RIEMANNIAN GEOMETRY CRITERION
FOR CLASSICAL CHAOS
Pavel Stránský
www.pavelstransky.cz
Instituto de 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
XLIII Escuela Latino Americana de Física, 1a ”Marcos Moshinsky“, Colegio Nacional, México
30th July 2013
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
1. Geometrical Method for Hamiltonian systems
Geometrical embedding
Hamiltonian of a free particle
in a curved space:
Hamiltonian in the flat Eucleidian
space with a potential:
A suitable metric gij
y
x
Geodesic
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)
Geodesics
-
Generalization of a straight line
Describe a ”free motion” in a curved space
“Shortest path” between two points
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
Flat space
Curved space
(dynamics)
(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
Choice of the metric
M. Pettini, Geometry and Topology in Hamiltonian Dynamics
and Statistical Mechanics, Springer, New York, 2007
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 incompatible connection form
metric compatible connection
L. Casetti, M. Pettini, E.D.G. Cohen, Phys. Rep. 337, 237 (2000)
L. Horwitz et al., Phys. Rev. Lett. 98, 234301 (2007)
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
•
R = const
(isotropic manifold)
•
R<0
•
dim = 2
Equation of the
geodetic deviation
stable R > 0
Equation of motion for
• harmonic oscillator with
frequency
• exponential divergence
with Lyapunov exponent
unstable
R>0
Unstable motion with estimated Lyapunov exponent
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
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)
2. Model
(Geometric collective model of nuclei)
Hamiltonian
It describes:
Collective motion of an atomic nucleus
(Bohr model)
… but also (for example):
Motion of a star around a galactic centre,
assuming the motion is cylindrically symmetric
(Hénon-Heiles model)
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
Geometric collective model
Surface of homogeneous nuclear matter:
Quadrupole deformations l = 2
Corresponding tensor of momenta
Quadrupole tensor of collective coordinates
(2 shape parameters, 3 Euler angles)
T…Kinetic term
V…Potential
Neglect higher
order terms
neglect
4 external parameters
G. Gneuss, U. Mosel, W. Greiner, Phys. Lett. 30B, 397 (1969)
Geometric collective model
Surface of homogeneous nuclear matter:
Quadrupole deformations l = 2
Corresponding tensor of momenta
Quadrupole tensor of collective coordinates
(2 shape parameters, 3 Euler angles)
T…Kinetic term
V…Potential
Neglect higher
order terms
neglect
4 external parameters
Scaling
properties
1 “shape” parameter
(order parameter)
Adjusting 3 independent scales
energy (Hamiltonian)
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)
Principal Axes System (PAS)
We focus only on the
nonrotating case J = 0!
g
Shape
variables:
b
Shape-phase structure
B
Phase
separatrix
V
V
A
b
C=1
Deformed ground state
b
Spherical ground state
3. Results and discussion
(Israeli geometry method applied to GCM)
Complete map of classical chaos in the GCM
Integrable limit
Integrable limit
deformed shape
regularity”
chaotic
Shape-phase transition
spherical shape
Veins of
regularity
regular
“Arc of
control
parameter
regular
Saddle point / local maximum
Israeli geometrical method
(stability / instability)
P. Stránský, P. Hruška, P. Cejnar, Phys. Rev. E 79 (2009), 046202
…calculated from trajectories…
y
x
…calculated from trajectories…
We plot a point every time
when a trajectory crosses a
given line (y = 0)
vx
y
x
vx
chaotic case – “fog”
(hypersensitivity of the motion
on the initial conditions)
Section at
y=0
ordered case – “circles”
x
… and Poincaré sections
Coexistence of quasiperiodic (ordered) and chaotic types of motion
…calculated from trajectories…
We plot a point every time
when a trajectory crosses a
given line (y = 0)
vx
y
x
vx
chaotic case – “fog”
(hypersensitivity of the motion
on the initial conditions)
Section at
y=0
ordered case – “circles”
x
… and Poincaré sections
Coexistence of quasiperiodic (ordered) and chaotic types of motion
Fraction of regularity
Measure of classical chaos
Surface of the section covered
with regular trajectories
Total kinematically
accessible surface of the
section
vx
REGULAR area
CHAOTIC area
x
freg=0.611
Quasiperiodic X unstable trajectories
1. Lyapunov exponent
Divergence of two neighboring trajectories
Regular: at most
polynomial divergence
Chaotic: exponential
divergence
2. SALI (Smaller Alignment Index)
• two divergencies
• fast convergence towards zero for chaotic trajectories
Ch. Skokos, J. Phys. A: Math. Gen 34, 10029 (2001); 37 (2004), 6269
Stability (Application of the Geometric method)
Integrable limit
Integrable limit
deformed shape
Shape-phase transition
regularity”
chaotic
spherical shape
Veins of
regularity
regular
“Arc of
control
parameter
regular
Saddle point / local maximum
Israeli geometrical method
(stability / instability)
Eigenvalues of the stability matrix V
Potential well
V
Stability-instability transition, as
predicted by the Israeli method
(e)
(b)
(a)
(f)
(c)
A=-1
B=1.09
b
Low-energy
regular region
Convex-concave
transition
Local energy
maximum
Contact of the red
and blue regions
Kinematically accessible area
Negative lower eigenvalue of V
Negative higher eigenvalue of V
Saddle point of
the potential
y
Concave-convex
transition
Vein of
regularity
x
Stability
(by Poincaré sections) (a)
Potential well
V
(e)
(b)
(a)
(f)
(c)
A=-1
B=1.09
b
Low-energy region where the regular
harmonic approximation is valid
(b) Stable-unstable transition according to
the geometry criterion
(e) Local maximum of the potential – sharp
minimum of regularity
(h) “Regular vein” – strongly pronounced
local maximum of regularity
Black points – regular
Red points – chaotic trajectories
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
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
convex border - all eigenvalues positive on the border
Another example:
Completely convex border, but
unstable – a region of negative
eigenvalues of V inside
Relation to stability
concave potential
surface - dispersing
convex potential
surface - focusing
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
INSTABILITY
Deviations from the geometry criterion
Integrable limit
Integrable limit
deformed shape
Shape-phase transition
chaotic
spherical shape
Parametric
instability?
control
parameter
Saddle point / local maximum
regular
Israeli geometrical method
(stability / instability)
Concave – convex transition of
the border of the kinematically
accessible region
Conclusions:
1. 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.
2. This indicator, 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 is given in
X. Wu, J. Geom. Phys. 59, 1357 (2009).
However, the systems presented there are not bound.
3. 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.
Conclusions:
1. 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.
2. This indicator, 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 is given in
X. Wu, J. Geom. Phys. 59, 1357 (2009).
However, the systems presented there are not bound.
3. 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
Conclusions:
1. 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.
2. This indicator, 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 is given in
X. Wu, J. Geom. Phys. 59, 1357 (2009).
However, the systems presented there are not bound.
3. 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