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Stability results in Celestial Mechanics: from perturbation theory to KAM theorem Alessandra Celletti Department of Mathematics University of Rome Tor Vergata 13 March 2019 Outline 1. Celestial Mechanics and Perturbation theory 2. KAM theory

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Stability results in Celestial Mechanics: from perturbation theory to KAM theorem

Alessandra Celletti

Department of Mathematics University of Rome Tor Vergata

13 March 2019

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Outline

  • 1. Celestial Mechanics and Perturbation theory
  • 2. KAM theory
  • 3. Symplectic/Conformally symplectic systems
  • 4. Some KAM applications to Celestial Mechanics
  • 5. Conclusions and perspectives
  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 2 / 43

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Celestial Mechanics

  • Celestial Mechanics studies the dynamics of natural and artificial object os

the Solar system.

Aristotle 384-322 AC Hypparcus 190-120 AC Ptolemy 85-165 Copernicus 1473-1543 Tycho Brahe 1546-1601 Galileo 1564-1642 Kepler 1571-1630 Newton 1642-1727 Laplace 1749-1827 Lagrange, Gauss, Delaunay, XIX century Poincarè 1854-1912 Kolmogorov, Arnold, Moser, Nekhoroshev XX century Celestial spheres epicycles epicycles/deferents heliocentric system

  • bservations

scientific method 2-body problem gravitation planetary motions Perturbation theories 3-body problem Stability theories

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 3 / 43

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Celestial Mechanics

  • Celestial Mechanics studies the dynamics of natural and artificial object os

the Solar system.

Aristotle 384-322 AC Hypparcus 190-120 AC Ptolemy 85-165 Copernicus 1473-1543 Tycho Brahe 1546-1601 Galileo 1564-1642 Kepler 1571-1630 Newton 1642-1727 Laplace 1749-1827 Lagrange, Gauss, Delaunay, XIX century Poincarè 1854-1912 Kolmogorov, Arnold, Moser, Nekhoroshev XX century Celestial spheres epicycles epicycles/deferents heliocentric system

  • bservations

scientific method 2-body problem gravitation planetary motions Perturbation theories 3-body problem Stability theories

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 3 / 43

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2-body problem

  • 2-body problem, e.g. Sun-Earth, is governed by Kepler’s laws, according to

which a planet moves around the Sun on an ellipse with the Sun in one focus ⇒ integrable problem ⇒ Hamiltonian formulation in action-angle coordinates: H(J, ϕ) = Z(J) , J ∈ Rn , ϕ ∈ Tn , whose Hamilton’s equations are:    ˙ J = −

∂H(J,ϕ) ∂ϕ

= − ∂Z(J)

∂ϕ

= 0 ⇒ J = J0 ˙ ϕ =

∂H(J,ϕ) ∂J

= ∂Z(J)

∂J

≡ ω(J) ⇒ ϕ = ω(J0)t + ϕ0 .

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 4 / 43

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2-body problem

  • 2-body problem, e.g. Sun-Earth, is governed by Kepler’s laws, according to

which a planet moves around the Sun on an ellipse with the Sun in one focus ⇒ integrable problem ⇒ Hamiltonian formulation in action-angle coordinates: H(J, ϕ) = Z(J) , J ∈ Rn , ϕ ∈ Tn , whose Hamilton’s equations are:    ˙ J = −

∂H(J,ϕ) ∂ϕ

= − ∂Z(J)

∂ϕ

= 0 ⇒ J = J0 ˙ ϕ =

∂H(J,ϕ) ∂J

= ∂Z(J)

∂J

≡ ω(J) ⇒ ϕ = ω(J0)t + ϕ0 .

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 4 / 43

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SLIDE 7

3-body problem

  • 3-body problem, e.g. Sun-Earth-Jupiter, is governed by a nearly-integrable

system: H(J, ϕ) = Z(J)

  • Sun-Earth

+ε R(J, ϕ)

Earth-Jupiter

, ε = mJupiter mSun ≃ 10−3

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 5 / 43

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3-body problem

  • 3-body problem, e.g. Sun-Earth-Jupiter, is governed by a nearly-integrable

system: H(J, ϕ) = Z(J)

  • Sun-Earth

+ε R(J, ϕ)

Earth-Jupiter

, ε = mJupiter mSun ≃ 10−3 whose Hamilton’s equations are: ˙ J = −ε∂R(J, ϕ) ∂ϕ ˙ ϕ = ω(J) + ε∂R(J, ϕ) ∂J

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 5 / 43

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3-body problem

  • 3-body problem, e.g. Sun-Earth-Jupiter, is governed by a nearly-integrable

system: H(J, ϕ) = Z(J)

  • Sun-Earth

+ε R(J, ϕ)

Earth-Jupiter

, ε = mJupiter mSun ≃ 10−3 whose Hamilton’s equations are: ˙ J = −ε∂R(J, ϕ) ∂ϕ ˙ ϕ = ω(J) + ε∂R(J, ϕ) ∂J WHICH IS THE SOLUTION?

