Laurette TUCKERMAN laurette@pmmh.espci.fr Hamiltonian Systems
Hamiltonian Systems H ( p 1 , . . . p N , q 1 , . . . q N ) q i = ∂ H p i = − ∂ H ˙ ˙ ∂p i ∂q i q i : positions p i : momenta convention for number of degrees of freedom: N (for dissipative systems, convention would be 2 N )
Conservation of volumes for ˙ x = f( x ) � V ( t + dt ) − V ( t ) = dt f · n da surface � = dt ∇ · f dv volume ∂ ˙ x i � � = dt dv ∂x i volume i � ∂ ˙ q i + ∂ ˙ p i � � � = dt dv ∂q i ∂p i volume i � ∂ 2 H ∂ 2 H � � � = dt − dv ∂q i ∂p i ∂q i ∂p i volume i = 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 = e z × ∇ ψ . Motion of particle at ( x, y ) : dx dt = u = − ∂ψ dy dt = v = ∂ψ ∂y ∂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 F j (p , q) such that dF j dt = 0 and � ∂F j ∂F k − ∂F j ∂F k � � [ F j , F k ] ≡ = 0 ∂q i ∂p i ∂p i ∂q i i Already have d H � ∂ H q i + ∂ H � ∂ H ∂ H − ∂ H ∂ H � � � � dt = ˙ p i ˙ = = 0 ∂q i ∂p i ∂q i ∂p i ∂p i ∂q i i i so any system with one degree of freedom is integrable,
� ∂F j ∂ H − ∂F j ∂ H � � [ F j , H ] = ∂q i ∂p i ∂p i ∂q i i � ∂F j q i + ∂F j � � = ˙ p i ˙ ∂q i ∂p i i = dF j dt so dF j dt = 0 = ⇒ [ F j , H ] = 0 N − 1 functions F j 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 I i = − ∂ H ′ θ i = ∂ H ′ ˙ ˙ = 0 = ω i (I) ∂θ i ∂I i I i ( t ) = I i (0) θ ( t ) = θ (0) + tω i Called twist map (I , θ ) are called action-angle variables
Classic pendulum H = 1 2 p 2 − cos q q : angle (position), p : (velocity = momentum) Phase portrait for the classic pendulum
I ≡ 1 � pdq 2 π Integral taken over a closed trajectory, on which H has the constant value H . For the pendulum: H = 1 2 p 2 − cos q p 2 = 2( H + cos q ) � p = 2( H + cos q ) � 2 π I = 1 � 2( H + cos q ) dq 2 π 0 = ⇒ 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 = d H /dI = 0 .
Non-integrable perturbations All Hamitonian systems with N = 1 are integrable Simplest non-integrable systems are of form: H ( p 1 , p 2 , q 1 , q 2 ) N = 2 H ( p, q, t ) sometimes called N = 1 . 5 Rotor in horizontal plane gets a “kick” with period τ = 2 π
H ( v, θ, t ) = v 2 � 2 + ǫ cos θ δ ( t − nτ ) n ˙ θ = ∂ H ∂v = v θ n +1 − θ n = v n mod 2 π v = − ∂ H ∂θ = ǫ sin θ � ˙ n δ ( t − nτ ) v n +1 − v n = ǫ sin θ n +1 H = H 0 ( θ, v ) + ǫ H 1 ( θ, v, t ) H 0 = v 2 2 H 0 = v 2 / 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 I j rotate with velocity I j Here I ∼ p ∼ v ∼ ω ∼ r
e or first return map arising from H 0 : Define the Poincar´ T 0 ( I, θ ) = ( I, θ )( t = 2 π ) = ( I, ( θ + 2 πI ) mod 2 π ) Each circle is invariant under T 0 , 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 T 0
Define n th iterate of T 0 . T n 0 ( I, θ ) = ( I, θ )( t = 2 πn ) = ( I, ( θ + 2 πn I ) mod 2 π ) If v = I = ω ( I ) = m/n , then θ + 2 π n m � � � � T n 0 ( I, θ ) = I, mod 2 π n = ( I, ( θ + 2 π m ) mod 2 π ) = ( I, θ ) so circles I = m/n consist of fixed points of T n 0 . Re-introduce the perturbation: H ǫ ≡ H 0 + ǫ H 1 and the corresponding maps T n ǫ ( I, θ ) ≡ ( I, θ )( t = 2 π n ) where I and θ evolve according to Hamiltonian H ǫ
Poincar´ e-Birkhoff Theorem ǫ on curve I = m Action of map T n 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 2 n 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 = 0 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 ) ≡ k →∞ T − k ( x ) = A x : lim � � W S ( B ) ≡ k →∞ T k ( x ) = B x : lim W U ( A ) = W S ( B ) intersect transfersely integrable H non-integrable H
⇒ W U ( A ) and W S ( B ) cross at C Non-integrable pert = 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 ǫ → 0 K ( ǫ ) = 0 lim and such that if w satisfies � � w − m � > K ( ǫ ) � � � ∀ m, n � � n 5 / 2 n 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 5 / 2 < w < m K K n − n + n 5 / 2 � ∞ ∞ ∞ 2 K ( ǫ ) 1 dx 1 � � n = 2 K ( ǫ ) n 3 / 2 ≤ 2 K n 5 / 2 x 3 / 2 n =1 n =1 n =1 � ∞ − 2 1 = 4 K ( ǫ ) � = 2 K ( ǫ ) < 1 for small ǫ x 1 / 2 3 3 1 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 ) r, µr 2 ˙ momentum p = ( p r , p φ ) = ( µ ˙ φ ) p 2 r + p 2 φ /r 2 H 0 (q , p) = | p | 2 2 µ − GMµ − GMµ = r 2 µ r p 2 dr dt = ∂ H 0 = p r dp r dt = − ∂ H 0 µr 3 − GMµ φ = r 2 ∂p r µ ∂r dφ dt = ∂ H 0 = p φ dp φ dt = − ∂ H 0 ∂φ = 0 µr 2 ∂p φ N = 2 degrees of freedom ( r, p r , φ, p φ ) angular momentum p φ conserved = ⇒ integrable
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