Computer Assisted Proof of Drift Orbits Along Normally Hyperbolic Manifolds
01 Mar 2022-Communications in Nonlinear Science and Numerical Simulation (Elsevier BV)-Vol. 106, pp 105970
TL;DR: In this article, the authors developed a methodology for computer assisted proofs of diffusion in a-priori chaotic systems based on hyperbolic invariant manifolds theory, which allows to validate the needed conditions in a finite number of steps, which can be performed by a computer by means of rigorous interval arithmetic computations.
About: This article is published in Communications in Nonlinear Science and Numerical Simulation.The article was published on 2022-03-01 and is currently open access. It has received 2 citations till now. The article focuses on the topics: Computer-assisted proof & Mathematical proof.
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TL;DR: In this paper , the existence of trajectories that drift along a sequence of cylinders with homoclinic connections was shown to be possible under certain conditions on the dynamics of the trajectories.
Abstract:
We develop a geometric mechanism to prove the existence of orbits that drift along a prescribed sequence of cylinders, under some general conditions on the dynamics. This mechanism can be used to prove the existence of Arnold diffusion for large families of perturbations of Tonelli Hamiltonians on
The main geometric objects in our framework are
Our main result is on the existence of diffusing orbits which drift along such chains of cylinders, under precise conditions on the dynamics on the cylinders – i.e., the existence of Poincaré sections with the return maps satisfying a tilt condition – and on the geometric properties of the intersections of the center-stable and center-unstable manifolds of the cylinders – i.e., certain compatibility conditions between the tilt map and the homoclinic maps associated to its essential invariant circles.
We give two proofs of our result, a very short and abstract one, and a more constructive one, aimed at possible applications to concrete systems.
2 citations
TL;DR: In this paper , a computer assisted proof or diffusion in the Planar Elliptic Restricted Three Body Problem (PEBP) is presented, based on shadowing of orbits along transversal intersections of stable/unstable manifolds of a normally hyperbolic cylinder.
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1,187 citations
Book•
01 Jan 1987
TL;DR: In this article, the authors introduce the notion of contact manifolds as a way to represent the local structure of a Poisson manifold and a Lie group on its cotangent bundle.
Abstract: I. Symplectic vector spaces and symplectic vector bundles.- 1: Symplectic vector spaces.- 1. Properties of exterior forms of arbitrary degree.- 2. Properties of exterior 2-forms.- 3. Symplectic forms and their automorphism groups.- 4. The contravariant approach.- 5. Orthogonality in a symplectic vector space.- 6. Forms induced on a vector subspace of a symplectic vector space.- 7. Additional properties of Lagrangian subspaces.- 8. Reduction of a symplectic vector space. Generalizations.- 9. Decomposition of a symplectic form.- 10. Complex structures adapted to a symplectic structure.- 11. Additional properties of the symplectic group.- 2: Symplectic vector bundles.- 12. Properties of symplectic vector bundles.- 13. Orthogonality and the reduction of a symplectic vector bundle.- 14. Complex structures on symplectic vector bundles.- 3: Remarks concerning the operator ? and Lepage's decomposition theorem.- 15. The decomposition theorem in a symplectic vector space.- 16. Decomposition theorem for exterior differential forms.- 17. A first approach to Darboux's theorem.- II. Semi-basic and vertical differential forms in mechanics.- 1. Definitions and notations.- 2. Vector bundles associated with a surjective submersion.- 3. Semi-basic and vertical differential forms.- 4. The Liouville form on the cotangent bundle.- 5. Symplectic structure on the cotangent bundle.- 6. Semi-basic differential forms of arbitrary degree.- 7. Vector fields and second-order differential equations.- 8. The Legendre transformation on a vector bundle.- 9. The Legendre transformation on the tangent and cotangent bundles.- 10. Applications to mechanics: Lagrange and Hamilton equations.