Summary

SummarySummaryN-body problemN-body problem

Globular Clusters Globular Clusters

Jackiw-Teitelboim TheoryJackiw-Teitelboim Theory

Poincare plotsPoincare plots

Chaotic ObservablesChaotic Observables

Symbolic DynamicsSymbolic Dynamics

Some quick mathSome quick math

Different OrbitsDifferent Orbits

ConclusionsConclusions

N-body problemN-body problemMethod of describing N-body Method of describing N-body

gravitational interactionsgravitational interactions

Only N=2 is known in closed form Only N=2 is known in closed form (Newtonian)(Newtonian)

N>2 can only be approximated N>2 can only be approximated numericallynumerically

In general relativity N=2 is still not In general relativity N=2 is still not known in closed formknown in closed form

Applications of this problem are quite Applications of this problem are quite necessary for cosmic study.necessary for cosmic study.

Globular clustersGlobular clusters

One mentioned application of N-body One mentioned application of N-body problemproblem

Newtonian systemNewtonian system

One defines a globular cluster as One defines a globular cluster as gravitationally bound concentrations gravitationally bound concentrations of approximately 1E4 – 1E6 stars of approximately 1E4 – 1E6 stars within a volume of 10-100 light years within a volume of 10-100 light years radiusradius

Relativistic 1D Self Relativistic 1D Self Gravitation (ROGS)Gravitation (ROGS)

This paper tackles ROGS' in 1+1 This paper tackles ROGS' in 1+1 spacetimespacetime

This models 3+1 spacetime using This models 3+1 spacetime using R=T theoryR=T theory

That is it includes dilaton theoryThat is it includes dilaton theory

This theory is consistent with This theory is consistent with nonrelativistic theorynonrelativistic theory

Also reduces to Jackiw-Teitelboim Also reduces to Jackiw-Teitelboim TheoryTheory

Jackiw-Teitelboim TheoryJackiw-Teitelboim Theory

2D action for gravity coupled to 2D action for gravity coupled to mattermatter

couples a dilatonic scalar field to the couples a dilatonic scalar field to the curvaturecurvature

MLRgxdS )(2

Poincare SectionPoincare SectionA surface of section as a way of A surface of section as a way of

presenting a trajectory of n-presenting a trajectory of n-dimensional phase space in an (n-1)-dimensional phase space in an (n-1)-

dimensional space.dimensional space.

One selects a phase element to be One selects a phase element to be constant and plotting the values of constant and plotting the values of the other elements each time the the other elements each time the selected element has the desired selected element has the desired value, an intersection surface is value, an intersection surface is

obtained.obtained.


Winding Number – a method of Winding Number – a method of tracking a trajectory around phase tracking a trajectory around phase

spacespace

If the winding number is not rational If the winding number is not rational we have a chaotic orbitwe have a chaotic orbit

Winding Number “R”Winding Number “R”

• The Winding Number is the average rotation The Winding Number is the average rotation angle per drive cycle.angle per drive cycle.

• The black line in the above picture displays The black line in the above picture displays a winding number of 2/5, since it is rational a winding number of 2/5, since it is rational the trajectory is periodicthe trajectory is periodic

• The winding number is defined as the The winding number is defined as the asymptotic limit over the entire trajectoryasymptotic limit over the entire trajectory


Lyapunov ExponentLyapunov Exponent

The Lyapunov exponent (or index) The Lyapunov exponent (or index) measures the rate of divergence measures the rate of divergence

between a trajectory with 2 different between a trajectory with 2 different initial conditionsinitial conditions

Lyapunov ExponentLyapunov Exponent

• The Lyapunov index measures the rate of The Lyapunov index measures the rate of divergence between a trajectory with 2 divergence between a trajectory with 2 different initial conditionsdifferent initial conditions

> 0 Divergent

= 0 Unchanging

< 0 Convergent

Logistic Equation and Logistic Equation and MapsMaps

Symbolic DynamicsSymbolic DynamicsA novel method of attempting to find A novel method of attempting to find

periodic orbitsperiodic orbits

One partitions the return map or One partitions the return map or poincare section and labels it poincare section and labels it

appropriatelyappropriately

Then one observes the location of the Then one observes the location of the points during a cycle or orbitpoints during a cycle or orbit