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 5 / 43

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3-body problem

  • 3-body problem, e.g. Sun-Earth-Jupiter, is governed by a nearly-integrable

system: H(J, ϕ) = Z(J)

  • Sun-Earth

+ε R(J, ϕ)

Earth-Jupiter

, ε = mJupiter mSun ≃ 10−3 whose Hamilton’s equations are: ˙ J = −ε∂R(J, ϕ) ∂ϕ ˙ ϕ = ω(J) + ε∂R(J, ϕ) ∂J WHICH IS THE SOLUTION? PERTURBATION THEORY

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 5 / 43

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Classical perturbation theory

  • Approximates solution by pushing the perturbation to higher orders in ε.

Theorem

Let H(J, ϕ) = Z(J) + εR(J, ϕ) with (J, ϕ) ∈ V × Tn for V ⊂ Rn open: ◮ R analytic and trigonometric (N Fourier coefficients) on V × Tn; ◮ non-resonance frequency for any J0 ∈ V: |ω(J0) · k| > 0 for all 0 < |k| ≤ N . Then, there exists ρ0 > 0, ε0 > 0 and for |ε| < ε0 there exists a canonical transformation (J, ϕ) → (J′, ϕ′) defined in S ρ0

2 (J0) × Tn ⊂ V × Tn and with

values in Sρ0(J0) × Tn, which transforms H as H′(J′, ϕ′) = Z′(J′) + ε2R′(J′, ϕ′) .

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 6 / 43

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Classical perturbation theory

  • Approximates solution by pushing the perturbation to higher orders in ε.

Theorem

Let H(J, ϕ) = Z(J) + εR(J, ϕ) with (J, ϕ) ∈ V × Tn for V ⊂ Rn open: ◮ R analytic and trigonometric (N Fourier coefficients) on V × Tn; ◮ non-resonance frequency for any J0 ∈ V: |ω(J0) · k| > 0 for all 0 < |k| ≤ N . Then, there exists ρ0 > 0, ε0 > 0 and for |ε| < ε0 there exists a canonical transformation (J, ϕ) → (J′, ϕ′) defined in S ρ0

2 (J0) × Tn ⊂ V × Tn and with

values in Sρ0(J0) × Tn, which transforms H as H′(J′, ϕ′) = Z′(J′) + ε2R′(J′, ϕ′) .

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 6 / 43

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Classical perturbation theory

  • Approximates solution by pushing the perturbation to higher orders in ε.

Theorem

Let H(J, ϕ) = Z(J) + εR(J, ϕ) with (J, ϕ) ∈ V × Tn for V ⊂ Rn open: ◮ R analytic and trigonometric (N Fourier coefficients) on V × Tn; ◮ non-resonance frequency for any J0 ∈ V: |ω(J0) · k| > 0 for all 0 < |k| ≤ N . Then, there exists ρ0 > 0, ε0 > 0 and for |ε| < ε0 there exists a canonical transformation (J, ϕ) → (J′, ϕ′) defined in S ρ0

2 (J0) × Tn ⊂ V × Tn and with

values in Sρ0(J0) × Tn, which transforms H as H′(J′, ϕ′) = Z′(J′) + ε2R′(J′, ϕ′) .

  • The proof is constructive and allows us to obtain

H′ = Z′ + ε2R′, H′′ = Z′′ + ε3R′′, H′′′ = Z′′′ + ε4R′′′, etc.

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 6 / 43

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Normal form with Lie series: constructive proof

Proof.

  • Canonical transformation (ϕ′, J′) = (eLχ ϕ, eLχ J), to transform

H = Z + εR = Z + ε(R + R) with R =average, R =non average: in H′(J′, ϕ′) = Z(J′) + εR(J′) + ε2R′(J′, ϕ′) , where Z′ = Z + εR is the normal form, Lχ ≡ {·, χ} is the Poisson bracket

  • perator, eLχ = ∞

k=0 1 k!Lk χ. Then:

H′ = eLχH = H + LχH + 1 2L2

χH + ...

= Z + εR + {H, χ} + 1 2L2

χH + ...

