Hamiltonian Systems
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Introduction
Hamilton’s Equations & the Hamiltonian
Phase Space
Constants of the Motion & Integrable Hamiltonians
Non-Integrable Systems, the KAM Theorem & PeriodDoubling
The Henon-Heiles Hamiltonian
The Chirikov Standard Map
The Arnold Cat Map
The Dissipative Standard Map
Applications
Introduction
• No dissipation = Conservative
– Phase space volume = constant
– No transients, no attractors
– Isolated (closed) system: Conservative
• Open system: Dissipative
• Hamiltonian system: Dynamics governed by H(q,p)
– Solar system:
• Gravitation: conservative
• Tidal forces, solar wind: dissipative
• Dissipation negligible for short times
– Microscopic (quantum) systems
• Integrable systems: non-chaotic
Hamilton’s Equations & the Hamiltonian
• Hamilton formulation: phase space = { qi, pi }
• Dynamics ( Hamilton’s equations ):
 H  q, p, t 
qi 
 pi
pi  
H
 qi
i  1, 2,
,Nf
Nf = degrees of freedom
E.g., n 3-D particles:
q = {qi} = { x1,y1,z1, x2,y2,z2, …, xn,yn,zn }
Nf = 3n
p = {pi} = { px1,py1,pz1, px2,py2,pz2, …, pxn,pyn,pzn }
Equivalent 1st order system of ODEs ( “DoF" = 2Nf ):

x  x j j  1,
,2 N f

H
x2i 1  f 2i 1  x  
 x2 i
x2i 1  qi
x2i  pi
H
x2 i  f 2 i  x   
 x2i 1
i  1,
,Nf
Symplectic structure: manifold with a (symplectic) 2-form as metric
For the Hamiltonian systems:   dH 
d f X f 
i
d
q
  dpi
i
  X g , X f    f , g  
j
  f g
g  f 



 q j  p j q j  p j 
(Poisson bracket)
Definition in terms of symplectic matrices ( see Goldstein ):
( M is symplectic if MT J M = J )
0
H
xJ
x
T
  f  g
f
,
g

    J
x
 x 
x  M x
I
J


I
0


 f
 f
df
f
  f H
 f H   f
 
qj 
pj  
 

 


dt

q

p

t

j 
 p j q j   t
j
j

j  q j  p j
f
  f , H 
t
d H H
H is conserved ( a constant of the motion )
→

if it’s not explicitly time dependent.
dt
t
Phase Space
For a conservative system, trajectories in phase space are
confined to a constant energy surface.
DoF = N → energy surface is (2N-1)-D
Volume in state space of autonomous system
1 dV
f
 i
V dt
i  xi
xi  fi  x 
See §3.13
For a conservative system H(q,p) :
 
1 dV
 
V dt
j 
 q j
 H  

 
  pj   pj
→ no attractors
 H

 q j

   0
 
→ transients persist
Liouville’s Theorem
Probability of finding a system in dV =
  q, p,t  dV
ρ= probability density of systems in phase space.
 
 
  H  H  
d

 
qj 
pj  
 

 



dt
 p j  t
 p j  q j  t
j   qj
j   qj  pj

  , H  
t
d
= material / hydrodynamic derivative → Lagrangian picture
dt

= change of ρ at fixed phase point → Eulerian picture
t
Liouville’s theorem:
d
0
dt
~ no-crossing theorem
for Hamiltonian systems
Integral Invariants
I t   

 t 
dx1
I t   
d I t 

dt
dxn   x, t 

dx J   x, t 
0
1
 0 

 0 
J  det J 
0
n
dx
0
1
dx
d I t 
0
dt
is an integral invariant if
d Jik  J
dJ
 xi i k

 0 J
d t i, k d t  J i k
i , k  xk
and Jik is the cofactor of Jik :
  x10
, xn 
xn0 
 x 
J  J i k    0i 
  xk 
d
dx
J  
dt
0
n
I(t) is an integral invariant →
 x1 ,
d J  d J
d

