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 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
Celestial Mechanics • Celestial Mechanics studies the dynamics of natural and artificial object os the Solar system. Aristotle Ptolemy epicycles heliocentric system 384-322 AC 85-165 Hypparcus Celestial Copernicus 190-120 AC epicycles/deferents 1473-1543 spheres scientific method gravitation Tycho Brahe Kepler 1546-1601 1571-1630 observations Galileo Newton 2-body 1564-1642 1642-1727 problem Perturbation Stability Laplace Poincarè 1749-1827 theories theories 1854-1912 Kolmogorov, 3-body Lagrange, planetary Arnold, Moser, Gauss, problem motions Nekhoroshev Delaunay, XX century XIX century A. Celletti (Univ. Rome Tor Vergata) Stability results in Celestial Mechanics 13 March 2019 3 / 43
Celestial Mechanics • Celestial Mechanics studies the dynamics of natural and artificial object os the Solar system. Aristotle Ptolemy epicycles heliocentric system 384-322 AC 85-165 Hypparcus Celestial Copernicus 190-120 AC epicycles/deferents 1473-1543 spheres scientific method gravitation Tycho Brahe Kepler 1546-1601 1571-1630 observations Galileo Newton 2-body 1564-1642 1642-1727 problem Perturbation Stability Laplace Poincarè 1749-1827 theories theories 1854-1912 Kolmogorov, 3-body Lagrange, planetary Arnold, Moser, Gauss, problem motions Nekhoroshev Delaunay, XX century XIX century A. Celletti (Univ. Rome Tor Vergata) Stability results in Celestial Mechanics 13 March 2019 3 / 43
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: J ∈ R n , ϕ ∈ T n , H ( J , ϕ ) = Z ( J ) , whose Hamilton’s equations are: ∂ H ( J ,ϕ ) = − ∂ Z ( J ) ˙ J = − = 0 ⇒ J = J 0 ∂ϕ ∂ϕ ∂ H ( J ,ϕ ) = ∂ Z ( J ) ϕ = ˙ ≡ ω ( J ) ⇒ ϕ = ω ( J 0 ) t + ϕ 0 . ∂ J ∂ J A. Celletti (Univ. Rome Tor Vergata) Stability results in Celestial Mechanics 13 March 2019 4 / 43
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: J ∈ R n , ϕ ∈ T n , H ( J , ϕ ) = Z ( J ) , whose Hamilton’s equations are: ∂ H ( J ,ϕ ) = − ∂ Z ( J ) ˙ J = − = 0 ⇒ J = J 0 ∂ϕ ∂ϕ ∂ H ( J ,ϕ ) = ∂ Z ( J ) ϕ = ˙ ≡ ω ( J ) ⇒ ϕ = ω ( J 0 ) t + ϕ 0 . ∂ J ∂ J A. Celletti (Univ. Rome Tor Vergata) Stability results in Celestial Mechanics 13 March 2019 4 / 43
3-body problem • 3-body problem, e.g. Sun-Earth-Jupiter, is governed by a nearly-integrable system: ε = m Jupiter ≃ 10 − 3 H ( J , ϕ ) = Z ( J ) + ε R ( J , ϕ ) , ���� � �� � m Sun Sun-Earth Earth-Jupiter A. Celletti (Univ. Rome Tor Vergata) Stability results in Celestial Mechanics 13 March 2019 5 / 43
3-body problem • 3-body problem, e.g. Sun-Earth-Jupiter, is governed by a nearly-integrable system: ε = m Jupiter ≃ 10 − 3 H ( J , ϕ ) = Z ( J ) + ε R ( J , ϕ ) , ���� � �� � m Sun Sun-Earth Earth-Jupiter whose Hamilton’s equations are: − ε∂ R ( J , ϕ ) ˙ J = ∂ϕ ω ( J ) + ε∂ R ( J , ϕ ) ϕ ˙ = ∂ J A. Celletti (Univ. Rome Tor Vergata) Stability results in Celestial Mechanics 13 March 2019 5 / 43
3-body problem • 3-body problem, e.g. Sun-Earth-Jupiter, is governed by a nearly-integrable system: ε = m Jupiter ≃ 10 − 3 H ( J , ϕ ) = Z ( J ) + ε R ( J , ϕ ) , ���� � �� � m Sun Sun-Earth Earth-Jupiter whose Hamilton’s equations are: − ε∂ R ( J , ϕ ) ˙ J = ∂ϕ ω ( J ) + ε∂ R ( J , ϕ ) ϕ ˙ = ∂ J WHICH IS THE SOLUTION? A. Celletti (Univ. Rome Tor Vergata) Stability results in Celestial Mechanics 13 March 2019 5 / 43
3-body problem • 3-body problem, e.g. Sun-Earth-Jupiter, is governed by a nearly-integrable system: ε = m Jupiter ≃ 10 − 3 H ( J , ϕ ) = Z ( J ) + ε R ( J , ϕ ) , ���� � �� � m Sun Sun-Earth Earth-Jupiter whose Hamilton’s equations are: − ε∂ R ( J , ϕ ) ˙ J = ∂ϕ ω ( J ) + ε∂ R ( J , ϕ ) ϕ ˙ = ∂ J WHICH IS THE SOLUTION? PERTURBATION THEORY A. Celletti (Univ. Rome Tor Vergata) Stability results in Celestial Mechanics 13 March 2019 5 / 43
