C����������� Dynamical S�� �����

CA L T EC H

Transport in HamiltonianSystems With Two or More

Degrees of Freedom

Shane D. RossControl and Dynamical Systems, Caltech

CDS 140, March 11, 2002

Overview

�Transport theory

� Time-independent Hamiltonian systems

� with 2 degrees of freedom

� with 3 (or N) degrees of freedom• Example: restricted three-body problem

� Some notes on computation

2

Chaotic Dynamics

3

Transport Theory

�Chaotic dynamics=⇒ statistical methods

4

Transport Theory

�Chaotic dynamics=⇒ statistical methods

�Transport theory

� Motion of ensembles of trajectories in phase space

� Asks: How long to move from one region to another?

� Determine transition probabilities, escape rates

4

Transport Theory

�Chaotic dynamics=⇒ statistical methods

�Transport theory

� Motion of ensembles of trajectories in phase space

� Asks: How long to move from one region to another?

� Determine transition probabilities, escape rates

� Applications:• Atomic ionization rates

• Chemical reaction rates

4

Transport Theory

�Chaotic dynamics=⇒ statistical methods

�Transport theory

� Motion of ensembles of trajectories in phase space

� Asks: How long to move from one region to another?

� Determine transition probabilities, escape rates

� Applications:• Atomic ionization rates

• Chemical reaction rates

• Comet transition rates

• Asteroid collision probabilities

4

Partition the Phase Space“Reactants” “Products”

5

Partition the Phase Space

�Systems with potential barriers• Electron near a nucleus

Nucleus

"Bound" "Free"

Potential Configuration Space(Rotating Frame)

6

Partition the Phase Space• Comet near the Sun and Jupiter

Ueff

JupiterSun

Potential Configuration Space(Rotating Frame)

7

Partition the Phase Space

�Partition is specific to problem

� We desire a way of describing dynamical boundariesthat represent the “frontier” between qualitativelydifferent types of behavior

8

Partition the Phase Space

�Partition is specific to problem

� We desire a way of describing dynamical boundariesthat represent the “frontier” between qualitativelydifferent types of behavior

�Example: motion of comet

� motion around the Sun

� motion around Jupiter

8

Statement of Problem� Suppose we study the motion on a manifold M� Suppose M is partitioned into disjoint regions

Ri, i = 1, . . . , NR,

such that

M =

NR⋃i=1

Ri.

� To keep track of the initial condition of a point, we saythat initially (at t = 0) region Ri is uniformly coveredwith species Si.

� Thus, species type of a point indicates the region inwhich it was located initially.

9

Statement of Problem� Statement of the transport problem:

Describe the distribution of speciesSi, i = 1, . . . , NR, throughout the regionsRj, j = 1, . . . , NR, for any time t > 0.

R1

R2

R3

R4

10

Statement of Problem� Some quantities we would like to compute are:

• Ti,j(t) = the total amount of species Si containedin region Rj at time t

• Fi,j(t) =dTi,j

dt (t) = the flux of species Si intoregion Rj at time t

R1

R2

R3

R4

11

Hamiltonian Systems

�Time-independent Hamiltonian H(q, p)

� N degrees of freedom

� Motion constrained to a (2N − 1)-dimensionalenergy surface ME corresponding to a valueH(q, p) = E = constant

� Symplectic area is conserved along the flow∮Lp · dq =

∫A

dp ∧ dq = constant

12

Symplectic Area Conserved

N∑i=1

σi

∫Ai

dpidqi = constant on an energy surface

13

Poincare Section� Suppose there is another (2N − 1)-dimensional

surface Q that is transverse (i.e., nowhere parallel) tothe flow in some local region.

� The Poincare section S is the (2N − 2)-dimensionalintersection of ME with Q.

14

Example for N = 2

�Circular restricted 3-body prob. (2D)

H =1

2((px + y)2 + (py − x)2) + U eff(x, y)

Sun Jupiter

y

x

CometRotatingFrame

Ueff

Position Space Effective Potential

15

3-Body Problem (2D)

�Look at fixed energy

-1 0 1

-1

0

1

-1 0 1

-1

0

1

-1 0 1

-1

0

1

-1 0 1

-1

0

1

S J S J

S JS J

Case 1 : E<E1 Case 2 : E

1<E<E

2

Case 4 : E3<E<E

4=E

5Case 3 : E

2<E<E

3

Position Space Projections

16

3-Body Problem (2D)

�Partition the energy surface

S JL1 L2

ExteriorRegion (X)

Interior (Sun) Region (S)

JupiterRegion (J)

ForbiddenRegion

Position Space Projection

17

3-Body Problem (2D)

�Look at motion near “saddle points”

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

x (nondimensional units, rotating frame)

y (nondimensional units, rotating frame)

S JL1

Zero velocity curve bounding forbidden

region

x (nondimensional units, rotating frame)xx

y (nondimensional units, rotating frame)

L2

(a) (b)

exteriorregion

interiorregion

Jupiterregion

forbiddenregion

L2

Position Space Projection

18

Potential Barriers� Hamiltonian systems with potential barriers give rise to

“saddle points” whose local form is given by

H(q, p) =ω

2(q2

1 + p21) + λq2p2, (1)

i.e., linearized vector field has eigenvalues ±iω, ±λ.

