Action-Angle Variables and KAM-Theory in General Relativity

Save this PDF as:

Size: px
Start display at page:

Download "Action-Angle Variables and KAM-Theory in General Relativity"

Transcription

1 Action-Angle Variables and KAM-Theory in General Relativity Daniela Kunst, Volker Perlick, Claus Lämmerzahl Center of Space Technology and Microgravity University of Bremen, Germany Workshop in Oldenburg March 2013

2 Outline Introduction Schwarzschild spacetime KAM-Theorem and non-degeneracy-conditions Kerr spacetime KAM-Theorem and non-degeneracy-conditions Conclusion and Outlook

3 Outline Introduction Schwarzschild spacetime KAM-Theorem and non-degeneracy-conditions Kerr spacetime KAM-Theorem and non-degeneracy-conditions Conclusion and Outlook

4 Outline Introduction Schwarzschild spacetime KAM-Theorem and non-degeneracy-conditions Kerr spacetime KAM-Theorem and non-degeneracy-conditions Conclusion and Outlook

5 Outline Introduction Schwarzschild spacetime KAM-Theorem and non-degeneracy-conditions Kerr spacetime KAM-Theorem and non-degeneracy-conditions Conclusion and Outlook

6 Outline Introduction Schwarzschild spacetime KAM-Theorem and non-degeneracy-conditions Kerr spacetime KAM-Theorem and non-degeneracy-conditions Conclusion and Outlook

7 Outline Introduction Schwarzschild spacetime KAM-Theorem and non-degeneracy-conditions Kerr spacetime KAM-Theorem and non-degeneracy-conditions Conclusion and Outlook

8 Outline Introduction Schwarzschild spacetime KAM-Theorem and non-degeneracy-conditions Kerr spacetime KAM-Theorem and non-degeneracy-conditions Conclusion and Outlook

9 Outline Introduction Schwarzschild spacetime KAM-Theorem and non-degeneracy-conditions Kerr spacetime KAM-Theorem and non-degeneracy-conditions Conclusion and Outlook

10 Introduction Integrable dynamical systems Dynamical systems of n-dof are described by the Hamiltonian H( q, p) and the equations of motion ṗ i = H(p i, q i ) q i q i = H(p i, q i ) p i The system is integrable if it has n integrals of motion F i, that are in involution. Its phase space is foliated into n-dimensional tori.

11 Introduction Integrable dynamical systems Dynamical systems of n-dof are described by the Hamiltonian H( q, p) and the equations of motion ṗ i = H(p i, q i ) q i q i = H(p i, q i ) p i The system is integrable if it has n integrals of motion F i, that are in involution. Its phase space is foliated into n-dimensional tori.

12 Introduction Integrable dynamical systems

13 Introduction Integrable dynamical systems Action-Angle Variables simplified description of periodic or quasi-periodic motion of integrable systems Action-Variable: specifies a particular torus Angle-Variable: describes location on this torus I i = γ i p i dq i, Equations of Motion Φ i = p i I i dq i H( I, Φ) İ i = H(Ii, Φ i ) = Φ i 0 and Φ i = H(Ii, Φ i ) = ω i (I i ) = const I i = H(Ii )

14 Introduction Integrable dynamical systems Action-Angle Variables simplified description of periodic or quasi-periodic motion of integrable systems Action-Variable: specifies a particular torus Angle-Variable: describes location on this torus I i = γ i p i dq i, Equations of Motion Φ i = p i I i dq i H( I, Φ) İ i = H(Ii, Φ i ) = Φ i 0 and Φ i = H(Ii, Φ i ) = ω i (I i ) = const I i = H(Ii )

15 Introduction Integrable dynamical systems Action-Angle Variables simplified description of periodic or quasi-periodic motion of integrable systems Action-Variable: specifies a particular torus Angle-Variable: describes location on this torus I i = γ i p i dq i, Equations of Motion Φ i = p i I i dq i H( I, Φ) İ i = H(Ii, Φ i ) = Φ i 0 and Φ i = H(Ii, Φ i ) = ω i (I i ) = const I i = H(Ii )

16 Introduction Integrable dynamical systems Action-Angle Variables simplified description of periodic or quasi-periodic motion of integrable systems Action-Variable: specifies a particular torus Angle-Variable: describes location on this torus I i = γ i p i dq i, Equations of Motion Φ i = p i I i dq i H( I, Φ) İ i = H(Ii, Φ i ) = Φ i 0 and Φ i = H(Ii, Φ i ) = ω i (I i ) = const I i = H(Ii )

17 Introduction The KAM-Theorem Integrable system + a small perturbation H(I, Φ) = H 0 (I) + εh pert (I, Φ) KAM-Theorem (Kolmogorov-Arnold-Moser)

18 Introduction The KAM-Theorem Integrable system + a small perturbation H(I, Φ) = H 0 (I) + εh pert (I, Φ) KAM-Theorem (Kolmogorov-Arnold-Moser) If an unperturbed system is non-degenerate, then for sufficiently small [...] perturbations, most non-resonant invariant tori do not vanish, [...] so that in phase space of the perturbed system, too, there are invariant tori densely filled with phase curves winding around them conditionally-periodically [...]. V.I. Arnold: Mathematical aspects of classical and celestial mechanics, Springer, Berlin (2006)

19 Introduction The KAM-Theorem

20 Introduction The KAM-Theorem Integrable system + a small perturbation H(I, Φ) = H 0 (I) + εh pert (I, Φ) KAM-Theorem (Kolmogorov-Arnold-Moser) Iso-Energetic Non-Degeneracy Condition (V. Arnold) D A = det ) ω 0 ω 0 ( ω I

21 Schwarzschild Spacetime General Hamiltonian H = p 0 = g0i p i g 00 + [ ( ) ] 1 g ij p i p j +m 2 2 g + 0i 2 p i g 00 g 00

22 Schwarzschild Spacetime... for Schwarzschild geometry (assume θ = π 2 ) [(1 H = r s ) ( r p 2 r + 1 p 2 r 2 θ + ) p2 ϕ sin( π 2 )2 ] (1 + m 2 r s ) r

23 Schwarzschild Spacetime... for Schwarzschild geometry (assume θ = π 2 ) [(1 H = r s ) ( r p 2 r + 1 p 2 r 2 θ + ) p2 ϕ sin( π 2 )2 ] (1 + m 2 r s ) r Action-Variables: I ϕ = p ϕ dϕ = 2πL γ φ I θ = 0 I r = p r dr = 2 γ r ra r p [ ( 1 1 r s ) r 1 ( 1 r s r ] )H L2 r 2 m2 dr

