HÉNON-HEILES HAMILTONIAN – CHAOS IN 2-DMODELING CHAOS & COMPLEXITY – 2008
YOUVAL DARUCD PHYSICS DEPT.
ABSTRACTChaos in two degrees of freedom (4 coordinates), demonstrated
by using the Hénon-Heiles Hamiltonian
INTRODUCTION
• In 1964, Michael Hénon and Carl Heiles were investigating the motion of a star around a galactic center
• Hamiltonian equations – typical physical problem
• We have energy conservation, so we do not want the phase space to contract
• Interesting dynamic with chaotic and non-chaotic regions
Introduction continued• The Hamiltonian governing this motion will have
three degrees of freedom (6 coordinates in phase space)
• Hénon and Heiles wanted to simplify the problem – Energy conservation – Angular momentum conservation (due to the
cylindrical symmetry) – Observation, motion is confined into two dimensions
• Chose to use a near elliptic motion Hamiltonian• Poincaré plots – as a tool to investigate dynamics
The Hamiltonian22
22
31 12 2 2
sin(3
3 )r ppH krm m
rr
222 32 21 ( )
2 21( )32
yxpp
H k x x yym
ym
2 2 2 2 321 ( 13
)2 x yH xp p x yy y
2 2
2
x
y
x
y
x py p
p x xy
p y x y
20, 0
10, , x is a func of E2
x xyx yor
x y
Fixed Points
Basic Analysis
0 0 1 00 0 0 1
1 2 2 0 02 1 2 0 0
Jy xx y
2 2( ) 1 4 4 4( ) 0
Det J x y yTr J
1,2
3,4
1 2 2
1 2 2
x y
x y
2 2 3
2
2( ) 1 4 4 , 0 23
1( ) 4 , 1/ 62
Det J y y with x E y y
Det J x with y E
Using the energy conservationFor x = 0 Det(J) = 1 only for y = 0For y = -1/2 |Det(J)|=1 for x = 0.25
Alarming since I expected the Det(J) to be 1 all the time
Sums to zero, can be only real or only imaginary
2 2 2 31 1( , ) ( 2 )2 3
V x y x y x y y
The Potential
• Show The Mathematica Potential plots• 3D Python• Poincaré - Python
Looking at the dynamics of the system using Poincaré, instead of LCE.
The extremum points I found were(0, 0), (0, 1), ( 3/2, -1/2), (- 3/2, -1/2)
Poincaré interpretation
• In the potential whale we can get both some regular orbit and chaotic once
• As we get closer to the saddle points line the percent of the chaotic region in phase space increases
• At the edges of the region we get elliptic orbits
• Chaos onset – around 1/9
2 3 22( 2 0)3xp E y y
More questions• Going back to the original star motion problem,
it would be interesting to know if stars are found in some typical energy or region of the phase space, and to make some Y vs. X plots
• Lyapunov Exponents analysis• Exploring more initial conditions• Use a different plane for the Poincaré plot• Improve program’s user interface, intersection
plane
