Hamiltonian Systems Hamiltonian Systems H ( p 1 , . . . p N , q 1 , - - PowerPoint PPT Presentation

Laurette TUCKERMAN laurette@pmmh.espci.fr

Hamiltonian Systems

Hamiltonian Systems

H(p1, . . . pN, q1, . . . qN) ˙ qi = ∂H ∂pi ˙ pi = −∂H ∂qi qi : positions pi : momenta convention for number of degrees of freedom: N (for dissipative systems, convention would be 2N)

Conservation of volumes for ˙ x = f(x) V (t + dt) − V (t) = dt

• surface

f · n da = dt

• volume

∇ · fdv = dt

• volume
• i

∂ ˙ xi ∂xi dv = dt

• volume
• i

∂ ˙ qi ∂qi + ∂ ˙ pi ∂pi

• dv

= dt

• volume
• i

∂2H ∂qi∂pi − ∂2H ∂qi∂pi

• dv

= 0

Developed in 18th century for celestial mechanics, now used in plasma physics (e.g. fields in a Tokamak) quantum systems (e.g. quantum optics, Bose-Einstein conden- sation) Fluid mechanics: 2D streamfunction ψ(x, y) u = ez × ∇ψ. Motion of particle at (x, y): dx dt = u = −∂ψ ∂y dy dt = v = ∂ψ ∂x Particles move along streamlines = curves of constant ψ ⇐ ⇒ In general, trajectories move along curves/surfaces of constant energy H

Integrable systems

Hamiltonian system with N degrees of freedom is integrable if ∃ N functions Fj(p, q) such that dFj dt = 0 and [Fj, Fk] ≡

• i

∂Fj ∂qi ∂Fk ∂pi − ∂Fj ∂pi ∂Fk ∂qi

• = 0

Already have dH dt =

• i

∂H ∂qi ˙ qi + ∂H ∂pi ˙ pi

• =
• i

∂H ∂qi ∂H ∂pi − ∂H ∂pi ∂H ∂qi

• = 0

so any system with one degree of freedom is integrable,

[Fj, H] =

• i

∂Fj ∂qi ∂H ∂pi − ∂Fj ∂pi ∂H ∂qi

• =
• i

∂Fj ∂qi ˙ qi + ∂Fj ∂pi ˙ pi

• = dFj

dt so dFj dt = 0 = ⇒ [Fj, H] = 0 N − 1 functions Fj are needed for an N-degree-of-freedom system to be integrable.

For an integrable system, there exists a transformation (I, θ) ← (p, q) H′(I, θ) = H(p, q) where, in fact, H′(I, θ) = H′(I) so that the dynamics in the (I, θ) variables are ˙ Ii = −∂H′ ∂θi = 0 ˙ θi = ∂H′ ∂Ii = ωi(I) Ii(t) = Ii(0) θ(t) = θ(0) + tωi Called twist map (I, θ) are called action-angle variables

Classic pendulum H = 1 2p2 − cos q q: angle (position), p: (velocity = momentum) Phase portrait for the classic pendulum

I ≡ 1 2π

• pdq

Integral taken over a closed trajectory, on which H has the constant value H. For the pendulum: H = 1 2p2 − cos q p2 = 2(H + cos q) p =

• 2(H + cos q)

I = 1 2π 2π

• 2(H + cos q) dq

= ⇒ I (canonical action variable) as a function of value of H (can be inverted to define H′ as function of I) Also define θ (canonical angle variable) such that ˙ θ = ω is constant in time

Left: H(I). Middle: pendulum configuration for H > 1 (repeated clock- wise or counterclockwise rotations) and H < 1 (small oscillations) Right: (q, p) trajectories, heteroclinic orbit at H = 1, ω = dH/dI = 0.