= Z + εR + {Z, χ} + ε{R, χ} + ... = Z + εR + ε R + {Z, χ} + ε{R, χ} + ε{ R, χ} + ... and we find χ (O(ε)) by solving the normal form equation: ε R + {Z, χ} = 0

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 7 / 43

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Normal form with Lie series

{Z, χ} = −ε R {Z, χ} =

✁ ✁ ✁ ❆ ❆ ❆

∂Z ∂ϕ ∂χ ∂J − ∂Z ∂J ∂χ ∂ϕ = −∂Z ∂J ∂χ ∂ϕ = −ω(J) ∂χ ∂ϕ= −ε R

  • Expand

R(J, ϕ) =

k=0

Rk(J)eik·ϕ and χ(J, ϕ) =

k

χk(J)eik·ϕ: −i

  • k

ω(J′) · k χk(J′)eik·ϕ′ = −ε

  • k=0, |k|≤N
  • Rk(J′)eik·ϕ′ ,

giving the solution for χ:

  • χk(J′) = ε
  • Rk(J′)

iω(J′) · k , where ω(J′) · k are the small divisors .

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 8 / 43

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Resonant normal form

  • A resonance occurs when ω(J) ·

k = 0 .

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 9 / 43

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Resonant normal form

  • A resonance occurs when ω(J) ·

k = 0 .

  • A resonant perturbation theory can be implemented to eliminate the

non–resonant terms.

  • Construct a change of variables C : (J, ϕ) → (J′, ϕ′), such that the new

Hamiltonian takes the form H′(J′, ϕ′) = Z′(J′, k · ϕ′) + ε2R′(J′, ϕ′) , where Z′ depends on ϕ′ only through the combinations k · ϕ′.

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 9 / 43

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Resonant normal form

  • A resonance occurs when ω(J) ·

k = 0 .

  • A resonant perturbation theory can be implemented to eliminate the

non–resonant terms.

  • Construct a change of variables C : (J, ϕ) → (J′, ϕ′), such that the new

Hamiltonian takes the form H′(J′, ϕ′) = Z′(J′, k · ϕ′) + ε2R′(J′, ϕ′) , where Z′ depends on ϕ′ only through the combinations k · ϕ′.

  • The combinations

k · ϕ are slow: d dt( k · ϕ) = k · ω = 0 .

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 9 / 43

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An application: Neptune’s discovery

1846: Jean Urbain Leverrier completed his mathematical studies to explain small discrepancies between theory and observations of the motion of Uranus

  • He postulated the existence of a new planet at the edge of the Solar system.
  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 10 / 43

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An application: Neptune’s discovery

1846: Jean Urbain Leverrier completed his mathematical studies to explain small discrepancies between theory and observations of the motion of Uranus

  • He postulated the existence of a new planet at the edge of the Solar system.
  • 18 September 1846: sent a letter to Johann Galle in Berlin
  • 23 September 1846: the same day the letter arrived, Galle discovered

Neptune.

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 10 / 43

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An application: Neptune’s discovery

1846: Jean Urbain Leverrier completed his mathematical studies to explain small discrepancies between theory and observations of the motion of Uranus

  • He postulated the existence of a new planet at the edge of the Solar system.
  • 18 September 1846: sent a letter to Johann Galle in Berlin
  • 23 September 1846: the same day the letter arrived, Galle discovered

Neptune.

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 10 / 43

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Modern applications of PT

Wide range of applications to study the dynamics of natural and artificial bodies. (Very) Partial list (in collaboration with C. Efthymiopoulos, F. Gachet, C. Gales, I. Gkolias, F. Paita, G. Pucacco, etc.):

  • Proper elements (a, e, i): through (high order) PT give quasi-integrals of the

normal form, unchanged over long time scales

  • Rotational dynamics: primary and secondary resonances in the spin-orbit

problem

  • Laplace resonance among the Galileian satellites of Jupiter (JUICE 2022):

λIo − 3λEu + 2λGan = 0

  • Halo orbits around the collinear Euler-Lagrangian points: reduction to the

center manifold + resonant normal form

  • Space debris: resonant PT for the geostationary orbit (1:1 tesseral

resonance) and GPS satellites (2:1 tesseral resonance).

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 11 / 43

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Poincaré non-integrability of the 3BP

  • 3BP non-integrable question: given 3 mass points, attracting each other

according to Newton’s law and not colliding, can we look for the coordinates

  • f each point at any time as the sum of a uniformly convergent series, whose

terms are known functions?

  • Poincaré showed that the series development of perturbation theory is in

general divergent ⇒ use asymptotic series: formal series, not necessarily convergent, whose truncation produces an approximation ⇒ optimal order.

  • Answer by Poincaré: negative + discovery of chaos + understanding

perturbations of integrable Hamiltonian systems is the fundamental problem

  • f the dynamics.
  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 12 / 43

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SLIDE 24

Poincaré non-integrability of the 3BP

  • 3BP non-integrable question: given 3 mass points, attracting each other

according to Newton’s law and not colliding, can we look for the coordinates

  • f each point at any time as the sum of a uniformly convergent series, whose

terms are known functions?

