introduction to kam theory - norbert wienervkaloshi/papers/lectures.pdfintegrable systems &...

Introduction to KAM theory

V. Kaloshin

July 3, 2013

V. Kaloshin (University of Maryland) Introduction to KAM theory July 3, 2013 1 / 20

The Global Plan

Integrable and nearly integrable systems, FPU paradox, KAMtheorem, Nekhoroshev stabilityWarm up: KAM theorem in low dimension, Newton’s method,Moser-Levi’s proof of KAMKAM theorem: Herman-Fejoz’s proof via an implicit functiontheoremApplication: stability of planetary motions, the semi-classicalasymptotics for Schrodinger operators

Plan of Lecture 1

Integrable integrable systems and action-angle coordinatesKey examples of nearly integrable systems: the standard map, theplanetary (N + 1) body problem, the Fermi-Pasta-Ulam model ofthe nonlinear spring.The Fermi-Pasta-Ulam paradox.KAM theorem: infinite time stability of most solutions.Nekhoroshev stability: exponentially long stability of all solutions.

Integrable systems & action-angles coordinates

Let (N2n, ω) be a symplectic manifold & H : N → R be a Hamiltonian.

Call F : N → R a first integral iff the Poisson bracket {F ,H} ≡ 0.

Functions F1 and F2 are in involution if their Poisson bracket is zero.

A Hamiltonian system is Arnold-Liouville integrable if there are nindependent first integrals in involution.

Arnold-Liouville integrability implies that on open setsU ⊂ T ∗Tn ∼= Tn × Rn 3 (θ, I) there exists a symplectic mapΦ : U ⊂ N 2n s. t. H ◦ Φ(I) depends only on I. In particular, Φ(U) isfoliated by invariant n-dim’l tori & on each torus Tn the flow is{

φ = ∂I(H ◦ Φ)(I) = ω(I),I = 0.

(φ, I)–action-angle coordinates

φ = ∂I(H ◦ Φ)(I) = ω(I),I = 0.

Action-angles coordinates

φ = ∂I(H ◦ Φ)(I) = ω(I),I = 0.

Integrable systems

Newtonial two body problem.

Pendulum H = I2

2 − cos 2πθ, (θ, I) ∈ T ∗T ' T× R.

Harmonic oscillator x = −kx , x ∈ R or in action-angleH = I2

2 , I2 = (x)2 + kx2, θ = arg(√

k x , x/√

k) (double check!).

Motion in a central force field x = F (‖x‖)x , x ∈ Rd .

Newtonian two center problem.

Lagrange’s top.

Toda lattice: chain of particles · · · < x0 < x1 < . . . with exponentialpotential

∑i exp(xi − xi+1)

A geodesic flow on an n-dimensional ellipsoid with differentprincipal axes.

A geodesic flow on a surface of revolution.V. Kaloshin (University of Maryland) Introduction to KAM theory July 3, 2013 6 / 20

Integrable systems

Pendulum H = I2

2 − cos 2πθ, (θ, I) ∈ T ∗T ' T× R.

2 , I2 = (x)2 + kx2, θ = arg(√

k x , x/√

k) (double check!).

Lagrange’s top.

Integrable systems

Pendulum H = I2

2 − cos 2πθ, (θ, I) ∈ T ∗T ' T× R.

2 , I2 = (x)2 + kx2, θ = arg(√

k x , x/√

k) (double check!).

Lagrange’s top.

Integrable systems

Pendulum H = I2

2 − cos 2πθ, (θ, I) ∈ T ∗T ' T× R.

2 , I2 = (x)2 + kx2, θ = arg(√

k x , x/√

k) (double check!).

Lagrange’s top.

Integrable systems

Pendulum H = I2

2 − cos 2πθ, (θ, I) ∈ T ∗T ' T× R.

2 , I2 = (x)2 + kx2, θ = arg(√

k x , x/√

k) (double check!).

Lagrange’s top.

Integrable systems

Pendulum H = I2

2 − cos 2πθ, (θ, I) ∈ T ∗T ' T× R.

2 , I2 = (x)2 + kx2, θ = arg(√

k x , x/√

k) (double check!).

Lagrange’s top.

Integrable systems

Pendulum H = I2

2 − cos 2πθ, (θ, I) ∈ T ∗T ' T× R.

2 , I2 = (x)2 + kx2, θ = arg(√

k x , x/√

k) (double check!).