24 Schwarzschild Spacetime KAM-Theorem and the iso-energetic non-degeneracy condition Action-Variables: I ϕ = p ϕ dϕ = 2πL γ φ I θ = 0 I r = p r dr = 2 γ r ra r p [ ( 1 1 r s ) r 1 ( 1 r s r ] )H L2 r 2 m2 dr The Hamiltonian is implicitly given as: H(I r, I ϕ )

25 Schwarzschild Spacetime KAM-Theorem and the iso-energetic non-degeneracy condition I ϕ = p ϕdϕ = 2πL γ φ I r = p rdr = 2 γ r ra r p = H(I r, I ϕ) implicitly [ ( r s ) ( r 1 r s r ) H L2 r 2 m2 ] dr Fundamental Frequencies defined as ω i = H by the equations of motion and applying the I i implicit function theorem yields: ω r = ( Ir H ) 1 ω ϕ = 1 2π I r L ω r (calculation based on W. Schmidt 2002)

26 Schwarzschild Spacetime KAM-Theorem and the iso-energetic non-degeneracy condition Fundamental Frequencies ω r = ( Ir H ) 1 ω ϕ = 1 I r 2π L ωr iso-energetic non-degeneracy condition D A = det ( ω I = ω3 r (2π) 2 ) ω = ω3 r 2 I r ω 0 (2π) 2 L 2 (( ) L c c L 1 K(k) L k k L ( )) E(k) 1 k K(k) 0 2 which means, that the map (I) ( ωϕ = 1 ) I r ω r 2π L bijective at fixed H must be

27 Schwarzschild Spacetime KAM-Theorem and the iso-energetic non-degeneracy condition D A ( L c c L 1)K(k) L k k L ( E(k) 1 k 2 K(k)) > 0 L c c L 1 < 0 L c c L 1 = 0 H = 1 bdries of bound motion

28 Schwarzschild Spacetime KAM-Theorem and the iso-energetic non-degeneracy condition c DA ( Lc L 1)K(k) L c c L L k E(k) ( k L 1 k2 K(k)) > 0 1<0 L c 1=0 c L H =1 bdries of bound motion H = const ωϕ ωr = const bdries of bound motion

29 KerrSpacetime... for Kerr geometry H = ar sp ϕ ρ 2 +r s(r 2 +a 2 )r r + 2 p 2 r + p2 θ + ρ2 rsr sin(θ) 2 p2 ϕ + ρ2 m 2 ρ 2 +r s+(r 2 +a 2 )r ( + with = r 2 + a 2 r sr and ρ 2 = r 2 + a 2 cos(θ) 2 ) 2 ar sp ϕ ρ r 2 +r s(r 2 +a 2 )r

30 KerrSpacetime... for Kerr geometry H = ar sp ϕ ρ 2 +r s(r 2 +a 2 )r r + 2 p 2 r + p2 θ + ρ2 rsr sin(θ) 2 p2 ϕ + ρ2 m 2 ρ 2 +r s+(r 2 +a 2 )r ( + with = r 2 + a 2 r sr and ρ 2 = r 2 + a 2 cos(θ) 2 ) 2 ar sp ϕ ρ r 2 +r s(r 2 +a 2 )r Action-Variables: I ϕ = p ϕ dϕ = 2πL z γ φ 1 2 C + Lz (1 H 2 )a 2 (1 y 2 ) L2 z y I θ = p θ dθ = 2 γ θ θ min (1 y) [(C + L 2 z )y (1 H 2 )a 2 (1 y)y L 2 z] dy ra 1 [ ] I r = p r dr = 2 ((r γ r 2 + a 2 )H al 2 z) 2 (r 2 + (L 2 z ah) 2 + C) dr r p

31 Kerr Spacetime KAM-Theorem and the iso-energetic non-degeneracy Condition Action-Variables: I ϕ = p ϕ dϕ = 2πL z γ φ 1 2 C + Lz (1 H 2 )a 2 (1 y 2 ) L2 z y I θ = p θ dθ = 2 γ θ θ min (1 y) [(C + L 2 z )y (1 H 2 )a 2 (1 y)y L 2 z] dy ra 1 [ ] I r = p r dr = 2 ((r γ r 2 + a 2 )H al 2 z) 2 (r 2 + (L 2 z ah) 2 + C) dr r p The Hamiltonian is implicitly given as: C(I θ, I ϕ, H) H(I r, I θ, I ϕ )

32 Kerr Spacetime KAM-Theorem and the iso-energetic non-degeneracy Condition I ϕ = p ϕdϕ = 2πL z γ φ 1 I θ = p θ dθ = 2 γ θ θ min I r = p r dr = 2 γ r ra rp = H(I r, I θ, I ϕ) implicitly C + L 2 z (1 H2 )a 2 (1 y 2 ) L2 z y (1 y) [ (C + L 2 z )y (1 H2 )a 2 (1 y)y L 2 z 1 2 [ ((r 2 + a 2 )H al 2 z ] dy ) 2 (r 2 + (L z ah) 2 + C)] dr Fundamental Frequencies ω r = I θ C Q I r C ω θ = Q ω ϕ = 1 [ Ir 2π L ω r + I ] θ L ω θ with Q = I θ I r Ir I θ C H C H (calculation based on W. Schmidt 2002)

33 Kerr Spacetime KAM-Theorem and the iso-energetic non-degeneracy Condition Fundamental Frequencies ω r = I θ C Q I r C ω θ = Q ω ϕ = 1 [ Ir 2π L ωr + I ] θ L ω θ iso-energetic non-degeneracy condition ( ω ) ω D A = det I 0 ω 0 which means, that the map (I) must be bijective at fixed H ( ωr, ω ) ( ϕ = 1 [ Iθ ω θ ω θ 2π L + I ] r ω r, ω ) r L ω θ ω θ

34 Conclusion and Outlook Particle with Spin around a Black Hole can be expressed as H = H 0 + SH pert Schwarzschild: D A 0 the motion of the particle with spin will mostly remain regular for small enough spin values Kerr: also expected but still to be shown What is the limit of the value of the spin for the non-resonant tori to break down? What is the rate of diffusion of the motion in Kerr spacetime?

35 Conclusion and Outlook Particle with Spin around a Black Hole can be expressed as H = H 0 + SH pert Schwarzschild: D A 0 the motion of the particle with spin will mostly remain regular for small enough spin values Kerr: also expected but still to be shown What is the limit of the value of the spin for the non-resonant tori to break down? What is the rate of diffusion of the motion in Kerr spacetime?

