Arnold diffusion for convex nearly integrable systems V. Kaloshin - PowerPoint PPT Presentation

Arnold diffusion for convex nearly integrable systems V. Kaloshin November 24, 2014 V. Kaloshin (University of Maryland) Arnold diffusion November 24, 2014 1 / 22

Plan of the talk Motivation: Ergodic and quasiergodic hypothesis. Nearly integrable systems and the problem of Arnold diffusion Results in 3, 4, and more degrees of freedom Indication of Arnold diffusion in the Solar system Stochastic aspects of Arnold diffusion V. Kaloshin (University of Maryland) Arnold diffusion November 24, 2014 2 / 22

Plan of the talk Motivation: Ergodic and quasiergodic hypothesis. Nearly integrable systems and the problem of Arnold diffusion Results in 3, 4, and more degrees of freedom Indication of Arnold diffusion in the Solar system Stochastic aspects of Arnold diffusion V. Kaloshin (University of Maryland) Arnold diffusion November 24, 2014 2 / 22

Plan of the talk Motivation: Ergodic and quasiergodic hypothesis. Nearly integrable systems and the problem of Arnold diffusion Results in 3, 4, and more degrees of freedom Indication of Arnold diffusion in the Solar system Stochastic aspects of Arnold diffusion V. Kaloshin (University of Maryland) Arnold diffusion November 24, 2014 2 / 22

Plan of the talk Motivation: Ergodic and quasiergodic hypothesis. Nearly integrable systems and the problem of Arnold diffusion Results in 3, 4, and more degrees of freedom Indication of Arnold diffusion in the Solar system Stochastic aspects of Arnold diffusion V. Kaloshin (University of Maryland) Arnold diffusion November 24, 2014 2 / 22

Plan of the talk Motivation: Ergodic and quasiergodic hypothesis. Nearly integrable systems and the problem of Arnold diffusion Results in 3, 4, and more degrees of freedom Indication of Arnold diffusion in the Solar system Stochastic aspects of Arnold diffusion V. Kaloshin (University of Maryland) Arnold diffusion November 24, 2014 2 / 22

Motivation: Ergodic Hypothesis Let H : R 2 n → R be a smooth function, ( q , p ) ∈ R n × R n . Let X H be the Hamiltonian flow associated to H . � ˙ q = ∂ p H (1) ˙ p = − ∂ q H Let S E = { ( q , p ) ∈ T ∗ M : H ( q , p ) = E } be an energy surface. Ergodic Hypothesis (Boltzmann, Maxwell) Is a generic Hamiltonian flow X H on a generic energy surface S E ergodic? Numerical doubts ( Fermi-Pasta-Ulam ) Chains of nonlinear springs V. Kaloshin (University of Maryland) Arnold diffusion November 24, 2014 3 / 22

Motivation: Ergodic Hypothesis Let H : R 2 n → R be a smooth function, ( q , p ) ∈ R n × R n . Let X H be the Hamiltonian flow associated to H . � ˙ q = ∂ p H (1) ˙ p = − ∂ q H Let S E = { ( q , p ) ∈ T ∗ M : H ( q , p ) = E } be an energy surface. Ergodic Hypothesis (Boltzmann, Maxwell) Is a generic Hamiltonian flow X H on a generic energy surface S E ergodic? Numerical doubts ( Fermi-Pasta-Ulam ) Chains of nonlinear springs V. Kaloshin (University of Maryland) Arnold diffusion November 24, 2014 3 / 22

Motivation: Ergodic Hypothesis Let H : R 2 n → R be a smooth function, ( q , p ) ∈ R n × R n . Let X H be the Hamiltonian flow associated to H . � ˙ q = ∂ p H (1) ˙ p = − ∂ q H Let S E = { ( q , p ) ∈ T ∗ M : H ( q , p ) = E } be an energy surface. Ergodic Hypothesis (Boltzmann, Maxwell) Is a generic Hamiltonian flow X H on a generic energy surface S E ergodic? Numerical doubts ( Fermi-Pasta-Ulam ) Chains of nonlinear springs V. Kaloshin (University of Maryland) Arnold diffusion November 24, 2014 3 / 22

Motivation: Ergodic Hypothesis Let H : R 2 n → R be a smooth function, ( q , p ) ∈ R n × R n . Let X H be the Hamiltonian flow associated to H . � ˙ q = ∂ p H (1) ˙ p = − ∂ q H Let S E = { ( q , p ) ∈ T ∗ M : H ( q , p ) = E } be an energy surface. Ergodic Hypothesis (Boltzmann, Maxwell) Is a generic Hamiltonian flow X H on a generic energy surface S E ergodic? Numerical doubts ( Fermi-Pasta-Ulam ) Chains of nonlinear springs V. Kaloshin (University of Maryland) Arnold diffusion November 24, 2014 3 / 22

Motivation: Ergodic Hypothesis Let H : R 2 n → R be a smooth function, ( q , p ) ∈ R n × R n . Let X H be the Hamiltonian flow associated to H . � ˙ q = ∂ p H ˙ p = − ∂ q H Let S E = { ( q , p ) ∈ T ∗ M : H ( q , p ) = E } be an energy surface. Ergodic Hypothesis (Boltzmann, Maxwell) Is a generic Hamiltonian flow X H on a generic energy surface S E ergodic? Numerical doubts ( Fermi-Pasta-Ulam ) Chains of nonlinear springs u n = k ( u n + 1 − u n ) − k ( u n − u n − 1 ) + α ( u n + 1 − u n ) 2 + α ( u n − u n − 1 ) 2 ¨ the α -term — nonlinearity. Most “small” solutions are almost periodic ! V. Kaloshin (University of Maryland) Arnold diffusion November 24, 2014 4 / 22