Lagrange’s top.

Integrable systems

Pendulum H = I2

2 − cos 2πθ, (θ, I) ∈ T ∗T ' T× R.

2 , I2 = (x)2 + kx2, θ = arg(√

k x , x/√

k) (double check!).

Lagrange’s top.

Integrable systems

Pendulum H = I2

2 − cos 2πθ, (θ, I) ∈ T ∗T ' T× R.

2 , I2 = (x)2 + kx2, θ = arg(√

k x , x/√

k) (double check!).

Lagrange’s top.

Integrable systems

Pendulum H = I2

2 − cos 2πθ, (θ, I) ∈ T ∗T ' T× R.

2 , I2 = (x)2 + kx2, θ = arg(√

k x , x/√

k) (double check!).

Lagrange’s top.

Nearly integrable systems: the kicked rotor

The (Chirikov) standard map (θ, I)→ (θ′, I′) = (θ + I′, I + ε sin 2πθ). It isoften called the kicked rotator.

The kicked rotator approximates systems studied in mechanics ofparticles, accelerator physics, plasma physics, and solid state physics.

This is a perturbation of time-one map of the Hamiltonian H0(I) = I2

2 ofa Harmonic oscillator.

Nearly integrable systems: a planetary (N + 1) bodyproblem

A planetary (N + 1) body problem is a small perturbation of the sum ofN two body problems.

Let q0,q1, . . . ,qN ∈ R3 — points, m0 � m1, . . . ,mN > 0 — masses.

mi qi =m0mi(q0 − qi)

|q0 − qi |3+∑

j 6=i,0mjmi(qj − qi)

|q0 − qi |3, i > 0.

|q0 − qi |3+∑

|q0 − qi |3, i > 0.

|q0 − qi |3+∑

|q0 − qi |3, i > 0.

Nearly integrable systems: Fermi-Pasta-Ulam model

The harmonic oscillator u = −ku or H = p2

2 + ku2

2 , where u isdisplacement and V (u) = ku2

2 is the potential of interaction.

A nonlinear spring: u = −u−αu2 or H = p2

2 + u2

2 +αu3

3 , where u isdisplacement and V (u) = u2

2 + αu3

A nonlinear spring: chain 0 = u0 < · · · < ui < ui+1 < · · · < uN = 1

the potential of interaction is V (u) =∑

exp(ui − ui+1) =

(1 + (ui − ui+1) +

(ui − ui+1)2

(ui − ui+1)3

3!+ . . .

2 + u2

2 +αu3

2 + αu3

A nonlinear spring: chain 0 = u0 < · · · < ui < ui+1 < · · · < uN = 1

the potential of interaction is V (u) =∑

exp(ui − ui+1) =

(1 + (ui − ui+1) +

(ui − ui+1)2

(ui − ui+1)3

3!+ . . .

2 + u2

2 +αu3

2 + αu3

For small u we have a small perturbation of the (linear) harmonicoscillator.

Chain of particules 0 = u0 < · · · < ui < ui+1 < · · · < uN = 1

the potential of interaction is V(u) =∑

V (ui − ui+1) =

(ui − ui+1)2

2+α(ui − ui+1)3

3+ . . . local min at 0!

2 + u2

2 +αu3

2 + αu3

For small u we have a small perturbation of the (linear) harmonicoscillator.

Chain of particules 0 = u0 < · · · < ui < ui+1 < · · · < uN = 1

the potential of interaction is V(u) =∑

V (ui − ui+1) =

(ui − ui+1)2

2+α(ui − ui+1)3

3+ . . . local min at 0!

Fermi-Pasta-Ulam paradox

Suppose u0 = 0, uN = 1 (fixed ends). For i = k , . . . ,N − 1 we have

uk = (uk+1 − 2uk + uk−1) + α((uk+1 − uk )2 + (uk − uk−1)2)

The action-angle coordinates (or normal modes)

{(uk , uk )}N−1k=1 → {(θk , Ik )}N−1

k=1 , where Ik = (A2k + ω2

k A2k )/2,

N∑n=1

uk sinnkπ

N + 1, ωk = 4 sin2(kπ/(2N + 2)).

Fermi-Pasta-Ulam experiment: N = 32 or 64 and α =small. Mostsolutions turn out to be almost periodic.