36 Conclusion and Outlook Particle with Spin around a Black Hole can be expressed as H = H 0 + SH pert Schwarzschild: D A 0 the motion of the particle with spin will mostly remain regular for small enough spin values Kerr: also expected but still to be shown What is the limit of the value of the spin for the non-resonant tori to break down? What is the rate of diffusion of the motion in Kerr spacetime?

37 Conclusion and Outlook Particle with Spin around a Black Hole can be expressed as H = H 0 + SH pert Schwarzschild: D A 0 the motion of the particle with spin will mostly remain regular for small enough spin values Kerr: also expected but still to be shown What is the limit of the value of the spin for the non-resonant tori to break down? What is the rate of diffusion of the motion in Kerr spacetime?

38 Conclusion and Outlook Particle with Spin around a Black Hole can be expressed as H = H 0 + SH pert Schwarzschild: D A 0 the motion of the particle with spin will mostly remain regular for small enough spin values Kerr: also expected but still to be shown What is the limit of the value of the spin for the non-resonant tori to break down? What is the rate of diffusion of the motion in Kerr spacetime?

39 Conclusion and Outlook Particle with Spin around a Black Hole can be expressed as H = H 0 + SH pert Schwarzschild: D A 0 the motion of the particle with spin will mostly remain regular for small enough spin values Kerr: also expected but still to be shown What is the limit of the value of the spin for the non-resonant tori to break down? What is the rate of diffusion of the motion in Kerr spacetime?

Dynamics of spinning particles in Schwarzschild spacetime

Dynamics of spinning particles in Schwarzschild spacetime Dynamics of spinning particles in Schwarzschild spacetime, Volker Perlick, Claus Lämmerzahl Center of Space Technology and Microgravity University of Bremen, Germany 08.05.2014 RTG Workshop, Bielefeld

More information

= 0. = q i., q i = E

= 0. = q i., q i = E Summary of the Above Newton s second law: d 2 r dt 2 = Φ( r) Complicated vector arithmetic & coordinate system dependence Lagrangian Formalism: L q i d dt ( L q i ) = 0 n second-order differential equations

More information

Perturbation theory, KAM theory and Celestial Mechanics 7. KAM theory

Perturbation theory, KAM theory and Celestial Mechanics 7. KAM theory Perturbation theory, KAM theory and Celestial Mechanics 7. KAM theory Alessandra Celletti Department of Mathematics University of Roma Tor Vergata Sevilla, 25-27 January 2016 Outline 1. Introduction 2.

More information

Aubry Mather Theory from a Topological Viewpoint

Aubry Mather Theory from a Topological Viewpoint Aubry Mather Theory from a Topological Viewpoint III. Applications to Hamiltonian instability Marian Gidea,2 Northeastern Illinois University, Chicago 2 Institute for Advanced Study, Princeton WORKSHOP

More information

Lecture 11 : Overview

Lecture 11 : Overview Lecture 11 : Overview Error in Assignment 3 : In Eq. 1, Hamiltonian should be H = p2 r 2m + p2 ϕ 2mr + (p z ea z ) 2 2 2m + eφ (1) Error in lecture 10, slide 7, Eq. (21). Should be S(q, α, t) m Q = β =

More information

Secular and oscillatory motions in dynamical systems. Henk Broer Johann Bernoulli Instituut voor Wiskunde en Informatica Rijksuniversiteit Groningen

Secular and oscillatory motions in dynamical systems. Henk Broer Johann Bernoulli Instituut voor Wiskunde en Informatica Rijksuniversiteit Groningen Secular and oscillatory motions in dynamical systems Henk Broer Johann Bernoulli Instituut voor Wiskunde en Informatica Rijksuniversiteit Groningen Contents 1. Toroidal symmetry 2. Secular (slow) versus

More information

Lectures on Dynamical Systems. Anatoly Neishtadt

Lectures on Dynamical Systems. Anatoly Neishtadt Lectures on Dynamical Systems Anatoly Neishtadt Lectures for Mathematics Access Grid Instruction and Collaboration (MAGIC) consortium, Loughborough University, 2007 Part 3 LECTURE 14 NORMAL FORMS Resonances

More information

Geometry and Dynamics of singular symplectic manifolds. Session 9: Some applications of the path method in b-symplectic geometry

Geometry and Dynamics of singular symplectic manifolds. Session 9: Some applications of the path method in b-symplectic geometry Geometry and Dynamics of singular symplectic manifolds Session 9: Some applications of the path method in b-symplectic geometry Eva Miranda (UPC-CEREMADE-IMCCE-IMJ) Fondation Sciences Mathématiques de

More information

KAM theory: a journey from conservative to dissipative systems

KAM theory: a journey from conservative to dissipative systems KAM theory: a journey from conservative to dissipative systems Alessandra Celletti Department of Mathematics University of Roma Tor Vergata 4 July 2012 Outline 1. Introduction 2. Qualitative description

More information

Hamiltonian Dynamics

Hamiltonian Dynamics Hamiltonian Dynamics CDS 140b Joris Vankerschaver jv@caltech.edu CDS Feb. 10, 2009 Joris Vankerschaver (CDS) Hamiltonian Dynamics Feb. 10, 2009 1 / 31 Outline 1. Introductory concepts; 2. Poisson brackets;

More information

Chaos in Hamiltonian systems

Chaos in Hamiltonian systems Chaos in Hamiltonian systems Teemu Laakso April 26, 2013 Course material: Chapter 7 from Ott 1993/2002, Chaos in Dynamical Systems, Cambridge http://matriisi.ee.tut.fi/courses/mat-35006 Useful reading:

More information

ENERGY SURFACES AND HIERARCHIES OF BIFURCATIONS. Instabilities in the forced truncated NLS.

ENERGY SURFACES AND HIERARCHIES OF BIFURCATIONS. Instabilities in the forced truncated NLS. ENERGY SURFACES AND HIERARCHIES OF BIFURCATIONS. Instabilities in the forced truncated NLS. Eli Shlizerman Faculty of mathematical and computer science Weizmann Institute, Rehovot 76100, Israel Vered Rom-Kedar

More information

Eva Miranda. UPC-Barcelona and BGSMath. XXV International Fall Workshop on Geometry and Physics Madrid

Eva Miranda. UPC-Barcelona and BGSMath. XXV International Fall Workshop on Geometry and Physics Madrid b-symplectic manifolds: going to infinity and coming back Eva Miranda UPC-Barcelona and BGSMath XXV International Fall Workshop on Geometry and Physics Madrid Eva Miranda (UPC) b-symplectic manifolds Semptember,

More information

Shadows of black holes

Shadows of black holes Shadows of black holes Volker Perlick ZARM Center of Applied Space Technology and Microgravity, U Bremen, Germany. 00 11 000 111 000 111 0000 1111 000 111 000 111 0000 1111 000 111 000 000 000 111 111