If the orbit is periodic or quasiperiodic If the orbit is periodic or quasiperiodic one will receive a perfectly periodic one will receive a perfectly periodic

set of symbols describing the set of symbols describing the trajectorytrajectory


The partitioning of the return map

A resulting trajectory in symbol space LRLRRRRLR…

Andrew Jason Penner

Symbolic dynamics started by a fellow named Hadamard. He was using this method to solve for allowed trajectories on spaces of negative curvature. Since then there have been several applications of this method to physical dynamical systems. The idea behind symbolic dynamics is that one can trace the path a trajectory takes in phase space, by the regions that it passes through. This has definite advantages to tracing the numerical coordinates of a trajectory especially in chaotic dynamics where a true representation of a trajectory would require infinite precision. The difficult part of this method lies in the determination of the partitions in phase space.

Andrew Jason Penner

The breaking of phase space can be made simple by the construction of a return map, a map that takes a coordinate of phase space and plots it against the same coordinate in the next evolutionary step. Using this map one could use the locat maxima and minima as partitions points, howver this method does not work all that well for systems with 2 dimensional tendencies, since these attractors tend to be double sheeted. If one were to use the local maxima and minima method, one may end up using more symbols than necessary to describe the system. This leads to difficulties in later calculations. The method of choice was the homoclinic points, where one maps the forward iterative map onto the attractor, and marks the points of tangency. This method ensures a minimal number of symbols used for a system.

Andrew Jason Penner

When calculating the symbolic dynamics of possible paths through symbol space the user only needs to produce a list of all possible permutations and combinations of the symbols used, to any period desired. Of course if we could leave the list like this, we would essentiually be saying that the trajectory of the system could go anywhere anytime. Physically we expect that this should not be the case, and we must consider a set of symbolic “limits” or “rules” that restrict the orbit in some way. These are referred to as fundamental forbidden zones. These are regions determined by a return map, that may never be entered by any part of a trajectory. These are determined in a 1D case by a forward iteration on the return map from a homoclinic point. For a 2D map one needs to also consider the possible reverse iterations (how did the trajectory get here points) that form the secondary restrictions to a symbolic dynamic orbit. Since these orbits are periodic one expect repetition in the orbit, and we find that we have to test all cyclic permutations of an orbit to determine its viability.


a = 3.9

xo = 0.30001

a = 3.9

xo = 0.29999

AttractorsAttractors

Chaotic systems are said to have Chaotic systems are said to have space filling trajectoriesspace filling trajectories

These trajectories always fall on what These trajectories always fall on what are known as chaotic attractorsare known as chaotic attractors

It is a slice through one of these It is a slice through one of these attractors which comprises the attractors which comprises the

Poincare sectionPoincare section

Periodic, Quasi-Periodic, Periodic, Quasi-Periodic, ChaoticChaotic

Periodic orbits -- exactly repeat their Periodic orbits -- exactly repeat their trajectories with no deviationstrajectories with no deviations

Quasi-Periodic orbits – exhibit small to Quasi-Periodic orbits – exhibit small to large deviations from a perfectly large deviations from a perfectly periodic trajectory however when periodic trajectory however when looking at their symbolic dynamics looking at their symbolic dynamics they do exhibit periodic behaviourthey do exhibit periodic behaviour

Chaotic orbits do not ever repeat Chaotic orbits do not ever repeat themselves, they may come very themselves, they may come very close to repeatingclose to repeating

Bifurcation DiagramsBifurcation Diagrams

A simple test for chaos to exist occurs A simple test for chaos to exist occurs in bifurcation diagramsin bifurcation diagrams

In regions where one finds single In regions where one finds single trajectories no chaos is expectedtrajectories no chaos is expected

3-body ROGS with 3-body ROGS with

No known nonrelativistic analogueNo known nonrelativistic analogue induces expansion or contraction induces expansion or contraction

of spacetime competing with of spacetime competing with gravitational self interactiongravitational self interaction

Large and positive Large and positive overcomes overcomes gravity but ?loses causality?gravity but ?loses causality?