Suppose u0 = 0, uN = 1 (fixed ends). For i = k , . . . ,N − 1 we have

The action-angle coordinates (or normal modes)

{(uk , uk )}N−1k=1 → {(θk , Ik )}N−1

k=1 , where Ik = (A2k + ω2

k A2k )/2,

N∑n=1

uk sinnkπ

N + 1, ωk = 4 sin2(kπ/(2N + 2)).

Fermi-Pasta-Ulam experiment: N = 32 or 64 and α =small. Mostsolutions turn out to be almost periodic.

Suppose u0 = 0, uN = 1 (fixed ends). For k = 1, . . . ,N − 1 we have

Initially, only mode A1 (blue) is excited. After flowing to other modes,A2 (green), A3 (red), etc., the energy almost fully returns to mode A1:this was a surprise!

Kolmogorov - Arnol’d - Moser Theorem (warm up)

Let H0(I) – integrable and Hε(θ, I) = H0(I) + εH1(θ, I) – a perturbation.

Kolmogorov, A. N. “On conservation of conditionally periodic motionsfor a small change in Hamilton’s function” Dokl. Akad. Nauk SSSR 98(1954), 527–530

KAM TheoremLet H0(I) be real analytic and nondegenerate, i.e. Hessiandet ∂2H0 6= 0. Let Hε(θ, I) be a real analytic perturbation. Then mostinitial conditions in an open bounded set Tn × B ⊂ Tn × Rn havequasiperiodic orbits filling an analytic n-dimensional torus. Inparticular, most solutions are bounded.

Kolmogorov - Arnol’d - Moser (KAM) Theorem

Let τ, γ > 0. ω ∈ Rn is called (τ, γ)-diophantine if for all k ∈ Zn \ 0 wehave |ω · k | ≥ γ|k |−(n−1)−τ . Denote this set Dτ,γ .

LemmaThere is c = c(n, τ) > 0 s. t. Leb (Dτ,γ ∩ B) ≥ (1− cγ) Leb(B).

KAM TheoremLet H0(I) be nondegenerate, i.e. Hessian det ∂2H0 6= 0. Let Hε(θ, I) bean analytic perturbation and ε small.

for each diophantine ω ∈ Dτ,γ the system Hε has an analytic invar.n-dim’l torus T n

ω with the flow on T nω conjugate to θ = ω.

The union of tori has measure 1−O(√ε) of the total measure.

Global normal form: There is a symplectic map Φε such thatKε(θ, I) = Hε ◦ Φε(θ, I) = K0(I) + exp(−Cε−1/%)R(θ, I),where % > 1 and R vanishes on Φ−1

ε (∪ω∈Dτ,γT nω ).

Nekhoroshev stability

Nekhoroshev Stability

Let H0(I) be nondegenerate, i.e. Hessian det ∂2H0 6= 0. Let Hε(θ, I) bean analytic perturbation. Then there is C = C(H0) > 0 such that foreach initial condition (θ0, I0) in a bounded set Tn × B ⊂ Tn × Rn fora solution (θ, I)(t) of Hε we have

‖I(t)− I0‖ ≤ Cε1/2n, ∀|t | ≤ C exp(Cε−1/2n).

Warning: For large dimensional systems exponential stability is not sopractical. For example, for the Solar system (the spacial 9 bodyproblem) 2n = 48.

RemarkKAM Theorem as well as Nekhoroshev stability hold for sufficientlysmooth systems with exponential stability replaced by polynomialstability.

‖I(t)− I0‖ ≤ Cε1/2n, ∀|t | ≤ C exp(Cε−1/2n).

Back to the Plan of Lecture 1

Integrable integrable systems and action-angle coordinates.Key examples of nearly integrable systems: the standard map, theplanetary (N + 1) body problem, the Fermi-Pasta-Ulam model ofthe nonlinear spring.The Fermi-Pasta-Ulam paradox: most solutions near equilibriumare almost periodic.KAM theorem: infinite time stability of most solutions.Nekhoroshev stability: exponentially long stability of all solutions.

The Global Plan

Integrable and nearly integrable systems, FPU paradox,KAM theorem, Nekhoroshev stabilityWarm up: KAM theorem in low dimension, Newton’s method,Moser-Levi’s proof of KAMKAM theorem: Herman-Fejoz’s proof via an implicit functiontheoremApplication: stability of planetary motions, the semi-classicalasymptotics for Schrodinger operators

introduction to kam theory - norbert wienervkaloshi/papers/lectures.pdfintegrable systems &...

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