Motivation: Ergodic Hypothesis Let H : R 2 n → R be a smooth function, ( q , p ) ∈ R n × R n . Let X H be the Hamiltonian flow associated to H . � ˙ q = ∂ p H ˙ p = − ∂ q H Let S E = { ( q , p ) ∈ T ∗ M : H ( q , p ) = E } be an energy surface. Ergodic Hypothesis (Boltzmann, Maxwell) Is a generic Hamiltonian flow X H on a generic energy surface S E ergodic? Numerical doubts ( Fermi-Pasta-Ulam ) Chains of nonlinear springs u n = k ( u n + 1 − u n ) − k ( u n − u n − 1 ) + α ( u n + 1 − u n ) 2 + α ( u n − u n − 1 ) 2 ¨ the α -term — nonlinearity. Most “small” solutions are almost periodic ! V. Kaloshin (University of Maryland) Arnold diffusion November 24, 2014 4 / 22

Motivation: Ergodic Hypothesis Let H : R 2 n → R be a smooth function, ( q , p ) ∈ R n × R n . Let X H be the Hamiltonian flow associated to H . � ˙ q = ∂ p H ˙ p = − ∂ q H Let S E = { ( q , p ) ∈ T ∗ M : H ( q , p ) = E } be an energy surface. Ergodic Hypothesis (Boltzmann, Maxwell) Is a generic Hamiltonian flow X H on a generic energy surface S E ergodic? Numerical doubts ( Fermi-Pasta-Ulam ) Chains of nonlinear springs u n = k ( u n + 1 − u n ) − k ( u n − u n − 1 ) + α ( u n + 1 − u n ) 2 + α ( u n − u n − 1 ) 2 ¨ the α -term — nonlinearity. Most “small” solutions are almost periodic ! V. Kaloshin (University of Maryland) Arnold diffusion November 24, 2014 4 / 22

Quasiergodic Hypothesis KAM theory Each nearly integrable systems has collections of invariant tori of positive measure = ⇒ no ergodicity! Quasiergodic Hypothesis (Birkhoff, Ehrenfest) Does a generic Hamiltonian flow on a generic energy surface S E have a dense orbit? V. Kaloshin (University of Maryland) Arnold diffusion November 24, 2014 5 / 22

Quasiergodic Hypothesis KAM theory Each nearly integrable systems has collections of invariant tori of positive measure = ⇒ no ergodicity! Quasiergodic Hypothesis (Birkhoff, Ehrenfest) Does a generic Hamiltonian flow on a generic energy surface S E have a dense orbit? V. Kaloshin (University of Maryland) Arnold diffusion November 24, 2014 5 / 22

Quasiergodic Hypothesis KAM theory Each nearly integrable systems has collections of invariant tori of positive measure = ⇒ no ergodicity! Quasiergodic Hypothesis (Birkhoff, Ehrenfest) Does a generic Hamiltonian flow on a generic energy surface S E have a dense orbit? V. Kaloshin (University of Maryland) Arnold diffusion November 24, 2014 6 / 22

Integrable systems & action-angles coordinates Let H : R 2 n → R be a Hamiltonian, ϕ ∈ T n be angle, I ∈ R n be action. A Hamiltonian system is Arnold-Liouville integrable if for an open set U ⊂ R n there exists a symplectic map Φ : T n × U → R 2 n s. t. H ◦ Φ( ϕ, I ) depends only on I and � ϕ = ∂ I ( H ◦ Φ)( I ) = ω ( I ) , ˙ ( ϕ, I ) –action-angle coordinates ˙ I = 0 . In particular, Φ( T n × U ) is foliated by invariant n -dim’l tori & on each torus T n the flow is linear. V. Kaloshin (University of Maryland) Arnold diffusion November 24, 2014 7 / 22

Integrable systems & action-angles coordinates Let H : R 2 n → R be a Hamiltonian, ϕ ∈ T n be angle, I ∈ R n be action. A Hamiltonian system is Arnold-Liouville integrable if for an open set U ⊂ R n there exists a symplectic map Φ : T n × U → R 2 n s. t. H ◦ Φ( ϕ, I ) depends only on I and � ϕ = ∂ I ( H ◦ Φ)( I ) = ω ( I ) , ˙ ( ϕ, I ) –action-angle coordinates ˙ I = 0 . In particular, Φ( T n × U ) is foliated by invariant n -dim’l tori & on each torus T n the flow is linear. V. Kaloshin (University of Maryland) Arnold diffusion November 24, 2014 7 / 22

Integrable systems & action-angles coordinates Let H : R 2 n → R be a Hamiltonian, ϕ ∈ T n be angle, I ∈ R n be action. A Hamiltonian system is Arnold-Liouville integrable if for an open set U ⊂ R n there exists a symplectic map Φ : T n × U → R 2 n s. t. H ◦ Φ( ϕ, I ) depends only on I and � ϕ = ∂ I ( H ◦ Φ)( I ) = ω ( I ) , ˙ ( ϕ, I ) –action-angle coordinates ˙ I = 0 . In particular, Φ( T n × U ) is foliated by invariant n -dimensional tori and on each torus T n the flow is linear. V. Kaloshin (University of Maryland) Arnold diffusion November 24, 2014 8 / 22