J
0
dt
dt
dt
d Jik
where
dt
J i k   J i m J
m
km

d   xi   xi
 0 0
d t   xk   xk

J
 J ik
 Jik
dJ
x
 xi  xm i k
 xi
 xi
ik
  0i J i k  
J

J
J

 i m J   x  J


mk
0
d t i , k  xk
i , k , m  xm  xk
i , k , m  xm
i , m  xm
d J  d J
d
 d

 


J
J
  x     J 
 x      x 
dt
dt
dt
 dt

 t

d I t 

dt

 0 
dx10
 

dxn0 J 
     x 
 t

I(t) is an integral invariant →


 t 
dx1
 

J
    x  
 t

 

dxn 
     x 
 t


     x  0
t
Liouville’s theorem:
 x  0
if ρ is the probability density of a Hamiltonian H(q,p,t).
(ρ acts like an incompressible fluid )
Proof:
J  t   det
q
q0
q
 p0
p
q0
p
 p0
I
J  t 
det
q0
t
q0
 p0
t
q0
q0
t
 p0
 p0
I
t
 p0
J  t 
J  t 
det
q
I  0 t
q0
 p0
t
q0
q0
t
 p0
I
 p0
t
 p0
1 



1 



1 
 det
  qi 0   pi 0 
i 1   q t 1   p t  
i0
i0



 q
p 
1    i 0  i 0  t 
 pi 0 
i   qi 0
  H
 H 
1  

 t 
 pi 0  qi 0 
i   qi 0  pi 0
→
dJ
 0    x  J
dt
→
 x  0
1  O  t 2 
&
d  

 x    0
dt
t
Alternatively:
 q p 
  H
 H 
 x   i  i   

 0
 pi 
 pi  qi 
i   qi
i   qi  pi
QED
In general:
1st order ODEs:

     x  0
t
xF
Hamiltonian systems
→

     F  0
t

 F    0
t
Equation is linear in ρ →
• Evolution of ρ can’t be chaotic, even when individual
trajectories are.
• No “divergence” of evolution of nearby ρ’s.
• No sensitivity to I.C.
• Prototype of chaotic Hamiltonian system: Arnold cat map.
Constants of Motion & Integrable Hamiltonians
Any quantity that is independent of t is a constant of the motion.
For a conservative system,
i.e.,
E  H  q, p is independent of t.
H q  t  , p  t    H  q 0 , p0 
t
 Each point on a possible trajectory {q(t),p(t)} has the same energy E.
 Many different trajectories may have the same energy.
 E is a constant of the motion.
 It is also an isolating integral that restricts the motion on
surface.
Let pj be a constant of the motion, then
0  pj  
i.e., H is not explicitly dependent on qj.
 Each trajectory is confined to an (2Nf-2)-D surface.
 Each trajectory can be characterized by { E, pj0 }.
some
H
q j
( qj is cyclic )
k independent isolating integrals
→ each trajectory is confined to an (2Nf-k)-D
surface
If k = Nf, the system is integrable. ( Trajectory Nf-D )
Isolating integrals are also called action variables Ji(q,p).
The variable conjugate to Ji(q,p) is called an angle variable.
Ji  
H
0
i
i 
H
 Ji
Θi can always be chosen as dimensionless so that
Ji has the dimension of action.
{ Θi, Ji } can be related to { q, p } by a canonical transformation.
If H can be written as H(J), the system is integrable.
Ji   Ji , H J   0
 i
can be satisfied iff
→ System is in involution.
 J , J  0
i
j
 i,j
( + independence )
Integrability can be examined by expressing the desired
canonical transformations in terms of a Birkhoff series.
Examples of integrable systems:
• All 1-D systems with analytic H → H = ωJ.
•
All systems with linear equations of motion → normal modes.
•
All systems that are completely separable.
•
Solitons
Let H = H(J), then
Inverse transformation:
i 
H
 i  J 
 Ji
→
qi  t   qi θ  t  , J  t  
i  t   i t  i  0
pi  t   pi θ  t  , J  t  
For a bounded system, q and p must be periodic functions of Θ.
Canonical perturbation theory:
Series diverges → non-integrable
q,p as series of Θ, J.
Simple Harmonic Oscillator
p2 1 2
H
 kq
2m 2
Nf = 1
→
q
Phase space: 2-D
dp p
k q