� Moser [1958] showed that the qualitative behavior of(1) carries over to the full nonlinear equations.

� In particular, the flow of (1) has form center × saddle.

19

Local Dynamics

� For fixed energy H = h, energy surface ' S2 × R.

� Other constants of motion: I1 = q21 + p2

1 and I2 = q2p2.

1p

q1

X

2p

q2

� Normally hyperbolic invariant manifold at q2 = p2 = 0,i.e.,

Mh =ω

2(q2

1 + p21) = h > 0.

Note that Mh ' S1, a periodic orbit.

20

Local Dynamics� Four cylinders of asymptotic orbits: the stable and

unstable manifolds W s±(Mh), W

u±(Mh).

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21

Transit and Nontransit Orbits� Cylinders separate transit from nontransit orbits.

� In three-body problem:

• These manifold tubes play an important role in what passesby Jupiter (transit orbits)

• and what bounces back (non-transit orbits)

• transit possible for objects “inside” the tube, otherwise notransit — this is important for transport issues

22

Tubes in the 3-Body Problem� Stable and unstable manifold tubes

• Control transport through the potential barrier.

x (rotating frame)

y (rotating frame)

L2 Periodic

OrbitStable

Manifold

Unstable

Manifold

Sun

Stable

Manifold

Forbidden Region

Unstable

Manifold

Jupiter

Forbidden Region

L2

23

Flux

�Tubes of transit orbits are the relevantobjects to study

� Tubes determine the flux between regions Fi,j(t).

� Note, net flux is zero for volume-preserving motion,so we consider the one-way flux.

� Example: FJ,S(t) = volume of trajectories that escapefrom the Jupiter region into the Sun region per unittime.

24

Transition Probabilities

�Fluxes give rates and probabilities

� Recently, Jaffe, Ross, Lo, Marsden, Farrelly, and Uzer[2002] computed the rate of escape of asteroids tem-porarily captured by Mars.

� Theory and numerical simulations agree well.

25

Transition Probabilities� Monte Carlo simultion (dashed) and theory (solid)

0 10 20 30 40Number of Periods

1

0.9

0.8

0.6

Surv

ival

Pro

babi

lity

26

Transition Probabilities

�More exotic transport between regions

� Look at the intersections between the interior of sta-ble and unstable tubes on the same energy surface.

� Could be from different potential barrier saddles.

Poincare

Section

Tube A

Tube B

27

Transition Probabilities� Example: Comet transport between outside and inside

of Jupiter

ExteriorRegion

InteriorRegion

JupiterRegion

ForbiddenRegion

StableManifold

UnstableManifold

JupiterSun

Rapid Transition

x (rotating frame)

y (rotating frame)

L2

(a) (b)

x (rotating frame)

y (rotating frame)

StableManifold

UnstableManifold

UnstableManifold

StableManifold

L2L1

CaptureOrbit

Jupiter

28

Transition Probabilities� Look at Poincare section intersected by both tubes.

� Choosing surface {x = constant; px < 0}, we look atthe canonical plane (y, py).

x (rotating frame)

y (

rota

ting f

ram

e)

L1 L2

Poincare

section

Stable�����e

Unstable�����e Unstable�����

e Slice

Stable�����e Slice

y

py

Position Space Canonical Plane (y, py)29

Transition Probabilities� Relative canonical area gives relative volume of orbits.

� Can be interpreted as the probability of transitionfrom one region to another.

Unstable�����e Slice

Stable�����e Slice

y

py Comet Orbits Passing �����

Exterior to Interior Region

Stable�����e Slice

Unstable��� �e Slice

Canonical Plane (y, py)30

Mixing� By keeping track of the intersections of the tubes, one

can describe the mixing of different regions (Ti,j(t)).

• It can get messy fast!

(from Jaffe, Farrelly and Uzer [1999])

31

Some Notes on Computation

�Computationally challenging!