More information

Physics 106b: Lecture 7 25 January, 2018

Physics 106b: Lecture 7 25 January, 2018 Physics 106b: Lecture 7 25 January, 2018 Hamiltonian Chaos: Introduction Integrable Systems We start with systems that do not exhibit chaos, but instead have simple periodic motion (like the SHO) with

More information

NEKHOROSHEV AND KAM STABILITIES IN GENERALIZED HAMILTONIAN SYSTEMS

NEKHOROSHEV AND KAM STABILITIES IN GENERALIZED HAMILTONIAN SYSTEMS NEKHOROSHEV AND KAM STABILITIES IN GENERALIZED HAMILTONIAN SYSTEMS YONG LI AND YINGFEI YI Abstract. We present some Nekhoroshev stability results for nearly integrable, generalized Hamiltonian systems

More information

Analytic solutions of the geodesic equation in static spherically symmetric spacetimes in higher dimensions

Analytic solutions of the geodesic equation in static spherically symmetric spacetimes in higher dimensions Analytic solutions of the geodesic equation in static spherically symmetric spacetimes in higher dimensions Eva Hackmann 2, Valeria Kagramanova, Jutta Kunz, Claus Lämmerzahl 2 Oldenburg University, Germany

More information

Im + α α. β + I 1 I 1< 0 I 1= 0 I 1 > 0

Im + α α. β + I 1 I 1< 0 I 1= 0 I 1 > 0 ON THE HAMILTONIAN ANDRONOV-HOPF BIFURCATION M. Olle, J. Villanueva 2 and J. R. Pacha 3 2 3 Departament de Matematica Aplicada I (UPC), Barcelona, Spain In this contribution, we consider an specic type

More information

Some Collision solutions of the rectilinear periodically forced Kepler problem

Some Collision solutions of the rectilinear periodically forced Kepler problem Advanced Nonlinear Studies 1 (2001), xxx xxx Some Collision solutions of the rectilinear periodically forced Kepler problem Lei Zhao Johann Bernoulli Institute for Mathematics and Computer Science University

More information

On the shadows of black holes and of other compact objects

On the shadows of black holes and of other compact objects On the shadows of black holes and of other compact objects Volker Perlick ( ZARM, Univ. Bremen, Germany) 1. Schwarzschild spacetime mass m photon sphere at r = 3m shadow ( escape cones ): J. Synge, 1966

More information

PERIODIC SOLUTIONS OF THE PLANETARY N BODY PROBLEM

PERIODIC SOLUTIONS OF THE PLANETARY N BODY PROBLEM 1 PERIODIC SOLUTIONS OF THE PLANETARY N BODY PROBLEM L. CHIERCHIA Department of Mathematics, Roma Tre University, Rome, I-146, Italy E-mail: luigi@mat.uniroma3.it The closure of periodic orbits in the

More information

ON JUSTIFICATION OF GIBBS DISTRIBUTION

ON JUSTIFICATION OF GIBBS DISTRIBUTION Department of Mechanics and Mathematics Moscow State University, Vorob ievy Gory 119899, Moscow, Russia ON JUSTIFICATION OF GIBBS DISTRIBUTION Received January 10, 2001 DOI: 10.1070/RD2002v007n01ABEH000190

More information

The restricted, circular, planar three-body problem

The restricted, circular, planar three-body problem The restricted, circular, planar three-body problem Luigi Chierchia Dipartimento di Matematica Università Roma Tre Largo S L Murialdo 1, I-00146 Roma (Italy) (luigi@matuniroma3it) March, 2005 1 The restricted

More information

Where is the PdV term in the first law of black-hole thermodynamics?

Where is the PdV term in the first law of black-hole thermodynamics? Where is the PdV term in the first law of black-hole thermodynamics? Brian P Dolan National University of Ireland, Maynooth, Ireland and Dublin Institute for Advanced Studies, Ireland 9th Vienna Central

More information

arxiv:gr-qc/ v1 11 May 2000

arxiv:gr-qc/ v1 11 May 2000 EPHOU 00-004 May 000 A Conserved Energy Integral for Perturbation Equations arxiv:gr-qc/0005037v1 11 May 000 in the Kerr-de Sitter Geometry Hiroshi Umetsu Department of Physics, Hokkaido University Sapporo,

More information

Magnetostatics III Magnetic Vector Potential (Griffiths Chapter 5: Section 4)

Magnetostatics III Magnetic Vector Potential (Griffiths Chapter 5: Section 4) Dr. Alain Brizard Electromagnetic Theory I PY ) Magnetostatics III Magnetic Vector Potential Griffiths Chapter 5: Section ) Vector Potential The magnetic field B was written previously as Br) = Ar), 1)

More information

A strong form of Arnold diffusion for two and a half degrees of freedom

A strong form of Arnold diffusion for two and a half degrees of freedom A strong form of Arnold diffusion for two and a half degrees of freedom arxiv:1212.1150v1 [math.ds] 5 Dec 2012 V. Kaloshin, K. Zhang February 7, 2014 Abstract In the present paper we prove a strong form

More information

INVARIANT TORI IN THE LUNAR PROBLEM. Kenneth R. Meyer, Jesús F. Palacián, and Patricia Yanguas. Dedicated to Jaume Llibre on his 60th birthday

INVARIANT TORI IN THE LUNAR PROBLEM. Kenneth R. Meyer, Jesús F. Palacián, and Patricia Yanguas. Dedicated to Jaume Llibre on his 60th birthday Publ. Mat. (2014), 353 394 Proceedings of New Trends in Dynamical Systems. Salou, 2012. DOI: 10.5565/PUBLMAT Extra14 19 INVARIANT TORI IN THE LUNAR PROBLEM Kenneth R. Meyer, Jesús F. Palacián, and Patricia

More information

GEOMETRIC QUANTIZATION

GEOMETRIC QUANTIZATION GEOMETRIC QUANTIZATION 1. The basic idea The setting of the Hamiltonian version of classical (Newtonian) mechanics is the phase space (position and momentum), which is a symplectic manifold. The typical

More information

Einstein-Maxwell-Chern-Simons Black Holes

Einstein-Maxwell-Chern-Simons Black Holes .. Einstein-Maxwell-Chern-Simons Black Holes Jutta Kunz Institute of Physics CvO University Oldenburg 3rd Karl Schwarzschild Meeting Gravity and the Gauge/Gravity Correspondence Frankfurt, July 2017 Jutta