Non-integrable perturbations

All Hamitonian systems with N = 1 are integrable Simplest non-integrable systems are of form: H(p1, p2, q1, q2) N = 2 H(p, q, t) sometimes called N = 1.5 Rotor in horizontal plane gets a “kick” with period τ = 2π

H(v, θ, t) = v2 2 + ǫ cos θ

• n

δ(t − nτ) ˙ θ = ∂H

∂v = v

θn+1 − θn = vn mod 2π ˙ v = −∂H

∂θ = ǫ sin θ n δ(t − nτ) vn+1 − vn = ǫ sin θn+1

H = H0(θ, v) + ǫ H1(θ, v, t) H0 = v2 2 H0 = v2/2 is integrable and already in action-angle variables = ⇒ phase space = set of concentric curves, each with its own (constant) angular velocity.

Unperturbed Twist Map

Points on circle Ij rotate with velocity Ij Here I ∼ p ∼ v ∼ ω ∼ r

Define the Poincar´ e or first return map arising from H0: T0(I, θ) = (I, θ)(t = 2π) = (I, (θ + 2πI) mod 2π) Each circle is invariant under T0, but its individual points are not necessarily invariant. Circle v = 0 circle consists of fixed points, circle v = 1/2 consists of 2-cycles, circle v = 1/3 consists of 3-cycles. Five-cycles of map T0

Define nth iterate of T0. T n

0 (I, θ) = (I, θ)(t = 2πn) = (I, (θ + 2πn I) mod 2π)

If v = I = ω(I) = m/n, then T n

0 (I, θ) =

• I,
• θ + 2π n m

n

• mod 2π
• = (I, (θ + 2π m) mod 2π) = (I, θ)

so circles I = m/n consist of fixed points of T n

0 .

Re-introduce the perturbation: Hǫ ≡ H0 + ǫ H1 and the corresponding maps T n

ǫ (I, θ) ≡ (I, θ)(t = 2π n)

where I and θ evolve according to Hamiltonian Hǫ

Poincar´ e-Birkhoff Theorem

Action of map T n

ǫ on curve I = m n :

–The image of (I, θ) under T n

ǫ is

T n

ǫ (I, θ) = (I′, θ)

The radius changes but not the angle. –Curves (I, θ) and (I′, θ) intersect each other a multiple of 2n times, creating alternating hyperbolic and elliptic points. –Area inside (I′, θ) is the same as that inside (I, θ).

Dynamics of T q near a circle with rational I = p/q

Left: Points on intermediate circle are fixed, those on outer (inner) circle rotate counterclockwise (clockwise) Right: P-B theorem implies curves (I, θ) and T n

ǫ (I, θ) = (I′, θ) intersect

at alternating set of elliptic and hyperbolic fixed points. Angular flow is counterclockwise outside and clockwise inside and radial flow alternates inwards and outwards.

Each new elliptic point is now surrounded by invariant circles, some of which have rational winding numbers. Poincar´ e-Birkhoff theorem applies recursively to each one! Fixed points of T n = ⇒ elliptic or hyperbolic points of T n

ǫ

What happens to new hyperbolic points?

Unstable and stable manifold of hyperbolic A and B are points approaching them in iterating backwards or forwards: W U(A) ≡

• x : lim

k→∞ T −k(x) = A

• W S(B) ≡
• x : lim

k→∞ T k(x) = B

• W U(A) = W S(B) intersect transfersely

integrable H non-integrable H

Non-integrable pert = ⇒ W U(A) and W S(B) cross at C T k and T −k map C into other points, all in W U(A)∩W S(B) Infinite number of intersections accumulate at A and B Intersections along decreasing distances + area conservation = ⇒ perpendicular directions increase = ⇒ homoclinic tangles near A and B = ⇒ chaos = separation of nearby points = sensitivity to initial conditions (SIC)

From E. Weisstein, Homoclinic Tangle, MathWorld: A Wolfram Web Resource http://mathworld.wolfram.com/HomoclinicTangle.html

From P. So, Unstable periodic orbits, Scholarpedia 2(2): 1353

The combined complexity of the chains of elliptic and hyper- bolic points and the homoclinic tangles was said by Poincar´ e to be too complicated to describe. Arnold tried:

Solid ellipses: surviving tori, whose winding numbers are sufficiently far from any ratio-