�Periodic orbits

� high order analytic expansion (see Llibre et al., 1985)

� normal form theory

� numerical continuation (AUTO2000 software)

32

Some Notes on Computation

�Stable and unstable manifolds

� Suppose ODE in Rn of form

x = f (x)

with periodic solution x(t) of period T .

� The variational equations are linearized equationsfor variations δx about x:

δx(t) = Df (x(t))δx(t)

= A(t)δx(t),

where A(t) is an n× n matrix of period T .

33

Some Notes on Computation� Solutions are known to be of the form

δx(t) = Φ(t, 0)δx(0),

where Φ(t, 0) is the state transition matrix (STM)from time 0 to t.

� The STM along a reference orbit is computed bynumerically integrating n(n + 1) ODEs:

˙x = f (x),

Φ(t, 0) = A(t)Φ(t, 0),

with initial conditions:

x(0) = x0,

Φ(0, 0) = I6.34

Some Notes on Computation� The monodromy matrix Φ(T, 0) has an unstable

and stable eigenvector. We can numerically integratethis linear approximation to the unstable (or stable)direction to obtain the unstable (or stable) manifold.

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Some Notes on Computation

�Poincare Sections

� This set of solutions approximating the unstable mani-fold can be numerically integrated until some stoppingcondition is reached (e.g., xj = constant).

Poincare

Section

Tube

36

Some Notes on Computation� Problems:

How to handle non-transversal intersections

37

N = 3 or More

�Extend to N ≥ 3 degrees of freedom

� Near equilibrium point, suppose linearizedHamiltonian vector field has eigenvalues±iωj, j = 1, . . . , N − 1, and ±λ.

� Assume the complexification is diagonalizable.

� Hamiltonian normal form theory tranformsHamiltonian into a lowest order form:

H(q, p) =

N−1∑i=1

ωi

2

(p2

i + q2i

)+ λqNpN .

� Equilibrium point is of typecenter× · · · × center× saddle (N − 1 centers).

38

N = 3 or More

�Multidimensional “saddle point”

� For fixed energy H = h, energy surface ' S2N−2 × R.

� Constants of motion:Ij = q2

j + p2j, j = 1, . . . , N − 1, and IN = qNpN .

1q

1p N-1

p

qN-1

X X X

Np

qN

The N Canonical Planes

39

N = 3 or More� Normally hyperbolic invariant manifold

at qN = pN = 0,

Mh =

n−1∑i=1

ωi

2

(p2

i + q2i

)= h > 0.

Note that Mh ' S2N−3, not a single trajectory.

� Four “cylinders” of asymptotic orbits: the stable andunstable manifolds W s

±(Mh), Wu±(Mh), which have the

structure S2N−3 × R.

40

N = 3 or More� Transport between regions is mediated by the

“higher dimensional tubes”

� Compute fluxes, transition probabilities, etc.

Σ

41

N = 3 or More• Example: restricted three-body problem (3D)

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

−0.02

0

0.02

0.04

0.06

0.08

0.1−0.01

0

0.01

x

z

y

3D Position Space

42

Future Directions

�Future Directions

� Compute fluxes, transition probabilitiesin 2 and 3 degree of freedom systems

� Add small dissipation◦ Can Hamiltonian methods still be used?

� Determine statistical laws for astronomical systems◦ Over a range of energies

◦ Is ergodic assumption valid?

◦ Obtain useful asteroid collision probabilities, etc.

� Combine with control for spacecraft navigation

43

References• Jaffe, C., S.D. Ross, M.W. Lo, J. Marsden, D. Farrelly and T. Uzer [2002]

Statistical theory of asteroid escape rates Phys. Rev. Let., submitted.

• Koon, W.S., M.W. Lo, J.E. Marsden and S.D. Ross [2000] Heteroclinic connec-tions between periodic orbits and resonance transitions in celestial mechanics,Chaos 10(2), 427–469.

• Gomez, G., W.S. Koon, M.W. Lo, J.E. Marsden, J. Masdemont and S.D. Ross[2001] Invariant manifolds, the spatial three-body problem and space missiondesign. AAS/AIAA Astrodynamics Specialist Conference.

• Jaffe, C., D. Farrelly and T. Uzer [1999] Transition state in atomic physics,Phys. Rev. A 60(5), 3833–3850.

• Meiss, J.D. [1992] Symplectic maps, variational principles, and transport, Rev.Mod. Phys. 64(3), 795–848.

•Wiggins, S. [1992] Chaotic Transport in Dynamical Systems, Springer-Verlag.

•Wiggins, S. [1994] Normally Hyperbolic Invariant Manifolds in Dy-namical Systems, Springer-Verlag.

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