More information

THE PLANETARY N BODY PROBLEM

THE PLANETARY N BODY PROBLEM THE PLANETARY N BODY PROBLEM LUIGI CHIERCHIA DIPARTIMENTO DI MATEMATICA UNIVERSITÀ ROMA TRE LARGO S L MURIALDO 1, I-00146 ROMA (ITALY) Keywords N body problem Three body problem Planetary systems Celestial

More information

A Glance at the Standard Map

A Glance at the Standard Map A Glance at the Standard Map by Ryan Tobin Abstract The Standard (Chirikov) Map is studied and various aspects of its intricate dynamics are discussed. Also, a brief discussion of the famous KAM theory

More information

Kerr black hole and rotating wormhole

Kerr black hole and rotating wormhole Kerr Fest (Christchurch, August 26-28, 2004) Kerr black hole and rotating wormhole Sung-Won Kim(Ewha Womans Univ.) August 27, 2004 INTRODUCTION STATIC WORMHOLE ROTATING WORMHOLE KERR METRIC SUMMARY AND

More information

CANONICAL PERTURBATION THEORY FOR THE ELLIPTIC RESTRICTED-THREE-BODY PROBLEM

CANONICAL PERTURBATION THEORY FOR THE ELLIPTIC RESTRICTED-THREE-BODY PROBLEM AAS 1-189 CANONICAL PERTURBATION THEORY FOR THE ELLIPTIC RESTRICTED-THREE-BODY PROBLEM Brenton Duffy and David F. Chichka The distinguishing characteristic of the elliptic restricted three-body problem

More information

Metric stability of the planetary N-body problem

Metric stability of the planetary N-body problem 1 Metric stability of the planetary N-body problem 3 Luigi Chierchia and Gabriella Pinzari 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 1 3 4 5 Abstract. The solution of the N-body problem (NBP) has challenged

More information

PH 451/551 Quantum Mechanics Capstone Winter 201x

PH 451/551 Quantum Mechanics Capstone Winter 201x These are the questions from the W7 exam presented as practice problems. The equation sheet is PH 45/55 Quantum Mechanics Capstone Winter x TOTAL POINTS: xx Weniger 6, time There are xx questions, for

More information

Math 350 Solutions for Final Exam Page 1. Problem 1. (10 points) (a) Compute the line integral. F ds C. z dx + y dy + x dz C

Math 350 Solutions for Final Exam Page 1. Problem 1. (10 points) (a) Compute the line integral. F ds C. z dx + y dy + x dz C Math 35 Solutions for Final Exam Page Problem. ( points) (a) ompute the line integral F ds for the path c(t) = (t 2, t 3, t) with t and the vector field F (x, y, z) = xi + zj + xk. (b) ompute the line

More information

Gevrey-smoothness of invariant tori for analytic nearly integrable Hamiltonian systems under Rüssmann s non-degeneracy condition

Gevrey-smoothness of invariant tori for analytic nearly integrable Hamiltonian systems under Rüssmann s non-degeneracy condition J Differential Equations 235 (27) 69 622 wwwelseviercom/locate/de Gevrey-smoothness of invariant tori for analytic nearly integrable Hamiltonian systems under Rüssmann s non-degeneracy condition Junxiang

More information

Journal of Differential Equations

Journal of Differential Equations J. Differential Equations 250 (2011) 2601 2623 Contents lists available at ScienceDirect Journal of Differential Equations www.elsevier.com/locate/jde A geometric mechanism of diffusion: Rigorous verification

More information

1 Schroenger s Equation for the Hydrogen Atom

1 Schroenger s Equation for the Hydrogen Atom Schroenger s Equation for the Hydrogen Atom Here is the Schroedinger equation in D in spherical polar coordinates. Note that the definitions of θ and φ are the exact reverse of what they are in mathematics.

More information

An introduction to General Relativity and the positive mass theorem

An introduction to General Relativity and the positive mass theorem An introduction to General Relativity and the positive mass theorem National Center for Theoretical Sciences, Mathematics Division March 2 nd, 2007 Wen-ling Huang Department of Mathematics University of

More information

Analytic Kerr Solution for Puncture Evolution

Analytic Kerr Solution for Puncture Evolution Analytic Kerr Solution for Puncture Evolution Jon Allen Maximal slicing of a spacetime with a single Kerr black hole is analyzed. It is shown that for all spin parameters, a limiting hypersurface forms

More information

Survey of strong normal-internal k : l resonances in quasi-periodically driven oscillators for l = 1, 2, 3.

Survey of strong normal-internal k : l resonances in quasi-periodically driven oscillators for l = 1, 2, 3. June, : WSPC - Proceedings Trim Size: in x in SPT-broer Survey of strong normal-internal k : l resonances in quasi-periodically driven oscillators for l =,,. H.W. BROER and R. VAN DIJK Institute for mathematics

More information

Rotating Black Holes in Higher Dimensions

Rotating Black Holes in Higher Dimensions Rotating Black Holes in Higher Dimensions Jutta Kunz Institute of Physics CvO University Oldenburg Models of Gravity in Higher Dimensions Bremen, 25.-29. 8. 2008 Jutta Kunz (Universität Oldenburg) Rotating

More information

Twisted Poisson manifolds and their almost symplectically complete isotropic realizations

Twisted Poisson manifolds and their almost symplectically complete isotropic realizations Twisted Poisson manifolds and their almost symplectically complete isotropic realizations Chi-Kwong Fok National Center for Theoretical Sciences Math Division National Tsing Hua University (Joint work

More information

Black Hole Astrophysics Chapters 7.5. All figures extracted from online sources of from the textbook.

Black Hole Astrophysics Chapters 7.5. All figures extracted from online sources of from the textbook. Black Hole Astrophysics Chapters 7.5 All figures extracted from online sources of from the textbook. Recap the Schwarzschild metric Sch means that this metric is describing a Schwarzschild Black Hole.

More information

Theory of Adiabatic Invariants A SOCRATES Lecture Course at the Physics Department, University of Marburg, Germany, February 2004

Theory of Adiabatic Invariants A SOCRATES Lecture Course at the Physics Department, University of Marburg, Germany, February 2004 Preprint CAMTP/03-8 August 2003 Theory of Adiabatic Invariants A SOCRATES Lecture Course at the Physics Department, University of Marburg, Germany, February 2004 Marko Robnik CAMTP - Center for Applied

More information

FFTs in Graphics and Vision. Homogenous Polynomials and Irreducible Representations

FFTs in Graphics and Vision. Homogenous Polynomials and Irreducible Representations FFTs in Graphics and Vision Homogenous Polynomials and Irreducible Representations 1 Outline The 2π Term in Assignment 1 Homogenous Polynomials Representations of Functions on the Unit-Circle Sub-Representations

More information

Control of chaos in Hamiltonian systems

Control of chaos in Hamiltonian systems Control of chaos in Hamiltonian systems G. Ciraolo, C. Chandre, R. Lima, M. Vittot Centre de Physique Théorique CNRS, Marseille M. Pettini Osservatorio Astrofisico di Arcetri, Università di Firenze Ph.