dq q
p/m
Fixed point:
→
H
p

p
m
2J
q t  
cos  (t )
m
p  t   2m J sin  (t )
H
 k q
q
m q  k q
→
Trajectory: 1-D (H = const → ellipse)
p d p  m k qd q
 q*, p *  0,0
Switching to (Θ,J):
p
p2 1
H
 m 2 q 2   J
2m 2
p
2m
k
m

= elliptic point
P
Ellipses:
periodic
1 2 1
p  mk q 2  C
2
2
Q
J
m
q
2
H

 P2  Q2
PQ 
1
pq
2
Trajectory is a circle of radius √J & area πJ in (P,Q) space
Q  t   J cos  (t )
P  t   J sin  (t )
H


J
  t   0   t
Area of ellipse:
 pd q   J d 
 2 J
E
1




 2mL g L   1  cos   
2mg L
2


p2
H
 mg L1  cos  
2
2mL
p
  2
p  mg L sin 
mL
 p

f 

 mgL sin    0
2
 mL  p
p   2mL2  E  mg L1  cos   
p
p

    sin 2
2
2mL g L
1
0.5
→ Conservative
4
3
2
2
-0.5
-1
3
4

0
J

 mgL cos 
g
 2   cos 
L
1 
mL2 

0 
 1 
v  c
2
  mL 
Trajectories in {θ, p } space for ε = 0.2, 0.6, 1.0, 1.4, 1.8
Elliptic points at
Hyperbolic points at
pθ = 0, θ = 2nπ
pθ = 0, θ = (2n+1)π
Separatrices: Stable & unstable manifolds of a hyperbolic point
Typical for integrable Hamiltonian systems
Elliptic Integrals & Elliptic Functions
Ref: M.Abramowitz, I.A.Stegun, “Handbook od Mathematical Functions”.
Complete Elliptic integrals:
(Incomplete) Elliptic integrals:

1st kind:
F  \ m   
0
2nd kind:


K m  F  \ m 
2

d
1  m sin 2 

E  \ m    d


E m  E  \ m 
2

1  m sin 
2
0
(Jacobian) Elliptic functions:
sin   sn  u | m  sn u
cos   cn u
u  F  \ m
  am u
1  m sin 2   dn u
Mathematica:
EllipticK[m] = EllipticF[π/2,m]
Sin[φ] = JacobiSN[EllipticF[φ,m],m]
Systems with N Degrees of Freedom
N
Integrable systems:
H  θ, J    i J i
i 1
Ji  
H
0
i
~ N uncoupled oscillators
i 
Q.M.:
equally
spaced
H
 i
 Ji
( simple harmonic if ω independent of J )
N constants of motion → trajectories on N-D torus in phase space
For N = 2: trajectories are on (invariant) torus
1 
H
 J1
2 
ωi incommensurate → q.p. → ergodic
→ time average = ensemble average
Non-integrable systems:
tori broken; J(θ) not lines
H
J2
The Kepler Problem (Integrable)
See H.Goldstein, "Classical Mechanics", 2nd ed., §10-7 ( with
minor variations )
2
p
pr2
k
H


2 2 r 2 r

m1m2
m1  m2
 r, p ,  , p    , J ,  , J 
r

1
1
2
2
Solution of the (separable) Hamilton-Jacobi eq gives
J 1  2 p
H  J1, J 2   
J 2   J1   k
2 2  k 2
 J1  J 2 
2
2
E
E
4 2  k 2
H