  • Poincaré showed that the series development of perturbation theory is in

general divergent ⇒ use asymptotic series: formal series, not necessarily convergent, whose truncation produces an approximation ⇒ optimal order.

  • Answer by Poincaré: negative + discovery of chaos + understanding

perturbations of integrable Hamiltonian systems is the fundamental problem

  • f the dynamics.
  • What about the stability of the 3BP?
  • J.J. Stoker: "Somewhat subjective character of the notion of stability [...] If

the definition of stability is rather general and simple, it is likely to include too many unstable cases as stable, while if it is too narrow, it may exclude too many motions which should be regarded as stable".

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 12 / 43

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SLIDE 25

Perturbation theory vs. KAM theory

Perturbation theory: nearly–integrable Hamiltonian systems, e.g. the 3–body problem (Sun-asteroid-Jupiter), based on the construction of a canonical transformation to an approximate solution. Convergence of indefinite iter- ation is prevented by the so-called small divisors. Kolmogorov-Arnold-Moser theory: concerns the persistence of quasi– periodic motions under small perturbations of an integrable system. Origi- nated by A.N. Kolmogorov in 1954, V.I. Arnold (1963) used a different ap- proach and generalized to Hamiltonian systems with degeneracies, while J.K. Moser (1962) covered the finitely differentiable case.

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 13 / 43

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Perturbation theory vs. KAM theory

Perturbation theory KAM theory

nearly–integrable Hamiltonian systems persistence of quasi–periodic motions efficient computational algorithms

Diophantine condition Non resonance condition Space debris 3-body problem

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 14 / 43

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SLIDE 27

Perturbation theory vs. KAM theory

Perturbation theory KAM theory

nearly–integrable Hamiltonian systems persistence of quasi–periodic motions efficient computational algorithms

Diophantine condition Non resonance condition Space debris 3-body problem

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 14 / 43

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Outline

  • 1. Celestial Mechanics and Perturbation theory
  • 2. KAM theory
  • 3. Symplectic/Conformally symplectic systems
  • 4. Some KAM applications to Celestial Mechanics
  • 5. Conclusions and perspectives
  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 15 / 43

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SLIDE 29

KAM theory

  • KAM theory: quasi–periodic motions and invariant tori in non–integrable

systems (Hamiltonian and Conformally Symplectic systems).

  • Calleja-Celletti-Llave (2013-): efficient KAM theory for conformally

symplectic (dissipative) systems, and other results.

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 16 / 43

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SLIDE 30

KAM theory

  • KAM theory: quasi–periodic motions and invariant tori in non–integrable

systems (Hamiltonian and Conformally Symplectic systems).

  • Calleja-Celletti-Llave (2013-): efficient KAM theory for conformally

symplectic (dissipative) systems, and other results.

  • Adding a dissipation to a Hamiltonian system is a very singular perturbation: many

quasi-periodic solutions for Hamiltonian systems, few attractors for dissipative systems, which need to include drift parameters.

  • A KAM theory with adjustment of parameters was developed in remarkable and

pioneer papers: [Moser1967], see also [Broer, Simó, etc.], with a parameter count different than in [CCL].

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 16 / 43

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SLIDE 31

KAM theory

  • KAM theory: quasi–periodic motions and invariant tori in non–integrable

systems (Hamiltonian and Conformally Symplectic systems).

  • Calleja-Celletti-Llave (2013-): efficient KAM theory for conformally

symplectic (dissipative) systems, and other results.

  • Adding a dissipation to a Hamiltonian system is a very singular perturbation: many

quasi-periodic solutions for Hamiltonian systems, few attractors for dissipative systems, which need to include drift parameters.

  • A KAM theory with adjustment of parameters was developed in remarkable and

pioneer papers: [Moser1967], see also [Broer, Simó, etc.], with a parameter count different than in [CCL].

  • KAM theory can be developed under two main assumptions:

frequency satisfying a Diophantine condition (to deal with small divisors); a non–degeneracy condition (on coordinates and parameters - to ensure the solution of the cohomological equations providing the approximate solutions).

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 16 / 43

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SLIDE 32

KAM theory

  • KAM theory motivated by stability in Celestial Mechanics (Laplace,

Lagrange, Poincaré, etc). Main problems modeled by:

  • nearly–integrable Hamiltonians;
  • nearly Hamiltonian systems.
  • Spin-orbit problem: H(y, x, t) = Z(y) + εR(y, x, t) with ε << 1 and

possibly a dissipation of the form −λ(y − µ) with λ << ε.

  • An application to the N–body problem in Celestial Mechanics was given by

Arnold: existence of a positive measure set of initial data providing quasi–periodic tori for e, i ≃ 0.

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 17 / 43

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SLIDE 33

Quantitative estimates

  • Hénon: quantitative estimates 3BP for a primaries mass-ratio 10−48 vs.