• nal. Others break into alternating elliptic and hyperbolic points. Around each elliptic

point is a set of elliptical trajectories. Each hyperbolic point is surrounded by a chaotic

• region. From V.I. Arnol’d, Small denominators and problems of stability of motion in classi-

cal and celestial mechanics, Russian Mathematical Surveys 18:6, 85–191 (1963). Reprinted in Hamiltonian Dynamical Systems: a reprint collection, ed. R.S. MacKay & J.D. Meiss

KAM Theorem

Kolmogorov (1954), Arnold (1961-3), Moser (1962) Poincar´ e-Birkhoff Theorem: tori with rational winding numbers w are destroyed by non-integrable perturbation What about tori with irrational w? If the perturbation is sufficiently small, some survive. A torus whose w is close to a rational with small denominator, (“not very irrational”) is destroyed by a small perturbation. A torus whose w is sufficiently far from all rationals (i.e. is “very irrational”) requires a large perturbation to be destroyed. “Most irrational” number is golden mean (1 + √ 5)/2, whose torus is last one destroyed, i.e. perturbation required is largest.

∃K(ǫ) such that lim

ǫ→0 K(ǫ) = 0

and such that if w satisfies ∀m, n

• w − m

n

• > K(ǫ)

n5/2 then torus with winding number w survives pert of size ǫ Estimate measure of interval of w of surviving tori: Each denominator n corresponds to ∼ n rationals 1

n, 2 n, . . . n−1 n

Surrounding each rational is w-interval of destroyed tori

m n − K n5/2 < w < m n + K n5/2 ∞

• n=1

2K(ǫ) n5/2 n = 2K(ǫ)

∞

• n=1

1 n3/2 ≤ 2K ∞

n=1

dx 1 x3/2 = 2K(ǫ)

• −2

3 1 x1/2 ∞

1

= 4K(ǫ) 3 < 1 for small ǫ For small ǫ, set of surviving w has finite (non-zero) measure!

Celestial Mechanics: I. Two-body problem

Mass µ has position q = (r, φ) momentum p = (pr, pφ) = (µ ˙ r, µr2 ˙ φ) H0(q, p) = |p|2 2µ − GMµ r = p2

r + p2 φ/r2

2µ − GMµ r dr dt = ∂H0 ∂pr = pr µ dpr dt = −∂H0 ∂r = p2

φ

µr3 − GMµ r2 dφ dt = ∂H0 ∂pφ = pφ µr2 dpφ dt = −∂H0 ∂φ = 0 N = 2 degrees of freedom (r, pr, φ, pφ) angular momentum pφ conserved = ⇒ integrable

Three-body problem

Historically, main motivation for studying celestial mechanics has been to determine whether the solar system is stable. Even three-body problem is known to be non-integrable. Euler, Lagrange, Jacobi, Hill, Poincar´ e, Levi-Civita and Birkhoff studied circular restricted three-body problem (CRTBP): M ≫ m ≫ µ m has circular orbit around M. Neither is affected by µ.

M (sun) ≫ m (planet) ≫ µ (asteroid) Integrable two-body problem: µ moves under influence of M Non-integrable perturbation: m perturbs motion of µ µ small: –neglect effect of µ on m and M –identify center of mass of M − m system with center of M In addition, assume: –m rotates about M in circular orbit –µ remains in plane containing M and m

Integrable Hamiltonian system H0 without mass m H0(q, p, t) = p2

r + p2 φ/r2

2µ − GMµ r where pφ is the angular momentum µr2 ˙ φ dr dt = ∂H0 ∂pr = pr µ dφ dt = ∂H0 ∂pφ = pφ µr2 = ˙ φ dpr dt = −∂H0 ∂r = p2

φ

µr3 − GMµ r2 = µ2r4 ˙ φ2 µr3 − GMµ r2 = µr ˙ φ2 − GMµ r2 dpφ dt = ∂H0 ∂φ = 0 = ⇒ conserved quantity