1 
 2
3
 J 1  J1  J 2 
→
1  2
k  Gm1m2
Nonintegrable Systems
Integrable systems: periodic / q.p.
→ non-chaotic
Transition to chaos: integrable → slightly non-integrable
Non-integrable systems: Df  2
For Df = 2, integrable → motion on 2-D torus
Non-integrable → motion on 3-D constant E surface
Poincare sections transverse to 3-D constant E surface is 2-D
Integrable system ( sets of nested tori with separatrices ):
• Series of discrete points → periodic
• Closed paths around point → quasi-periodic orbits around
elliptic point
• Hyperbolic orbits near hyperbolic points
Chaos ?
Henon-Heiles, almost integrable
KAM Theorem
Let
H  J, θ  H0  J    H1  J, θ
H0 integrable
KAM theorem (criteria dropped): Tori that survive perturbation satisfy
W
g  
m

n
n5
g(ε) increases monotonically with ε
Implication:
• For ε > 0, all tori with rational W break up
• (KAM) tori with irrational W persist, then break up 1 by 1 as ε
• Last to be destroyed has golden mean ratio ( most irrational )
Qualitative explanation:
• W rational → motion sustained by strong resonances between
overlapping harmonics
•
→ any perturbation will remove overlappings
•
→ rapid break up of tori
increases
Sequential break-up of KAM tori
Band of width g   
around m/n
n5
Tori within band dissolve
Resonance
structure
Chaos ~ overlap of resonances
Df = N → (2N-1)-D const E surface, N-D tori
A torus can partition E surface only if N = 2N-1 or (2N-1)-1
→ N = 1 or 2
( KAM tori partition phase space )
For N > 2, tori break up → stochastic web ( no partitioning )
Poincare-Birkhoff Theorem
H
H ~ twisted map (area preserving)
0
Break up of n/M tori
→ n/M pairs of 2n/2M
elliptic & hyperbolic fixed
points
→ Period-doubling
Insets & outsets of hyperbolic point dissolve first
→ homoclinic & heteroclinc tangles → chaos?
Different I.C. may lead to q.p./chaotic motion
For Df  2, surviving KAM tori confine chaotic motion
near broken tori
For Df  3, chaos from any broken torus can roam mostly
freely (Arnold diffusion).
Lyapunov exponent:
•
Σλi = 0 for conservative system
•
Chaos: at least 1 λi > 0
Monodromy matrix M :
 z  t   M   z  0
Eigenvalues μi of M ~ Floquet multipliers:
μi comes in pairs of (μ, μ-1 )
z = periodic orbit
Πμi = 1
i  e
i
Period-Doubling
Break up of m/n tori
• m/n pairs of 2m/2n elliptic & hyperbolic fixed points
• Period doubling
• ε increases → further period doubling …
• δH = 8.721097…, αH = -4.01807…
Period-n-tuplings are common in Hamiltonian systems
Cause: resonances among constituent nonlinear oscillators
Meyer's theorem: 5 types of bifurcations
Singularities in H can also cause non-integrability
E.g., billiard balls
Henon-Heiles Problem
A star in axially symmetric galaxy;
• Nf = 3
• Known integrals of motion: E, Lz
• No known analytic form of 3rd integral
• If 3rd integral not exist → σ(vρ)  σ(vz)
• Observed: σ(vρ) : σ(vz) = 2 : 1
Henon-Heiles model:
1 2 1 3
r  r sin 3
2
3
1
1
V  x, y   x 2  y 2  x 2 y  y 3
2
3
V  r,   


Henon-Heiles Hamiltonian
1 2
1 2
1 3
2
2
2
H   px  p y    x  y   x y  y
2
2
3
V
V
2
0.4
0.3
1.5
0.2
1
0.1
-2
-1
1
2
y
0.5
-0.1
-0.2
2
x
-2
0
1
-1
1
-1
0
-1
-2
2
1
-2
2
-2
7.5
-1
5
2.5
0
0
y
1
-2
-1
0
1
2
2
V
0.4
0.3
Height of potential well around 0 is 1/6
→ bound orbits for E < 1/6
1 2
1 2
1 3
2
2
2
H

