • “Shortest path” between two points Geodesics Paris -> Mexico In reality, other effects are taken into account – winds, jet stream, air traffic Visualisation of a curved space - mapping onto the flat space

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 Equation of the geodesic deviation (Jacobi equation) Tangent dynamics equation Lyapunov exponent

7. M. Pettini, Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics, Springer, New York, 2007 Choice of the metric 1. Jacobi metric - conformal metric - arc-length - nonzero scalar curvature (negative only when DV < 0) 2. Eisenhart metric - space with 2 extra dimensions - arc-length equivalent with time - only one nonzero Christoffel’s symbol and vanishing scalar curvature 3. Israeli metric (Horwitz et al.) - conformal metric - arc-length proportional to time metric compatible connection - metric incompatible connection form L. Casetti, M. Pettini, E.D.G. Cohen, Phys. Rep. 337, 237 (2000) L. Horwitz et al., Phys. Rev. Lett. 98, 234301(2007)

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 4th power of dimension 2. Scalar curvature stable R > 0 • R = const Equation of the geodetic deviation • Equation of motion for • harmonic oscillator with frequency • exponential divergence with Lyapunov exponent (isotropic manifold) unstable R > 0 • R < 0 Unstable motion with estimated Lyapunov exponent • dim = 2 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

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 metric and connection form, the equation of the geodesic deviation is expressed 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. Example of unstable configuration Kinematically accessible area Negative lower eigenvalue of V Negative higher eigenvalue of V L. Horwitz et al., Phys. Rev. Lett. 98, 234301 (2007)

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

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

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

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

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

15. Principal Axes System (PAS) We focus only on the nonrotating case J = 0! g Shape variables: b Shape-phase structure Phase separatrix B V V A b C=1 b Deformedground state Spherical ground state

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

17. Complete map of classical chaos in the GCM Integrablelimit Integrablelimit deformed shape spherical shape Veins of regularity chaotic Shape-phase transition regular “Arc of regularity” control parameter Saddle point / local maximum regular Israeli geometrical method (stability / instability) P. Stránský, P. Hruška, P. Cejnar, Phys. Rev. E79 (2009), 046202

18. …calculated from trajectories… y x

19. chaotic case – “fog” (hypersensitivity of the motion on the initial conditions) Section at y = 0 vx ordered case – “circles” x …calculated from trajectories… We plot a point every time when a trajectory crosses a given line (y = 0) vx y x … and Poincaré sections Coexistence of quasiperiodic (ordered) and chaotic types of motion

20. chaotic case – “fog” (hypersensitivity of the motion on the initial conditions) Section at y = 0 vx ordered case – “circles” x …calculated from trajectories… We plot a point every time when a trajectory crosses a given line (y = 0) vx y x … and Poincaré sections Coexistence of quasiperiodic (ordered) and chaotic types of motion

21. vx x Fraction of regularity Measure of classical chaos Surface of the section covered with regular trajectories Total kinematically accessible surface of the section REGULARarea CHAOTICarea freg=0.611

22. 1. Lyapunov exponent Quasiperiodic X unstable trajectories Divergence of two neighboring trajectories Regular: at most polynomial divergence Chaotic: exponential divergence 2. SALI (Smaller Alignment Index) • twodivergencies • fast convergence towardszero for chaotic trajectories Ch. Skokos, J. Phys. A: Math. Gen 34, 10029 (2001); 37 (2004), 6269

23. Stability (Application of the Geometric method) Integrablelimit Integrablelimit deformed shape spherical shape Veins of regularity chaotic Shape-phase transition regular “Arc of regularity” control parameter Saddle point / local maximum regular Israeli geometrical method (stability / instability)

24. Potential well V Eigenvalues of the stability matrix V (e) (f) (b) (c) Stability-instability transition, as predicted by the Israeli method (a) b A=-1 B=1.09 Kinematically accessible area Negative lower eigenvalue of V Negative higher eigenvalue of V y x Low-energy regular region Convex-concave transition Saddle point of the potential Local energy maximum Contact of the red and blue regions Concave-convex transition Vein of regularity

25. Potential well V Stability(by Poincaré sections) (a) (b) (e) (h) Low-energy region where the regular harmonic approximation is valid Stable-unstable transition according to the geometry criterion Local maximum of the potential – sharp minimum of regularity “Regular vein” – strongly pronounced local maximum of regularity (e) (f) (b) (c) (a) b A=-1 B=1.09 Black points – regular Red points – chaotic trajectories (a) (b) (c) (d) (e) (f) (g) (h)

26. Curvature of the equipotential surfaces Kinematically accessible area Negative lower eigenvalue of V Negative higher eigenvalue of V concave border - at least one of the eigenvalues negative on the border all eigenvalues positive on the border convex border - Completely convex border, but unstable – a region of negative eigenvalues of V inside Another example: Relation to stability In the case of the GCM, the existence of (partly) concave potential surfaces is equivalent with the existence of negative eigenvalues of V inside the accessible area concave potential surface - dispersing convex potential surface - focusing INSTABILITY

27. Deviations from the geometry criterion Integrablelimit Integrablelimit deformed shape spherical shape Parametric instability? chaotic Shape-phase transition control parameter Saddle point / local maximum Israeli geometrical method (stability / instability) regular Concave – convex transition of the border of the kinematically accessible region

28. Conclusions: • The “Israeli geometry criterion” gives a fast indicator of stability of a Hamiltonian system without the need of solving equations of motion. In the GCM it exactly corresponds to the curvature of equipotential surfaces. • This indicator, althoughonly approximate, works wellin 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 is given in X. Wu, J. Geom. Phys. 59, 1357 (2009). However, the systems presented there are not bound. The complete study of the dynamics in the GCM shows only small deviations from the criterion (chaotic dynamics penetration into stable region, completely regular appearing in unstable region). These deviations may be caused by an effect similar to the Parametric instability. The Riemannian geometry indicator gives a good estimate on the stability, but it does not capture the full richness of the inner dynamics of a Hamiltonian system.

29. Conclusions: • The “Israeli geometry criterion” gives a fast indicator of stability of a Hamiltonian system without the need of solving equations of motion. In the GCM it exactly corresponds to the curvature of equipotential surfaces. • This indicator, althoughonly approximate, works wellin 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 is given in X. Wu, J. Geom. Phys. 59, 1357 (2009). However, the systems presented there are not bound. The complete study of the dynamics in the GCM shows only small deviations from the criterion (chaotic dynamics penetration into stable region, completely regular appearing in unstable region). These deviations may be caused by an effect similar to the Parametric instability. The Riemannian geometry indicator gives a good estimate on the stability. However, it does not capture the full richness of the inner dynamics of a Hamiltonian system. Thank you for your attention And special thanks to Roelof Bijker, Octavio Castaños and all the organizers of ELAF 2013

30. Conclusions: • The “Israeli geometry criterion” gives a fast indicator of stability of a Hamiltonian system without the need of solving equations of motion. In the GCM it exactly corresponds to the curvature of equipotential surfaces. • This indicator, althoughonly approximate, works wellin 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 is given in X. Wu, J. Geom. Phys. 59, 1357 (2009). However, the systems presented there are not bound. The complete study of the dynamics in the GCM shows only small deviations from the criterion (chaotic dynamics penetration into stable region, completely regular appearing in unstable region). These deviations may be caused by an effect similar to the Parametric instability. The Riemannian geometry indicator gives a good estimate on the stability. However, it does not capture the full richness of the inner dynamics of a Hamiltonian system. Thank you for your attention And special thanks to Roelof Bijker, Octavio Castaños and all the organizers of ELAF 2013