summary
DESCRIPTION
Summary. N-body problem Globular Clusters Jackiw-Teitelboim Theory Poincare plots Chaotic Observables Symbolic Dynamics Some quick math Different Orbits Conclusions. N-body problem. Method of describing N-body gravitational interactions Only N=2 is known in closed form (Newtonian) - PowerPoint PPT PresentationTRANSCRIPT
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.
Chaotic ObservablesChaotic Observables
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
Chaotic ObservablesChaotic Observables
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
Symbolic DynamicsSymbolic Dynamics
The partitioning of the return map
A resulting trajectory in symbol space LRLRRRRLR…
Symbolic DynamicsSymbolic Dynamics
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.