p

p

x

y

x
y

y
Hamilton's eqs:
x
y
2
2
3
-2



H
x
 px
 px
H
px  
 x  2x y
x
H
y
 py
 py
py  
0.2
0.1
-1
1
-0.1
-0.2

H
  y  x2  y2
y
Nf = 2 → 3-D const E surface
→ 2-D Poincare section
→
E
1 2
1 2 1 3
2
p

p

y  y

x
y
2
2
3
Choice: y-py plane at x = 0
2
y
py
y
0.3
0.3
0.2
0.2
0.1
0.1
-0.3
-0.2
-0.1
0.1
0.2
0.3
x
-0.3
-0.2
-0.1
0.1
0.2
0.3
-0.1
-0.1
-0.2
-0.2
-0.3
-0.3
E = 0.06, x0 = 0, y0 = -0.1475, px0 = 0.3101, py0 = 0.
Distorted torus: Quasi-periodic
N = 1000
y
y
py
0.4
0.3
0.3
0.2
0.2
0.1
0.1
-0.3
-0.2
-0.1
0.1
0.2
0.3
x
-0.3
-0.2
-0.1
0.1
-0.1
-0.1
-0.2
-0.2
-0.3
-0.3
E = 0.06, x0 = 0, y0 = 0.1563, px0 = 0.18876, py0 = -0.25
Hyperbolic points:
•
separatrices
• Heteroclinic tangles → Stochastic layers (webs)
0.2
0.3
0.4
y
N = 2000
y
py
0.3
0.2
0.1
-0.3
-0.2
0.1
-0.1
0.2
0.3
0.4
-0.1
x
-0.2
-0.3
Outside separtrix → Qualitatively different
•
x-y orbits wraps y-axis
•
Bounds allowed region
E = 0.06, x0 = 0, y0 = 0, px0 = -0.0428, py0 = -0.3438
N = 200
y
0.4
0.3
0.2
0.2
0.1
-0.3
-0.2
-0.1
0.1
0.2
0.3
0.4 -0.4
-0.2
0.2
-0.1
-0.2
-0.2
-0.3
E = 0.06
-0.4
N = 10000
E = 0.1
•
Orbits near separatrices easily disturbed
•
Breakup of KAM torus → necklace ----
(Birkhoff thm)
( Associated hyperbolic points not shown )
•
Remaining KAM tori block chaotic roaming
N = 40000
0.4
0.6
0.4
0.4
0.2
-0.4
-0.2
0.2
0.2
0.4
0.6
-0.4
-0.2
0.2
0.4
0.6
-0.2
-0.2
-0.4
-0.4
-0.6
E = 0.14
N = 40000
E = 0.166
N = 40000
•
Single trajectory roams through most places
•
Lyapunov exponents: +,0,0,-
0.8
The Chirikov Standard Map
Aliases: Taylor-Greene-Chirikov map, the standard map.
K
sin 2 n mod 1
2
n1  n  rn1 mod 1
K
  n  rn 
sin 2 n mod 1
2
rn 1  rn 
(Moser) Twist map:
n1  n
n1  n  W  n  mod 1
W = winding number
J
1
0
W 1
1
area preserving
J
1
 K cos 2
1 1  K cos 2
area preserving
1
Fixed points ( with r → J ):
K
→
Jn 
sin 2 n  J n  m
2
K
n  J n 
sin 2 n   n  p →
2
m = 0 and θ* = 0, ½
K < 2π →
Near fixed point:
  rn1  1  K cos 2 *    rn 
    1 1  K cos 2 *    
 n 
 n1  
Floquet multipliers:
1   
2
2 m
m,p = integers
sin 2 * 
K
K
J* 
sin 2 *  p  m  p  0, 1
2
1 
 K cos 2 *
1
1  K cos 2 *  
0
 1    K cos 2 *  K cos 2 *  0
1
1  
K cos 2 *  K 2 cos 2 2 * 4 K cos 2 * 