Jupiter-Sun 10−3: “Ainsi, ces théorèmes, bien que d’un très grand intérêt théorique, ne semblent pas pouvoir en leur état actuel être appliqués á des problèmes pratiques”.

  • J. Moser: "Speaking as a mathematician

the emphasis will naturally be on the theoretical aspects and much remains to be done in the applications of these results to realistic problems".

  • Nowadays, computer-assisted estimates

give realistic results in simple model problems.

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 18 / 43

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SLIDE 34

Conservative/Dissipative Standard Map

  • Conservative standard map (discrete analogue of the spin-orbit problem):

y′ = y + ε sin x y ∈ R , x ∈ T x′ = x + y′ .

  • Dissipative standard map (discrete analogue of the spin-orbit problem with

tidal torque): y′ = λy + µ + ε sin x y ∈ R , x ∈ T x′ = x + y′ .

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 19 / 43

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SLIDE 35

Conservative/Dissipative Standard Map

  • Conservative standard map (discrete analogue of the spin-orbit problem):

y′ = y + ε sin x y ∈ R , x ∈ T x′ = x + y′ .

  • Dissipative standard map (discrete analogue of the spin-orbit problem with

tidal torque): y′ = λy + µ + ε sin x y ∈ R , x ∈ T x′ = x + y′ . For ε = 0, ω and µ are related by ω ≡ µ 1 − λ .

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 19 / 43

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SLIDE 36
  • 3
  • 2
  • 1

1 2 3 1 2 3 4 5 6 y x epsilon=0

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 20 / 43

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SLIDE 37
  • 3
  • 2
  • 1

1 2 3 1 2 3 4 5 6 y x epsilon=0.2

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 20 / 43

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SLIDE 38
  • 3
  • 2
  • 1

1 2 3 1 2 3 4 5 6 y x epsilon=0.4

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 20 / 43

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SLIDE 39
  • 3
  • 2
  • 1

1 2 3 1 2 3 4 5 6 y x epsilon=0.6

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 20 / 43

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SLIDE 40
  • 3
  • 2
  • 1

1 2 3 1 2 3 4 5 6 y x epsilon=0.8

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 20 / 43

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SLIDE 41
  • 3
  • 2
  • 1

1 2 3 1 2 3 4 5 6 y x epsilon=1

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 20 / 43

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SLIDE 42

1 2 3 4 1 2 3 4 5 6 y x epsilon=0

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 21 / 43

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SLIDE 43

1 2 3 4 1 2 3 4 5 6 y x epsilon=0.2

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 21 / 43

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SLIDE 44

1 2 3 4 1 2 3 4 5 6 y x epsilon=0.4

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 21 / 43

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SLIDE 45

1 2 3 4 1 2 3 4 5 6 y x epsilon=0.6

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 21 / 43

slide-46
SLIDE 46

1 2 3 4 1 2 3 4 5 6 y x epsilon=0.8

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 21 / 43

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SLIDE 47

1 2 3 4 1 2 3 4 5 6 y x epsilon=1

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 21 / 43

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SLIDE 48

KAM goal

  • The set of Diophantine frequencies

D(C, τ) = {ω ∈ Rn : |ω·q−p|−1 ≤ C|k|τ , ∀p ∈ Z, q ∈ Zn\{0} , C, τ > 0 } has full Lebesgue measure for τ > n − 1.

  • Fix a Diophantine frequency ω and look for an invariant torus for the

nearly-integrable system.

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 22 / 43

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SLIDE 49

Outline

  • 1. Celestial Mechanics and Perturbation theory
  • 2. KAM theory
  • 3. Symplectic/Conformally symplectic systems
  • 4. Some KAM applications to Celestial Mechanics
  • 5. Conclusions and perspectives
  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 23 / 43

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SLIDE 50

Symplectic/Conformally symplectic systems

  • We consider maps, but the results can be easily extended to flows.
  • Let M = U × Tn be the phase space with U ⊆ Rn open, simply connected

domain with smooth boundary; M is endowed with the standard scalar product and a symplectic form Ω.

Definition

A map f on M is symplectic, if f ∗Ω = Ω . A family of maps fµ, µ ∈ Rn, is conformally symplectic, if there exists a function λ : M → R such that f ∗

µΩ = λΩ .

(Notation: no underline vectors)

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 24 / 43

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SLIDE 51

Dissipative effects in Celestial Mechanics:

⊲ PLANETS: Poynting-Robertson effect, Stokes drag (primordial solar nebula), tides

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 25 / 43

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SLIDE 52

Dissipative effects in Celestial Mechanics:

⊲ PLANETS: Poynting-Robertson effect, Stokes drag (primordial solar nebula), tides ⊲ SATELLITES: tidal torque, Yarkowski/YORP effects

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 25 / 43

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SLIDE 53

Dissipative effects in Celestial Mechanics:

⊲ PLANETS: Poynting-Robertson effect, Stokes drag (primordial solar nebula), tides ⊲ SATELLITES: tidal torque, Yarkowski/YORP effects ⊲ SPACECRAFT: atmospheric drag (decreasing exponentially with the altitude), sloshing, mass consumption.