- 11. Lagrange equations and the calculus of variations.- 12. The Poincare-Cartan integral invariant.- 13. Mechanical systems with time dependent Hamiltonian or Lagrangian functions.- III. Symplectic manifolds and Poisson manifolds.- 1. Symplectic manifolds definition and examples.- 2. Special submanifolds of a symplectic manifold.- 3. Symplectomorphisms.- 4. Hamiltonian vector fields.- 5. The Poisson bracket.- 6. Hamiltonian systems.- 7. Presymplectic manifolds.- 8. Poisson manifolds.- 9. Poisson morphisms.- 10. Infinitesimal automorphisms of a Poisson structure.- 11. The local structure of Poisson manifolds.- 12. The symplectic foliation of a Poisson manifold.- 13. The local structure of symplectic manifolds.- 14. Reduction of a symplectic manifold.- 15. The Darboux-Weinstein theorems.- 16. Completely integrable Hamiltonian systems.- 17. Exercises.- IV. Action of a Lie group on a symplectic manifold.- 1. Symplectic and Hamiltonian actions.- 2. Elementary properties of the momentum map.- 3. The equivariance of the momentum map.- 4. Actions of a Lie group on its cotangent bundle.- 5. Momentum maps and Poisson morphisms.- 6. Reduction of a symplectic manifold by the action of a Lie group.- 7. Mutually orthogonal actions and reduction.- 8. Stationary motions of a Hamiltonian system.- 9. The motion of a rigid body about a fixed point.- 10. Euler's equations.- 11. Special formulae for the group SO(3).- 12. The Euler-Poinsot problem.- 13. The Euler-Lagrange and Kowalevska problems.- 14. Additional remarks and comments.- 15. Exercises.- V. Contact manifolds.- 1. Background and notations.- 2. Pfaffian equations.- 3. Principal bundles and projective bundles.- 4. The class of Pfaffian equations and forms.- 5. Darboux's theorem for Pfaffian forms and equations.- 6. Strictly contact structures and Pfaffian structures.- 7. Protectable Pfaffian equations.- 8. Homogeneous Pfaffian equations.- 9. Liouville structures.- 10. Fibered Liouville structures.- 11. The automorphisms of Liouville structures.- 12. The infinitesimal automorphisms of Liouville structures.- 13. The automorphisms of strictly contact structures.- 14. Some contact geometry formulae in local coordinates.- 15. Homogeneous Hamiltonian systems.- 16. Time-dependent Hamiltonian systems.- 17. The Legendre involution in contact geometry.- 18. The contravariant point of view.- Appendix 1. Basic notions of differential geometry.- 1. Differentiable maps, immersions, submersions.- 2. The flow of a vector field.- 3. Lie derivatives.- 4. Infinitesimal automorphisms and conformai infinitesimal transformations.- 5. Time-dependent vector fields and forms.- 6. Tubular neighborhoods.- 7. Generalizations of Poincare's lemma.- Appendix 2. Infinitesimal jets.- 1. Generalities..- 2. Velocity spaces.- 3. Second-order differential equations.- 4. Sprays and the exponential mapping.- 5. Covelocity spaces.- 6. Liouville forms on jet spaces.- Appendix 3. Distributions, Pfaffian systems and foliations.- 1. Distributions and Pfaffian systems.- 2. Completely integrable distributions.- 3. Generalized foliations defined by families of vector fields.- 4. Differentiable distributions of constant rank.- Appendix 4. Integral invariants.- 1. Integral invariants of a vector field.- 2. Integral invariants of a foliation.- 3. The characteristic distribution of a differential form.- Appendix 5. Lie groups and Lie algebras.- 1. Lie groups and Lie algebras generalities.- 2. The exponential map.- 3. Action of a Lie group on a manifold.- 4. The adjoint and coadjoint representations.- 5. Semi-direct products.- 6. Notions regarding the cohomology of Lie groups and Lie algebras.- 7. Affine actions of Lie groups and Lie algebras.- Appendix 6. The Lagrange-Grassmann manifold.- 1. The structure of the Lagrange-Grassmann manifold.- 2. The signature of a Lagrangian triplet.- 3. The fundamental groups of the symplectic group and of the Lagrange-Grassmann manifold.- Appendix 7. Morse families and Lagrangian submanifolds.- 1. Lagrangian submanifolds of a cotangent bundle.- 2. Hamiltonian systems and first-order partial differential equations.- 3. Contact manifolds and first-order partial differential equations.- 4. Jacobi's theorem.- 5. The Hamilton-Jacobi equation for autonomous systems.- 6. The Hamilton-Jacobi equation for non autonomous systems.
