Summary

1
Summary

• N-body problem
• Globular Clusters
• Jackiw-Teitelboim Theory
• Poincare plots
• Chaotic Observables
• Symbolic Dynamics
• Some quick math
• Different Orbits
• Conclusions

2
N-body problem

• Method of describing N-body gravitational
  interactions
• Only N2 is known in closed form (Newtonian)
• Ngt2 can only be approximated numerically
• In general relativity N2 is still not known in
  closed form
• Applications of this problem are quite necessary
  for cosmic study.

3
Globular clusters

• One mentioned application of N-body problem
• Newtonian system
• One defines a globular cluster as gravitationally
  bound concentrations of approximately 1E4 1E6
  stars within a volume of 10-100 light years radius

4
Relativistic 1D Self Gravitation (ROGS)

• This paper tackles ROGS' in 11 spacetime
• This models 31 spacetime using RT theory
• That is it includes dilaton theory
• This theory is consistent with nonrelativistic
  theory
• Also reduces to Jackiw-Teitelboim Theory

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Jackiw-Teitelboim Theory

• 2D action for gravity coupled to matter
• couples a dilatonic scalar field to the curvature

6
Poincare Section

• A surface of section as a way of presenting a
  trajectory of n-dimensional phase space in an
  (n-1)-dimensional space.
• One selects a phase element to be constant and
  plotting the values of the other elements each
  time the selected element has the desired value,
  an intersection surface is obtained.

7
Chaotic Observables

• Winding Number a method of tracking a
  trajectory around phase space
• If the winding number is not rational we have a
  chaotic orbit

8
Winding Number R

• The Winding Number is the average rotation angle
  per drive cycle.
• The black line in the above picture displays a
  winding number of 2/5, since it is rational the
  trajectory is periodic
• The winding number is defined as the asymptotic
  limit over the entire trajectory

9
Chaotic Observables

• Lyapunov Exponent
• The Lyapunov exponent (or index) measures the
  rate of divergence between a trajectory with 2
  different initial conditions

10
Lyapunov Exponent

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

l gt 0 Divergent l 0 Unchanging l lt 0 Convergent
11
Logistic Equation and Maps
L
12
Symbolic Dynamics

• A novel method of attempting to find periodic
  orbits
• One partitions the return map or poincare section
  and labels it appropriately
• Then one observes the location of the points
  during a cycle or orbit
• If the orbit is periodic or quasiperiodic one
  will receive a perfectly periodic set of symbols
  describing the trajectory

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Symbolic Dynamics
The partitioning of the return map
A resulting trajectory in symbol space LRLRRRRLR
14
Symbolic Dynamics
a 3.9 xo 0.30001
a 3.9 xo 0.29999
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Attractors

• Chaotic systems are said to have space filling
  trajectories
• These trajectories always fall on what are known
  as chaotic attractors
• It is a slice through one of these attractors
  which comprises the Poincare section

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Periodic, Quasi-Periodic, Chaotic

• Periodic orbits -- exactly repeat their
  trajectories with no deviations
• Quasi-Periodic orbits exhibit small to large
  deviations from a perfectly periodic trajectory
  however when looking at their symbolic dynamics
  they do exhibit periodic behaviour
• Chaotic orbits do not ever repeat themselves,
  they may come very close to repeating

17
Bifurcation Diagrams

• A simple test for chaos to exist occurs in
  bifurcation diagrams
• In regions where one finds single trajectories no
  chaos is expected

18
3-body ROGS with L

• No known nonrelativistic analogue
• L-- induces expansion or contraction of spacetime
  competing with gravitational self interaction
• Large and positive L overcomes gravity but
  ?loses causality?

19
EoM

• We start with the well known action

20
EoM
That leaves us with the following equations of
motion

• And the stress energy for the point masses

21
Some change of variables

• Using the ADM formalism, and canonical variables
  the action may be re-written as
• This leads to a longer set of first order field
  equations
• Then finally reducing the problem further we get
  a nice simple action with 2 constraint equations

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Conjugate Momenta

• With the Hamiltonian in the action we can
• calculate the conjugate momenta for the system
• p_i diff(L,x_i)
• Rearranging the canonical variables and
  corresponding conjugate momenta we have a system
  with sixfold symmetry (find this symmetry)
• Since Z is arbitrary (chooses a plane) and p_Z 0
  in the center of inertia frame, we are left with
  a 4D phase space

23
Potential Well

• The relativistic potential well is defined as the
  difference between the Hamiltonian and the
  relativistic kinetic energy
• For low momenta the potential wall becomes that
  of the non-relativistic system

24
Annulus Orbits

• Particles Never cross same bisector twice in
  succession (re-word)
• Their claim is periodic orbits are difficult to
  find
• Insert figure 4 and description on page 19

25
Pretzel Orbits

• Particles oscillate around a bisector
  corresponding to a stable or quasistable bound
  subsystem of 2 particles (classical analogue)
• Found Characteristsic are similar at different
  energies
• Change of cosmo const also changes these orbits
  as they did in the annulus case

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Chaotic Orbits

• Particles wander between A and B motion in an
  irregular fashion (direct quote)
• Poincare section shows dark regions
• Chaos exhibits space filling.
• Claims to occure in transition regions between
  annuli and pretzel orbits
• These depend strongly on cosmo const
• These orbits are hard to find due to sensitivity
  to IC's (no kidding)
• Increase/Decrease cosmo const expand/shrink phase
  space stretch
• Most significant change occurs when cosmo const
  goes negative

27
Conclusions

• Not much was really concluded, general
  relationships between the chaos exhibited and the
  cosmological constant were drawn, but nothing
  quanitative.

28
Comments

• Looking for periodic orbit theory one can
  easilydetermine full chaotic constants for the
  system.