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 25 / 43

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SLIDE 54

KAM surfaces and norms

Definition

A KAM surface with frequency ω ∈ D(C, τ) for a family fµ of conformally symplectic maps depending on a real parameter µ is an n–dimensional invariant surface described parametrically by an embedding K : Tn → M and a drift µ = µ∗, solutions of the invariance equation: fµ∗ ◦ K(θ) = K(θ + ω) .

Definition

Analytic norm. Given ρ > 0, use the norms in the set of analytic functions: fρ = sup

θ∈Tn

ρ

|f(θ)| Tn

ρ = {θ ∈ Cn/(2πZ)n : Re(θ) ∈ Tn, |Im(θj)| ≤ ρ , j = 1, ..., n} .

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 26 / 43

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SLIDE 55

KAM Theorem (references: Llave-Gonzalez-Jorba-Villanueva & Calleja-Llave-Celletti)

Theorem (CCL 2013)

  • Let ω ∈ D(C, τ), fµ conformally symplectic.
  • (K0, µ0) approximate solution:

fµ0 ◦ K0(θ) − K0(θ + ω) = E0(θ) .

  • Assume that the solution is sufficiently approximate, i.e. E0ρ small.
  • Assume a suitable non–degeneracy condition (twist+non degeneracy w.r.t.

parameters).

  • Then, there exists an exact solution (Ke, µe), such that for 0 < δ < ρ

2 :

fµe ◦ Ke(θ) − Ke(θ + ω) = 0 and Ke − K0ρ−2δ ≤ C1 C2 δ−2τ E0ρ , |µe − µ0| ≤ C2 E0ρ (C1, C2 > 0) .

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 27 / 43

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SLIDE 56

Summary of the Proof: a-posteriori approach

Step 1: Linearization of approximate solution fµ0 ◦ K0(θ) − K0(θ + ω) = E0(θ) Step 2: determine the new approximation K1 = K0 + W0, µ1 = µ0 + σ0, with corrections W0, σ0 with E1 = O(E02) Step 3: solve the cohomological equation for W0 and σ0, involving small divisors and requiring the non-degeneracy condition Step 4: convergence of the iterative step (Newton quadratic iteration method): E0 → E1 = O(E02) → E2 = O(E12) = O(E04) → ... Step 5: local uniqueness.

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 28 / 43

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SLIDE 57

Main ingredients of the proof

  • Complex extension: allows to get Cauchy estimates; given an analytic

function on a domain, Cauchy estimates provide a bound on the norm of the partial derivatives over a smaller domain.

  • Diophantine condition: to control the small divisors to get Cauchy estimates.
  • Non-degeneracy: to solve the cohomological equations and to ensure that

the structure of the frequency space is preserved in the action space.

  • Newton quadratic iteration method: Hamiltonian case

H = Z + εR → → H′ = Z′ + ε2R′ → → H′′ = Z′′ + ε4R′′ → → H′′′ = Z′′′ + ε8R′′′ ...

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 29 / 43

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SLIDE 58

Efficient (computer-assisted) algorithm

  • Efficient:

⊲ Each step needs:

  • shifting functions,
  • multiplying, composing and differentiating functions,
  • solving difference equations with constant coefficients.

⊲ Starting with an FFT with N Fourier modes, the Newton step needs O(N) storage and O(N log N) operations.

  • Computer-assisted:

⊲ The KAM proof requires very long computations (initial approximation, KAM algorithm, etc): rounding-off and propagation errors → interval arithmetic: represent any real number as an interval and perform elementary operations

  • n intervals, rather than on real numbers (computer time increases).
  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 30 / 43

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SLIDE 59

Outline

  • 1. Celestial Mechanics and Perturbation theory
  • 2. KAM theory
  • 3. Symplectic/Conformally symplectic systems
  • 4. Some KAM applications to Celestial Mechanics
  • 5. Conclusions and perspectives
  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 31 / 43

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SLIDE 60

Effective estimates for the standard maps

  • Conservative standard map, ω = 2π

√ 5−1 2

: ⊲ [A.C., L. Chierchia ’90] = ⇒ 86% of the numerical breakdown value ⊲ [R. de la Llave, D. Rana ’90] = ⇒ 93% of the numerical breakdown value ⊲ [J.-L. Figueras, A. Haro, A. Luque, 2016] symplectic (twist and non-twist) standard maps = ⇒ 99.9% of Greene’s value

  • Dissipative standard map, ω = 2π

√ 5−1 2

: ⊲ [R. Calleja, A.C.,R. de la Llave 2019] λ = 0.9 = ⇒ 99.9% of the numerical breakdown value, provided µ is suitably tuned.