1,095 citations
Book•
31 Jan 1971
TL;DR: The three-body problem was studied in this paper, where Covarinace of Lagarangian Derivatives and Canonical Transformation were applied to the problem of estimating the perimeter and the velocity of the system.
Abstract: The Three-Body Problem: Covarinace of Lagarangian Derivatives.- Canonical Transformation.- The Hamilton-Jacobi Equation.- The Cauchy-Existence Theorem.- The n-Body Poblem.- Collision.- The Regularizing Transformation.- Application to the Three-Bdy Problem.- An Estimate of the Perimeter.- An Estimate of the Velocity.- Sundman's Theorem.- Triple Collision.- Triple-Collision Orbits.- Periodic Solutions: The Solutions of Lagrange.- Eigenvalues.- An Existence Theorem.- The Convergence Proof.- An Application to the Solution of Lagrange.- Hill's Problem.- A Generalization of Hill's Problem.- The Continuation Method.- The Fixed-Point Theorem.- Area-Preserving Analytic Transformations.- The Birkhoff Fixed-Point Theorem.- Stability: The Function-Theoretic Center Problem.- The Convergence Proof.- The Poincare Center Problem.- The Theorem of Liapunov.- The Theorem of Dirichlet.- The Normal Form of Hamiltonian Systems.- Area-Preserving Transformations.- Existence of Invariant Curves.- Proof of Lemma.- Application to the Stability Problem.- Stability of Equilibrium Solutions.- Quasi-Periodic Motion and Systems of Several Degrees of Freedom.- The Recurrence Theorem.
1,075 citations
TL;DR: In this paper, the authors consider the problem of proper degeneracy and prove the existence of a non-degeneracy of diffeomorphisms with respect to a constant number of vertices.
Abstract: CONTENTSIntroduction § 1. Results § 2. Preliminary results from mechanics § 3. Preliminary results from mathematics § 4. The simplest problem of stability § 5. Contents of the paperChapter I. Theory of perturbations § 1. Integrable and non-integrable problems of dynamics § 2. The classical theory of perturbations § 3. Small denominators § 4. Newton's method § 5. Proper degeneracy § 6. Remark 1 § 7. Remark 2 § 8. Application to the problem of proper degeneracy § 9. Limiting degeneracy. Birkhoff's transformation § 10. Stability of positions of equilibrium of Hamiltonian systemsChapter II. Adiabatic invariants § 1. The concept of an adiabatic invariant § 2. Perpetual adiabatic invariance of action with a slow periodic variation of the Hamiltonian § 3. Adiabatic invariants of conservative systems § 4. Magnetic traps § 5. The many-dimensional caseChapter III. The stability of planetary motions § 1. Picture of the motion § 2. Jacobi, Delaunay and Poincare variables §3. Birkhoff's transformation § 4. Calculation of the asymptotic behaviour of the coefficients in the expansion of § 5. The many-body problemChapter IV. The fundamental theorem § 1. Fundamental theorem § 2. Inductive theorem § 3. Inductive lemma § 4. Fundamental lemma § 5. Lemma on averaging over rapid variables § 6. Proof of the fundamental lemma § 7. Proof of the inductive lemma § 8. Proof of the inductive theorem § 9. Lemma on the non-degeneracy of diffeomorphisms § 10. Averaging over rapid variables § 11. Polar coordinates § 12. The applicability of the inductive theorem § 13. Passage to the limit § 14. Proof of the fundamental theoremChapter V. Technical lemmas § 1. Domains of type D § 2. Arithmetic lemmas § 3. Analytic lemmas § 4. Geometric lemmas § 5. Convergence lemmas § 6. NotationChapter VI. Appendix § 1. Integrable systems § 2. Unsolved problems § 3. Neighbourhood of an invariant manifold §4. Intermixing § 5. Smoothing techniquesReferences
1,057 citations
Journal Article•
756 citations