More information

Chaotic transport through the solar system

Chaotic transport through the solar system The Interplanetary Superhighway Chaotic transport through the solar system Richard Taylor rtaylor@tru.ca TRU Math Seminar, April 12, 2006 p. 1 The N -Body Problem N masses interact via mutual gravitational

More information

Instructions: No books. No notes. Non-graphing calculators only. You are encouraged, although not required, to show your work.

Instructions: No books. No notes. Non-graphing calculators only. You are encouraged, although not required, to show your work. Exam 3 Math 850-007 Fall 04 Odenthal Name: Instructions: No books. No notes. Non-graphing calculators only. You are encouraged, although not required, to show your work.. Evaluate the iterated integral

More information

Black Holes ASTR 2110 Sarazin. Calculation of Curved Spacetime near Merging Black Holes

Black Holes ASTR 2110 Sarazin. Calculation of Curved Spacetime near Merging Black Holes Black Holes ASTR 2110 Sarazin Calculation of Curved Spacetime near Merging Black Holes Test #2 Monday, November 13, 11-11:50 am Ruffner G006 (classroom) Bring pencils, paper, calculator You may not consult

More information

Orbital dynamics in the tidally-perturbed Kepler and Schwarzschild systems. Sam R. Dolan

Orbital dynamics in the tidally-perturbed Kepler and Schwarzschild systems. Sam R. Dolan Orbital dynamics in the tidally-perturbed Kepler and Schwarzschild systems Sam R. Dolan Gravity @ All Scales, Nottingham, 24th Aug 2015 Sam Dolan (Sheffield) Perturbed dynamics Nottingham 1 / 67 work in

More information

QUARK EFFECT ON H-ATOM SPECTRA(1S)

QUARK EFFECT ON H-ATOM SPECTRA(1S) QUARK EFFECT ON H-ATOM SPECTRA1S March 8, 18 1 A QUARK MODEL OF THE PROTON The purpose of this paper is to test a theory of how the quarks in the proton might affect the H-atom spectra. There are three

More information

arxiv: v1 [math.ds] 19 Dec 2012

arxiv: v1 [math.ds] 19 Dec 2012 arxiv:1212.4559v1 [math.ds] 19 Dec 2012 KAM theorems and open problems for infinite dimensional Hamiltonian with short range Xiaoping YUAN December 20, 2012 Abstract. Introduce several KAM theorems for

More information

The Stark Effect. a. Evaluate a matrix element. = ee. = ee 200 r cos θ 210. r 2 dr. Angular part use Griffiths page 139 for Yl m (θ, φ)

The Stark Effect. a. Evaluate a matrix element. = ee. = ee 200 r cos θ 210. r 2 dr. Angular part use Griffiths page 139 for Yl m (θ, φ) The Stark Effect a. Evaluate a matrix element. H ee z ee r cos θ ee r dr Angular part use Griffiths page 9 for Yl m θ, φ) Use the famous substitution to find that the angular part is dφ Y θ, φ) cos θ Y

More information

Problem Set 5 Math 213, Fall 2016

Problem Set 5 Math 213, Fall 2016 Problem Set 5 Math 213, Fall 216 Directions: Name: Show all your work. You are welcome and encouraged to use Mathematica, or similar software, to check your answers and aid in your understanding of the

More information

The Geometry of Relativity

The Geometry of Relativity The Geometry of Relativity Tevian Dray Department of Mathematics Oregon State University http://www.math.oregonstate.edu/~tevian PNWMAA 4/11/15 Tevian Dray The Geometry of Relativity 1/25 Books The Geometry

More information

Transcendental cases in stability problem. Hamiltonian systems

Transcendental cases in stability problem. Hamiltonian systems of Hamiltonian systems Boris S. Bardin Moscow Aviation Institute (Technical University) Faculty of Applied Mathematics and Physics Department of Theoretical Mechanics Hamiltonian Dynamics and Celestial

More information

arxiv: v2 [gr-qc] 8 Nov 2017

arxiv: v2 [gr-qc] 8 Nov 2017 Proceedings of RAGtime?/?,??/?? September,????/????, Opava, Czech Republic 1 S. Hledík and Z. Stuchlík, editors, Silesian University in Opava,????, pp. 1 8 arxiv:1711.02442v2 [gr-qc] 8 Nov 2017 Chaotic

More information

A parameterization method for Lagrangian tori of exact symplectic maps of R 2r

A parameterization method for Lagrangian tori of exact symplectic maps of R 2r A parameterization method for Lagrangian tori of exact symplectic maps of R 2r Jordi Villanueva Departament de Matemàtiques, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona (Spain.

More information

A Cantor set of tori with monodromy near a focus focus singularity

A Cantor set of tori with monodromy near a focus focus singularity INSTITUTE OF PHYSICS PUBLISHING Nonlinearity 17 (2004) 1 10 NONLINEARITY PII: S0951-7715(04)65776-8 A Cantor set of tori with monodromy near a focus focus singularity Bob Rink Mathematics Institute, Utrecht

More information

Angular Momentum. Classically the orbital angular momentum with respect to a fixed origin is. L = r p. = yp z. L x. zp y L y. = zp x. xpz L z.

Angular Momentum. Classically the orbital angular momentum with respect to a fixed origin is. L = r p. = yp z. L x. zp y L y. = zp x. xpz L z. Angular momentum is an important concept in quantum theory, necessary for analyzing motion in 3D as well as intrinsic properties such as spin Classically the orbital angular momentum with respect to a

More information

Quantum Mechanics in Three Dimensions

Quantum Mechanics in Three Dimensions Physics 342 Lecture 21 Quantum Mechanics in Three Dimensions Lecture 21 Physics 342 Quantum Mechanics I Monday, March 22nd, 21 We are used to the temporal separation that gives, for example, the timeindependent

More information

Integrals in cylindrical, spherical coordinates (Sect. 15.7)

Integrals in cylindrical, spherical coordinates (Sect. 15.7) Integrals in clindrical, spherical coordinates (Sect. 15.7 Integration in spherical coordinates. Review: Clindrical coordinates. Spherical coordinates in space. Triple integral in spherical coordinates.