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 32 / 43

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SLIDE 61

The rotation of the Moon: 1:1 resonance

  • Spin–orbit problem:

⊲ triaxial satellite S (with I1 < I2 < I3); ⊲ satellite moving on a Keplerian orbit around a central planet P; ⊲ spin–axis perpendicular to orbit plane and coinciding with shortest physical axis; ⊲ tidal torque, due to the non–rigidity.

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 33 / 43

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SLIDE 62

Modeling the spin–orbit problem

  • Spin–orbit problem:

⊲ triaxial satellite S (with I1 < I2 < I3); ⊲ satellite moving on a Keplerian orbit around a central planet P; ⊲ spin–axis perpendicular to orbit plane and coinciding with shortest physical axis; ⊲ NO tidal torque, due to the non–rigidity.

  • Conservative case: equation of motion:

¨ x + εVx(x, t; e) = 0 , ε = 3 2 I2 − I1 I3 corresponding to a 1–dim, time–dependent Hamiltonian: H(y, x, t) = y2 2 − ε 2V(x, t; e) .

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 34 / 43

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SLIDE 63

Modeling the spin–orbit problem

  • Spin–orbit problem:

⊲ triaxial satellite S (with I1 < I2 < I3); ⊲ satellite moving on a Keplerian orbit around a central planet P; ⊲ spin–axis perpendicular to orbit plane and coinciding with shortest physical axis; ⊲ tidal torque, due to the non–rigidity.

  • Dissipative case: equation of motion with tidal torque averaged over an orbital

period: ¨ x + εVx(x, t; e) = −λ(˙ x − µ) , where λ = CD λ(e), µ = µ(e), CD depending on the physical properties.

  • λ plays the role of conformal factor;
  • µ plays the role of drift.
  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 35 / 43

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SLIDE 64

KAM confinement stability

  • Confinement in 1-dim, time-dependent conservative system H(y, x, t):

dim(phase space)=3, dim(invariant tori)=2 → confinement in phase space for ∞ times between bounding invariant tori.

2 4 6 2 4 6 0.669 0.6695 0.67 0.6705 0.671 2 4 6 0.669 0.6695 0.67 0.6705

  • Confinement no more valid for n > 2: the motion can diffuse through

invariant tori, reaching arbitrarily far regions (Arnold’s diffusion).

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 36 / 43

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SLIDE 65

KAM for the conservative/dissipative spin–orbit problem

  • Synchronous frequency = ω = 1, (Diophantine) frequencies of the bounding tori =

ω± ≡ 1 ±

1 2+

√ 5−1 2

.

Theorem [Conservative, A.C. (1990)]

Consider the spin–orbit Hamiltonian defined in U × T2 with U ⊂ R open set. Then, for the true eccentricity of the Moon e = 0.0549, there exist invariant tori, bounding the motion of the Moon, for any ε ≤ εMoon = 3.45 · 10−4 (astronomical value).

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 37 / 43

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SLIDE 66

KAM for the conservative/dissipative spin–orbit problem

  • Synchronous frequency = ω = 1, (Diophantine) frequencies of the bounding tori =

ω± ≡ 1 ±

1 2+

√ 5−1 2

.

Theorem [Conservative, A.C. (1990)]

Consider the spin–orbit Hamiltonian defined in U × T2 with U ⊂ R open set. Then, for the true eccentricity of the Moon e = 0.0549, there exist invariant tori, bounding the motion of the Moon, for any ε ≤ εMoon = 3.45 · 10−4 (astronomical value).

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 37 / 43

slide-67
SLIDE 67

KAM for the conservative/dissipative spin–orbit problem

  • Synchronous frequency = ω = 1, (Diophantine) frequencies of the bounding tori =

ω± ≡ 1 ±

1 2+

√ 5−1 2

.

Theorem [Conservative, A.C. (1990)]

Consider the spin–orbit Hamiltonian defined in U × T2 with U ⊂ R open set. Then, for the true eccentricity of the Moon e = 0.0549, there exist invariant tori, bounding the motion of the Moon, for any ε ≤ εMoon = 3.45 · 10−4 (astronomical value).

Theorem [Dissipative, AC-Chierchia (2009), Calleja-AC-Llave (2013)]

Let λ0 ∈ R+, ω Diophantine. There exists 0 < ε0 < 1, such that for any ε ∈ [0, ε0] and any λ ∈ [−λ0, λ0] there exists a unique function K = K(θ, t) and a drift term µ which is the solution of the invariance equation for the dissipative spin-orbit model.