Introduce intermediate mass m, at distance |q − Rm| from µ: H(q, p, t) = |p|2 2µ − GMµ r − Gmµ |q − Rm(t)| Mass m follows circular orbit = ⇒ Rm(t) changes orientation = ⇒ H is non-autonomous (depends explicitly on time) Autonomous H via rotating frame φ → φ′ but retain pφ: φ′ = φ − Ωt ∂H′ ∂pφ = dφ′ dt = dφ dt − Ω = pφ µr2 − Ω H′

0(q′, p) =

p2

r + p2 φ/r2

2µ − Ωpφ − GMµ r H′(q′, p) = p2

r + p2 φ/r2

2µ − Ωpφ − GMµ r − Gmµ |q′ − Rm| where Rm is now constant. Ωpφ is Coriolis term.

Action variable Iφ = 1 2π 2π pφ dφ = pφ Action variable Ir p2

r = 2µ(H0 + ΩIφ + GMµ/r) − I2 φ/r2

Ir = 1 2π

• prdr = −Iφ +

GMµ2

• −2µ(H0 + ΩIφ)

1 Ir + Iφ =

• −2µ(H0 + ΩIφ)

GMµ2 H0 = −ΩIφ − 1 2µ GMµ2 Ir + Iφ 2 ≡ H0(Ir, Iφ) Frequencies ω0r = ∂H0 ∂Ir = (GM)2µ3 (Ir + Iφ)3 ≡ ωµ ω0φ = ∂H0 ∂Iφ = −Ω + (GM)2µ3 (Ir + Iφ)3 = −Ω + ωµ

H′(q′, p) = p2

r + p2 φ/r2

2µ − Ωpφ − GMµ r − Gmµ |q′ − Rm| H′ depends on φ′ through q′ so dpφ dt = −∂H′ ∂φ′ = − ∂H′ ∂|q′ − Rm| ∂|q′ − Rm| ∂φ = 0 H′ non-integrable = ⇒ m destroys tori with rational w of H′ Winding number w is ratio ω0φ ω0r = −Ω + ωµ ωµ = 1 − Ω ωµ For M=sun and m=Jupiter, Poincar´ e-Birkhoff theorem = ⇒ Kirkwood gaps in frequencies (orbital paths) of asteroids µ For M=Saturn and m=moon of Saturn, P-B theorem = ⇒ gaps in the particles µ in the rings of Saturn

Kirkwood gaps

Gaps in the distribution of main belt asteroids as function of semi-major axis (equivalent to orbital period). They correspond to the location of orbital resonances with Jupiter. From Wikipedia, Kirkwood gap

Billiards

Generalized billiard table bounded by closed curve r(φ)

From H.J. Korsch & F. Zimmer, Chaotic Billiards, http://kluedo.ub.uni-kl.de/frontdoor.php?source opus=1202 M.V. Berry, Regularity and chaos in classical mechanics, illustrated by three deformations of a circular billiard, Eur. J. Phys 2, 91-102 (1981)

Trajectory = sequence of bounces = (φn, αn) Bounce = (angular position φ, angle α with tangent to table) (φn, αn) = ⇒ (φn+1, αn+1) Alternately, use arclength S along table and/or p = cos(α) (Sn, pn) = ⇒ (Sn+1, pn+1)

Circle r = c = ⇒ integrable dynamics since α is conserved Orbits are

• periodic

if α/2π is rational quasiperiodic if α/2π is irrational Elliptical billiards are also integrable.

Trajectories in elliptical billiards. From H.J. Korsch & F. Zimmer, Chaotic Billiards http://kluedo.ub.uni-kl.de/frontdoor.php?source opus=1202.

Stadium (Bunimovich) = two line segments connected by two semi-circles. Dynamics are ergodic: all points are visited.