2
1
  1   K cos 2 *  K 2 cos2 2 * 4 K cos 2 * 
2

1
  1  K  K 2  4K 

2
For (J*,θ*) = (0,0):
Reμ
|μ|
Imμ
1
2
1
1.75
1
2
3
4
5
6
1.5
0.5
1.25
-1
1
3
-2
4
5
6
0.75
-0.5
0.5
-3
0.25
-1
1
2
3
4
5
→ (0,0) is a stable spiral for K < 4
1
  1   K  K 2  4K 
For (J*,θ*) = (0, ½ ):
2

8
6
μ
→ (0, ½) is a saddle point for all K
4
2
1
2
3
4
5
6
6
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.54
0.2
0.4
0.6
0.8
0.2
1
0.4
0.6
0.8
1
0.52
K = 0
K = 0.2
0.5
c.f. Henon-Heiles
0.48
θ = 0, J = ½
Period 2
0.46
0.2
0.4
0.6
0.8
1
The Arnold Cat Map
xn 1  2 xn  yn mod 1
J
yn 1  xn  yn mod 1
 2   1     1  0
2 1
1 1
  3  1  0
2
1
→ Area-preserving

1
  3 5
2
→ All fixed points are saddle points
Fixed points:
x *  y*  m
x* k
 xn 1   2 1  xn 
 y    1 1  y  mod 1
 n 
 n1  
 2 1  1   2 
 1 1  0    1 

   
 2 1   0   1
 1 1   1    1

   
→
y* 0
x* 0

2.618
0.382
→
↓
←
Ex 8.8-1: Fixed points of f(n) have rational coordinates
Evolution of 1-D conservative system:
d
0
dt
→
  qn , pn , n     q0 , p0 ,0
pn 1  2 pn  qn mod 1
qn 1  pn  qn mod 1
( n ~ tn )
  qn , pn , n    Ajnk exp 2 i  jqn  k pn  
Fourier analysis:
jk
  qn 1 , pn 1 , n  1   Aj k
 n 1
Aj k   A jk
exp 2 i  jqn 1  k pn 1  
n *
n
jk
  Ajnk1 exp 2 i  j  qn  pn   k  2 pn  qn  
jk
  Ajnk1 exp 2 i  j  k  qn  j  2k  pn 
jk
j  j  k
k   j  2k
j  2 j  k 
k  k   j
  qn 1 , pn 1 , n  1   A2 nj1k, k  j exp 2 i  jqn  k pn 
   qn , pn , n 
j k 
→
 n 1
n
A2 j k , k  j  Aj , k
 n 1
n 
Aj , k  Aj  k , j  2 k
Spread to new modes,
But not IC sensitive
The Dissipative Standard Map
G.Schmidt, B.H.Wang, PR A 32, 2994 (85)
rn 1  J D rn 
K
sin 2 n mod 1
2
n1  n  rn1 mod 1
K
  n  J D rn 
sin 2 n mod 1
2
JD
 K cos 2
1
1  K cos 2
JD = 0 :
 JD
Dissipative for JD < 1
K
 n 1   n 
sin 2 n mod 1
2
( Sine circle map with Ω = 0)
1
JD = 0 ~ Circle map
0.8
0.6
0.4
0.2
1
2
3
4
5
JD = 0 , K > 1 → period-doubling route to chaos beyond K∞
Channels overlap for JD > 0
Bifurcated 2n orbits ~
period-doubling 2n : p1, p2
periodic orbits ~
periodic windows: p2', p3'
2n chaotic bands disappear at
univeral values of JD
Applications
• Billiards (elastic collisions, piecewise linear)
–
–
–
–
Rectangular or circular walls → periodic / q.p.
stadium / Sinai billiards ( round obstacle ) → chaotic for some orbits
2 balls + gravity: All kinds of behavior
Quantum chaos
• Astronomical Dynamics
– Orbits of Pluto & some asteroids may be chaotic
– Kuiper objects
• Particle Accelerators
– Avoid possible chaotic trajectories in accelerator design
• Superconductivity
– Vortex structures under magnetic field (type II superconductors) phaselocking, Arnold tongues, Farey tree, devil's staircase
• Optics
– small dielectric spheres: whispering gallery lasers
– Spheres distorted → chaos