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 37 / 43

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SLIDE 68

Figure: Spin-orbit attractor for e = 10−3, e = 0.0549, e = 0.2056.

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 38 / 43

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SLIDE 69

Conservative three–body problem

  • Planar, circular, restricted three–body problem in action–angle Delaunay

variables (L, G) ∈ R2, (ℓ, g) ∈ T2: H(L, G, ℓ, g) = − 1 2L2 − G + εR(L, G, ℓ, g) . ε = 0 Keplerian motion, actions: L = √a, G = L √ 1 − e2.

  • h(L, G) = − 1

2L2 − G: degenerate Hessian h′′, but Arnold’s isoenergetic

non–degenerate (persistence of invariant tori on a fixed energy surface): det h′′(L, G) h′(L, G) h′(L, G)T

  • = 3

L4 = 0 for all L = 0 .

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 39 / 43

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SLIDE 70

Conservative three–body problem

  • Concrete example: Sun, Jupiter, asteroid 12 Victoria
  • aV = 0.449 [Jupiter–Sun unit distance] and eV = 0.22,
  • fix energy level E∗

V = − 1 2L2

V − GV + 10−3R(LV, GV, ℓ, g)

  • ≃ −1.769,
  • prove the existence of two (2–dim) trapping tori with frequencies ω±.
  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 40 / 43

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SLIDE 71

Conservative three–body problem

  • Concrete example: Sun, Jupiter, asteroid 12 Victoria
  • aV = 0.449 [Jupiter–Sun unit distance] and eV = 0.22,
  • fix energy level E∗

V = − 1 2L2

V − GV + 10−3R(LV, GV, ℓ, g)

  • ≃ −1.769,
  • prove the existence of two (2–dim) trapping tori with frequencies ω±.

Proposition [three–body problem, A.C., L. Chierchia (2007)]

Let E = E∗

  • V. Then, for |ε| ≤ 10−3 the unperturbed tori with trapping

frequencies ω± can be analytically continued into KAM tori for the perturbed system on the energy level H−1 E∗

V) keeping fixed the ratio of the

frequencies.

  • Due to the link between a, e and L, G, this result guarantees that a, e remain

close to the unperturbed values within an interval of size of order ε.

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 40 / 43

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SLIDE 72

Outline

  • 1. Celestial Mechanics and Perturbation theory
  • 2. KAM theory
  • 3. Symplectic/Conformally symplectic systems
  • 4. Some KAM applications to Celestial Mechanics
  • 5. Conclusions and perspectives
  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 41 / 43

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SLIDE 73

Conclusions and perspectives

  • Analytical methods based on PT in space debris and Celestial Mechanics:

⊲ classify space debris according to their dynamical features; ⊲ find regular and chaotic regions (dissipation at low altitudes); ⊲ design disposal orbits.

  • Computer-assisted KAM theory gives efficient estimates in practical

problems.

  • KAM theory in conservative and dissipative systems:

⊲ give quantitative estimates on the dissipative spin-orbit problem; ⊲ compute invariant tori and attractors for other realistic problems and space missions (e.g., other 3BP, Lissajous, halo, etc.).

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 42 / 43

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SLIDE 74

References.

  • R. Calleja, A.C., Breakdown of invariant attractors for the dissipative standard map, CHAOS

(2010)

  • R. Calleja, A.C., R. de la Llave, A KAM theory for conformally symplectic systems:

efficient algorithms and their validation, J. Differential Equations 255, n. 5 (2013)

  • R. Calleja, A.C., R. de la Llave, Domains of analyticity of Lindstedt expansions of KAM tori

in dissipative perturbations of Hamiltonian systems, Nonlinearity 30 (2017)

  • R. Calleja, A.C., R. de la Llave, KAM estimates for the dissipative standard map, Preprint

(2019)

  • A.C., Analysis of resonances in the spin-orbit problem in Celestial Mechanics (Part I and II),

ZAMP (1990)

  • A.C., Stability and Chaos in Celestial Mechanics, Springer (2010)
  • A.C., L. Chierchia, KAM Stability and Celestial Mechanics, Memoirs AMS (2007)
  • A.C., C. Gale¸

s, On the dynamics of space debris: 1:1 and 2:1 resonances,

  • J. Nonlinear Science (2014)
  • A.C., C. Gale¸

s, G. Pucacco, Bifurcation of lunisolar secular resonances for space debris

  • rbits, SIAM J. Appl. Dyn. Syst. (2016)
  • F. Gachet, A.C., G. Pucacco, C, Efthymiopoulos, Geostationary secular dynamics revisited:

application to high area-to-mass ratio objects, CM&DA (2017)

  • A. Celletti (Univ. Rome Tor Vergata)

Stability results in Celestial Mechanics 13 March 2019 43 / 43