More information

Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell 9.2 The Blackbody as the Ideal Radiator A material that absorbs 100 percent of the energy incident on it from all directions

More information

The Geometry of Relativity

The Geometry of Relativity The Geometry of Relativity Tevian Dray Department of Mathematics Oregon State University http://www.math.oregonstate.edu/~tevian OSU 4/27/15 Tevian Dray The Geometry of Relativity 1/27 Books The Geometry

More information

7a3 2. (c) πa 3 (d) πa 3 (e) πa3

7a3 2. (c) πa 3 (d) πa 3 (e) πa3 1.(6pts) Find the integral x, y, z d S where H is the part of the upper hemisphere of H x 2 + y 2 + z 2 = a 2 above the plane z = a and the normal points up. ( 2 π ) Useful Facts: cos = 1 and ds = ±a sin

More information

3 a = 3 b c 2 = a 2 + b 2 = 2 2 = 4 c 2 = 3b 2 + b 2 = 4b 2 = 4 b 2 = 1 b = 1 a = 3b = 3. x 2 3 y2 1 = 1.

3 a = 3 b c 2 = a 2 + b 2 = 2 2 = 4 c 2 = 3b 2 + b 2 = 4b 2 = 4 b 2 = 1 b = 1 a = 3b = 3. x 2 3 y2 1 = 1. MATH 222 LEC SECOND MIDTERM EXAM THU NOV 8 PROBLEM ( 5 points ) Find the standard-form equation for the hyperbola which has its foci at F ± (±2, ) and whose asymptotes are y ± 3 x The calculations b a

More information

ON THE STABILITY OF SOME PROPERLY DEGENERATE HAMILTONIAN SYSTEMS WITH TWO DEGREES OF FREEDOM. Luca Biasco. Luigi Chierchia

ON THE STABILITY OF SOME PROPERLY DEGENERATE HAMILTONIAN SYSTEMS WITH TWO DEGREES OF FREEDOM. Luca Biasco. Luigi Chierchia DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Volume 9 Number March 3 pp. 33 6 ON THE STABILITY OF SOME PROPERLY DEGENERATE HAMILTONIAN SYSTEMS WITH TWO DEGREES OF FREEDOM Luca

More information

Note: Each problem is worth 14 points except numbers 5 and 6 which are 15 points. = 3 2

Note: Each problem is worth 14 points except numbers 5 and 6 which are 15 points. = 3 2 Math Prelim II Solutions Spring Note: Each problem is worth points except numbers 5 and 6 which are 5 points. x. Compute x da where is the region in the second quadrant between the + y circles x + y and

More information

Physics 115C Homework 2

Physics 115C Homework 2 Physics 5C Homework Problem Our full Hamiltonian is H = p m + mω x +βx 4 = H +H where the unperturbed Hamiltonian is our usual and the perturbation is H = p m + mω x H = βx 4 Assuming β is small, the perturbation

More information

KAM theory and Celestial Mechanics

KAM theory and Celestial Mechanics KAM theory and Celestial Mechanics Luigi Chierchia Mathematics Department Università Roma Tre Largo San L. Murialdo, 1 I-00146 Roma Italy E-mail: luigi@mat.uniroma3.it January 20, 2005 Key words Hamiltonian

More information

Hamiltonian chaos and dark matter

Hamiltonian chaos and dark matter Hamiltonian chaos and dark matter Martin D. Weinberg UMass Astronomy weinberg@astro.umass.edu July 15, 2013 Monterey 07/15/13 slide 1 Goals Goals Bars Post-modern dynamics galactic dynamics Progress report

More information

Averaging II: Adiabatic Invariance for Integrable Systems (argued via the Averaging Principle)

Averaging II: Adiabatic Invariance for Integrable Systems (argued via the Averaging Principle) Averaging II: Adiabatic Invariance for Integrable Systems (argued via the Averaging Principle In classical mechanics an adiabatic invariant is defined as follows[1]. Consider the Hamiltonian system with

More information

Time-Periodic Solutions of the Einstein s Field Equations II: Geometric Singularities

Time-Periodic Solutions of the Einstein s Field Equations II: Geometric Singularities Science in China Series A: Mathematics 2009 Vol. 52 No. 11 1 16 www.scichina.com www.springerlink.com Time-Periodic Solutions of the Einstein s Field Equations II: Geometric Singularities KONG DeXing 1,

More information

The abundant richness of Einstein-Yang-Mills

The abundant richness of Einstein-Yang-Mills The abundant richness of Einstein-Yang-Mills Elizabeth Winstanley Thanks to my collaborators: Jason Baxter, Marc Helbling, Eugen Radu and Olivier Sarbach Astro-Particle Theory and Cosmology Group, Department

More information

Non-associative Deformations of Geometry in Double Field Theory

Non-associative Deformations of Geometry in Double Field Theory Non-associative Deformations of Geometry in Double Field Theory Michael Fuchs Workshop Frontiers in String Phenomenology based on JHEP 04(2014)141 or arxiv:1312.0719 by R. Blumenhagen, MF, F. Haßler, D.

More information

Numerical Relativity in Spherical Polar Coordinates: Calculations with the BSSN Formulation

Numerical Relativity in Spherical Polar Coordinates: Calculations with the BSSN Formulation Numerical Relativity in Spherical Polar Coordinates: Calculations with the BSSN Formulation Pedro Montero Max-Planck Institute for Astrophysics Garching (Germany) 28/01/13 in collaboration with T.Baumgarte,

More information

M2A2 Problem Sheet 3 - Hamiltonian Mechanics

M2A2 Problem Sheet 3 - Hamiltonian Mechanics MA Problem Sheet 3 - Hamiltonian Mechanics. The particle in a cone. A particle slides under gravity, inside a smooth circular cone with a vertical axis, z = k x + y. Write down its Lagrangian in a) Cartesian,

More information

AVERAGING AND RECONSTRUCTION IN HAMILTONIAN SYSTEMS

AVERAGING AND RECONSTRUCTION IN HAMILTONIAN SYSTEMS AVERAGING AND RECONSTRUCTION IN HAMILTONIAN SYSTEMS Kenneth R. Meyer 1 Jesús F. Palacián 2 Patricia Yanguas 2 1 Department of Mathematical Sciences University of Cincinnati, Cincinnati, Ohio (USA) 2 Departamento

More information

Black holes with AdS asymptotics and holographic RG flows

Black holes with AdS asymptotics and holographic RG flows Black holes with AdS asymptotics and holographic RG flows Anastasia Golubtsova 1 based on work with Irina Aref eva (MI RAS, Moscow) and Giuseppe Policastro (ENS, Paris) arxiv:1803.06764 (1) BLTP JINR,

More information

MAY THE FORCE BE WITH YOU, YOUNG JEDIS!!!