EoMEoM

We start with the well known action:We start with the well known action:

EoMEoM

And the stress energy for the point And the stress energy for the point masses:masses:

That leaves us with the following That leaves us with the following equations of motion:equations of motion:

Some change of variablesSome change of variablesUsing the ADM formalism, and Using the ADM formalism, and

canonical variables the action may be canonical variables the action may be re-written as:re-written as:

This leads to a longer set of first order This leads to a longer set of first order field equationsfield equations

Then finally reducing the problem Then finally reducing the problem further we get a nice simple action further we get a nice simple action

with 2 constraint equationswith 2 constraint equations

11

00

2 ))(( RNRNtzxzpxdSa

aaa

Conjugate MomentaConjugate MomentaWith the Hamiltonian in the action we With the Hamiltonian in the action we

cancan

calculate the conjugate momenta for calculate the conjugate momenta for the system:the system:

p_i = diff(L,x_i)p_i = diff(L,x_i)

Rearranging the canonical variables Rearranging the canonical variables and corresponding conjugate and corresponding conjugate

momenta we have a system with momenta we have a system with sixfold symmetry (find this symmetry)sixfold symmetry (find this symmetry)

Since Z is arbitrary (chooses a plane) Since Z is arbitrary (chooses a plane) and p_Z =0 in the center of inertia and p_Z =0 in the center of inertia frame, we are left with a 4D phase frame, we are left with a 4D phase

spacespace

Potential WellPotential Well

The relativistic potential well is The relativistic potential well is defined as the difference between the defined as the difference between the

Hamiltonian and the relativistic Hamiltonian and the relativistic kinetic energykinetic energy

For low momenta the potential wall For low momenta the potential wall becomes that of the non-relativistic becomes that of the non-relativistic

systemsystem

Annulus OrbitsAnnulus Orbits

Particles Never cross same bisector Particles Never cross same bisector twice in succession (re-word)twice in succession (re-word)

Their claim is periodic orbits are Their claim is periodic orbits are difficult to find difficult to find

Insert figure 4 and description on Insert figure 4 and description on page 19page 19

Pretzel OrbitsPretzel Orbits

Particles oscillate around a bisector Particles oscillate around a bisector corresponding to a stable or corresponding to a stable or

quasistable bound subsystem of 2 quasistable bound subsystem of 2 particles (classical analogue)particles (classical analogue)

Found Characteristsic are similar at Found Characteristsic are similar at different energiesdifferent energies

Change of cosmo const also changes Change of cosmo const also changes these orbits as they did in the annulus these orbits as they did in the annulus

casecase

Chaotic OrbitsChaotic Orbits

Particles wander between A and B Particles wander between A and B motion in an irregular fashion (direct motion in an irregular fashion (direct

quote)quote)

Poincare section shows dark regionsPoincare section shows dark regions

Chaos exhibits space filling. Chaos exhibits space filling.

Claims to occure in transition regions Claims to occure in transition regions between annuli and pretzel orbitsbetween annuli and pretzel orbits

These depend strongly on cosmo These depend strongly on cosmo constconst

These orbits are hard to find due to These orbits are hard to find due to sensitivity to IC's (no kidding)sensitivity to IC's (no kidding)

Increase/Decrease cosmo const Increase/Decrease cosmo const expand/shrink phase space stretchexpand/shrink phase space stretch

Most significant change occurs when Most significant change occurs when cosmo const goes negativecosmo const goes negative

ConclusionsConclusions

Not much was really concluded, Not much was really concluded, general relationships between the general relationships between the

chaos exhibited and the cosmological chaos exhibited and the cosmological constant were drawn, but nothing constant were drawn, but nothing

quanitative.quanitative.

CommentsComments

Looking for periodic orbit theory one Looking for periodic orbit theory one can “easily”determine full chaotic can “easily”determine full chaotic

constants for the system.constants for the system.

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Summary