Cosine Billiards

r(φ) = 1 + ǫ cos(φ) ǫ = 0 ǫ = 0.1 ǫ = 0.2 ǫ = 0.3 ǫ = 0: integrable circle case ǫ increases: rational tori break up, destroying irrational tori close to small-denominator fractions ǫ = ǫc: last torus, with w = w∗ golden mean, breaks up

Low-denominator fractions m/n with angles α

m/n α p = cos(α) m/n α p = cos(α) 1/2 90◦ 1/5 36◦ 0.809 1/3 60◦ 1/2 2/5 72◦ 0.309 2/3 120◦ −1/2 3/5 108◦ −0.309 1/4 45◦ 1/ √ 2 4/5 144◦ −0.809 3/4 135◦ −1/ √ 2 √ 2 127.279◦ −0.6057 w∗ = 0.618 291.262◦ 0.3626

ǫ = 0.1 ǫ = 0.2 p = cos α ǫ = 0.3 ǫ = 0.4 p = cos α S (arclength) S (arclength)

From H.J. Korsch & F. Zimmer, Chaotic Billiards http://kluedo.ub.uni-kl.de/frontdoor.php?source opus=1202.

Fluid dynamics

Fluid-dynamical streamfunction can play role of Hamiltonian: dx dt = u = −∂ψ ∂y dy dt = v = ∂ψ ∂x In fluid context, chaos can be desirable: promotes mixing Non-integrable perturbation: blinking vortex Vortex at (+a, 0) is switched on for time T then vortex at (−a, 0) is switched on for time T

• H. Aref: theory, 1984

Ottino: experiments, 1989

Rotate ∆θ around (+a, 0): x′ y′

• =

a

• +

cos ∆θ − sin ∆θ sin ∆θ cos ∆θ x − a y

• where ∆θ+(x, y) = κT/((x − a)2 + y2)

Rotate ∆θ around (−a, 0): x′′ y′′

• =

−a

• +

cos ∆θ − sin ∆θ sin ∆θ cos ∆θ x′ + a y′

• where ∆θ−(x′, y′) = κT/((x′ + a)2 + y′2)

Non-dimensional parameter controlling non-integrability: µ = κT/a2

Simulation of blinking vortex flow shows increasing degree of chaos as µ increases. From H. Aref, Stirring by chaotic advection, J. Fluid Mech. 143, 1–21 (1984).

Hamiltonian saddle-node bifurcation

center and saddle meet and annihilate δ = 0.2 δ = 0

• 0.2
• 0.1

0.1 0.2

• 0.075
• 0.05
• 0.025

0.025 0.05 0.075

• 0.2
• 0.1

0.1 0.2

• 0.075
• 0.05
• 0.025

0.025 0.05 0.075

• 0.2
• 0.1

0.1 0.2

• 0.075
• 0.05
• 0.025

0.025 0.05 0.075

• 0.4
• 0.2

0.2 0.4

• 0.002
• 0.001

0.001 0.002

q p q B p q p A C D

q ø

A B C

Q Q Q Q Q +

• Q-

+ +

• δ = 0.1

Φ(q)

Normal form: ¨ q = δ − q2 ⇐ ⇒

• ˙

q = p = ∂H

∂p

˙ p = δ − q2 = −∂H

∂q = −Φ′(q)

where H = p2 2 + Φ(q) = p2 2 + q3 3 − δq Fixed points are extrema of Φ p = 0 q = ± √ δ for δ > 0 Stability is determined by

• 1

−2q 0

• ⇐

⇒ λ(−λ) = 2q q = + √ δ ⇐ ⇒ −λ2 > 0 ⇐ ⇒ λ = ±iω ⇐ ⇒ q a center q = − √ δ ⇐ ⇒ −λ2 < 0 ⇐ ⇒ λ = ±σ ⇐ ⇒ q a saddle At δ = 0, a saddle and center are created simultaneously

Hamiltonian pitchfork bifurcation

Saddle ⇐ ⇒ Saddle-Center-Saddle (or Center ⇐ ⇒ Center-Saddle-Center)

µ = −1 µ = +1

H = v2 2 − Φ(u) = v2 2 − u4 4 + µu2 2 ¨ u = u3 − µu ⇐ ⇒      ˙ u = v = ∂H ∂v ˙ v = u3 − µu = −∂H ∂u = ∂Φ ∂u