Classical perturbation theory • Approximates solution by pushing the perturbation to higher orders in ε . Theorem Let H ( J , ϕ ) = Z ( J ) + ε R ( J , ϕ ) with ( J , ϕ ) ∈ V × T n for V ⊂ R n open: ◮ R analytic and trigonometric ( N Fourier coefficients) on V × T n ; ◮ non-resonance frequency for any J 0 ∈ V : | ω ( J 0 ) · k | > 0 for all 0 < | k | ≤ N . Then, there exists ρ 0 > 0, ε 0 > 0 and for | ε | < ε 0 there exists a canonical 2 ( J 0 ) × T n ⊂ V × T n and with transformation ( J , ϕ ) → ( J ′ , ϕ ′ ) defined in S ρ 0 values in S ρ 0 ( J 0 ) × T n , which transforms H as H ′ ( J ′ , ϕ ′ ) = Z ′ ( J ′ ) + ε 2 R ′ ( J ′ , ϕ ′ ) . A. Celletti (Univ. Rome Tor Vergata) Stability results in Celestial Mechanics 13 March 2019 6 / 43
Classical perturbation theory • Approximates solution by pushing the perturbation to higher orders in ε . Theorem Let H ( J , ϕ ) = Z ( J ) + ε R ( J , ϕ ) with ( J , ϕ ) ∈ V × T n for V ⊂ R n open: ◮ R analytic and trigonometric ( N Fourier coefficients) on V × T n ; ◮ non-resonance frequency for any J 0 ∈ V : | ω ( J 0 ) · k | > 0 for all 0 < | k | ≤ N . Then, there exists ρ 0 > 0, ε 0 > 0 and for | ε | < ε 0 there exists a canonical 2 ( J 0 ) × T n ⊂ V × T n and with transformation ( J , ϕ ) → ( J ′ , ϕ ′ ) defined in S ρ 0 values in S ρ 0 ( J 0 ) × T n , which transforms H as H ′ ( J ′ , ϕ ′ ) = Z ′ ( J ′ ) + ε 2 R ′ ( J ′ , ϕ ′ ) . A. Celletti (Univ. Rome Tor Vergata) Stability results in Celestial Mechanics 13 March 2019 6 / 43
Classical perturbation theory • Approximates solution by pushing the perturbation to higher orders in ε . Theorem Let H ( J , ϕ ) = Z ( J ) + ε R ( J , ϕ ) with ( J , ϕ ) ∈ V × T n for V ⊂ R n open: ◮ R analytic and trigonometric ( N Fourier coefficients) on V × T n ; ◮ non-resonance frequency for any J 0 ∈ V : | ω ( J 0 ) · k | > 0 for all 0 < | k | ≤ N . Then, there exists ρ 0 > 0, ε 0 > 0 and for | ε | < ε 0 there exists a canonical 2 ( J 0 ) × T n ⊂ V × T n and with transformation ( J , ϕ ) → ( J ′ , ϕ ′ ) defined in S ρ 0 values in S ρ 0 ( J 0 ) × T n , which transforms H as H ′ ( J ′ , ϕ ′ ) = Z ′ ( J ′ ) + ε 2 R ′ ( J ′ , ϕ ′ ) . • The proof is constructive and allows us to obtain H ′ = Z ′ + ε 2 R ′ , H ′′ = Z ′′ + ε 3 R ′′ , H ′′′ = Z ′′′ + ε 4 R ′′′ , etc. A. Celletti (Univ. Rome Tor Vergata) Stability results in Celestial Mechanics 13 March 2019 6 / 43
Normal form with Lie series: constructive proof Proof. • Canonical transformation ( ϕ ′ , J ′ ) = ( e L χ ϕ, e L χ J ) , to transform H = Z + ε R = Z + ε ( R + � R ) with R = average, � R = non average: in H ′ ( J ′ , ϕ ′ ) = Z ( J ′ ) + ε R ( J ′ ) + ε 2 R ′ ( J ′ , ϕ ′ ) , where Z ′ = Z + ε R is the normal form , L χ ≡ {· , χ } is the Poisson bracket operator, e L χ = � ∞ k ! L k 1 χ . Then: k = 0 e L χ H = H + L χ H + 1 H ′ 2 L 2 = χ H + ... Z + ε R + {H , χ } + 1 2 L 2 = χ H + ... = Z + ε R + { Z , χ } + ε { R , χ } + ... Z + ε R + 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
Normal form with Lie series { Z , χ } = − ε � R ∂ Z ❆ ✁ ∂χ ∂ J − ∂ Z ∂ϕ = − ∂ Z ∂χ ∂χ ∂ϕ = − ω ( J ) ∂χ ∂ϕ = − ε � { Z , χ } = ✁ R ❆ ∂ϕ ∂ J ∂ J ✁ ❆ R ( J , ϕ ) = � R k ( J ) e ik · ϕ and χ ( J , ϕ ) = � • Expand � k � = 0 � χ k ( J ) e ik · ϕ : k � � � χ k ( J ′ ) e ik · ϕ ′ = − ε R k ( J ′ ) e ik · ϕ ′ , ω ( J ′ ) · k � � − i k k � = 0 , | k |≤ N giving the solution for χ : R k ( J ′ ) � χ k ( J ′ ) = ε � i ω ( 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
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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