MAY THE FORCE BE WITH YOU, YOUNG JEDIS!!! Final Exam Math 222 Spring 2011 May 11, 2011 Name: Recitation Instructor s Initials: You may not use any type of calculator whatsoever. (Cell phones off and away!) You are not allowed to have any other

More information

Solution to Problem Set No. 6: Time Independent Perturbation Theory

Solution to Problem Set No. 6: Time Independent Perturbation Theory Solution to Problem Set No. 6: Time Independent Perturbation Theory Simon Lin December, 17 1 The Anharmonic Oscillator 1.1 As a first step we invert the definitions of creation and annihilation operators

More information

Nonlinear Single-Particle Dynamics in High Energy Accelerators

Nonlinear Single-Particle Dynamics in High Energy Accelerators Nonlinear Single-Particle Dynamics in High Energy Accelerators Part 4: Canonical Perturbation Theory Nonlinear Single-Particle Dynamics in High Energy Accelerators There are six lectures in this course

More information

Functional RG methods in QCD

Functional RG methods in QCD methods in QCD Institute for Theoretical Physics University of Heidelberg LC2006 May 18th, 2006 methods in QCD motivation Strong QCD QCD dynamical symmetry breaking instantons χsb top. dofs link?! deconfinement

More information

IV. Electronic Spectroscopy, Angular Momentum, and Magnetic Resonance

IV. Electronic Spectroscopy, Angular Momentum, and Magnetic Resonance IV. Electronic Spectroscopy, Angular Momentum, and Magnetic Resonance The foundation of electronic spectroscopy is the exact solution of the time-independent Schrodinger equation for the hydrogen atom.

More information

Dynamics of interacting vortices on trapped Bose-Einstein condensates. Pedro J. Torres University of Granada

Dynamics of interacting vortices on trapped Bose-Einstein condensates. Pedro J. Torres University of Granada Dynamics of interacting vortices on trapped Bose-Einstein condensates Pedro J. Torres University of Granada Joint work with: P.G. Kevrekidis (University of Massachusetts, USA) Ricardo Carretero-González

More information

Printed from file Manuscripts/Stark/stark.tex on February 24, 2014 Stark Effect

Printed from file Manuscripts/Stark/stark.tex on February 24, 2014 Stark Effect Printed from file Manuscripts/Stark/stark.tex on February 4, 4 Stark Effect Robert Gilmore Physics Department, Drexel University, Philadelphia, PA 94 February 4, 4 Abstract An external electric field E

More information

SOLUTIONS TO THE GINZBURG LANDAU EQUATIONS FOR PLANAR TEXTURES IN SUPERFLUID 3 He

SOLUTIONS TO THE GINZBURG LANDAU EQUATIONS FOR PLANAR TEXTURES IN SUPERFLUID 3 He SOLUTIONS TO THE GINZBURG LANDAU EQUATIONS FOR PLANAR TEXTURES IN SUPERFLUID 3 He V. L. GOLO, M. I. MONASTYRSKY, AND S. P. NOVIKOV Abstract. The Ginzburg Landau equations for planar textures of superfluid

More information

Solvable Lie groups and the shear construction

Solvable Lie groups and the shear construction Solvable Lie groups and the shear construction Marco Freibert jt. with Andrew Swann Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel 19.05.2016 1 Swann s twist 2 The shear construction The

More information

Physics 139B Solutions to Homework Set 4 Fall 2009

Physics 139B Solutions to Homework Set 4 Fall 2009 Physics 139B Solutions to Homework Set 4 Fall 9 1. Liboff, problem 1.16 on page 594 595. Consider an atom whose electrons are L S coupled so that the good quantum numbers are j l s m j and eigenstates

More information

Wave extraction using Weyl scalars: an application

Wave extraction using Weyl scalars: an application Wave extraction using Weyl scalars: an application In collaboration with: Chris Beetle, Marco Bruni, Lior Burko, Denis Pollney, Virginia Re Weyl scalars as wave extraction tools The quasi Kinnersley frame

More information

Invariant Tori in Hamiltonian Systems with High Order Proper Degeneracy

Invariant Tori in Hamiltonian Systems with High Order Proper Degeneracy Ann. Henri Poincaré 99 (9999), 1 18 1424-0637/99000-0, DOI 10.1007/s00023-003-0000 c 2010 Birkhäuser Verlag Basel/Switzerland Annales Henri Poincaré Invariant Tori in Hamiltonian Systems with High Order

More information

PROPERLY DEGENERATE KAM THEORY (FOLLOWING V. I. ARNOLD) Luigi Chierchia. Gabriella Pinzari

PROPERLY DEGENERATE KAM THEORY (FOLLOWING V. I. ARNOLD) Luigi Chierchia. Gabriella Pinzari DISCRETE AND CONTINUOUS doi:10.3934/dcdss.010.3.545 DYNAMICAL SYSTEMS SERIES S Volume 3, Number 4, December 010 pp. 545 578 PROPERLY DEGENERATE KAM THEORY FOLLOWING V. I. ARNOLD Luigi Chierchia Dipartimento

More information

Theory. V H Satheeshkumar. XXVII Texas Symposium, Dallas, TX December 8 13, 2013

Theory. V H Satheeshkumar. XXVII Texas Symposium, Dallas, TX December 8 13, 2013 Department of Physics Baylor University Waco, TX 76798-7316, based on my paper with J Greenwald, J Lenells and A Wang Phys. Rev. D 88 (2013) 024044 with XXVII Texas Symposium, Dallas, TX December 8 13,

More information

Superintegrability in a non-conformally-at space

Superintegrability in a non-conformally-at space (Joint work with Ernie Kalnins and Willard Miller) School of Mathematics and Statistics University of New South Wales ANU, September 2011 Outline Background What is a superintegrable system Extending the

More information

Pentahedral Volume, Chaos, and Quantum Gravity

Pentahedral Volume, Chaos, and Quantum Gravity Pentahedral Volume, Chaos, and Quantum Gravity Hal Haggard May 30, 2012 Volume Polyhedral Volume (Bianchi, Doná and Speziale): ˆV Pol = The volume of a quantum polyhedron Outline 1 Pentahedral Volume 2

More information

2 A. Jorba and J. Villanueva coordinates the Hamiltonian can be written as H( ; x; I; y) =h! ;Ii hz; B( )zi + H 1( ; x; I; y); (1) where z =(x;

2 A. Jorba and J. Villanueva coordinates the Hamiltonian can be written as H( ; x; I; y) =h! ;Ii hz; B( )zi + H 1( ; x; I; y); (1) where z =(x; The fine geometry of the Cantor families of invariant tori in Hamiltonian systems y Angel Jorba and Jordi Villanueva Abstract. This work focuses on the dynamics around a partially elliptic, lower dimensional

More information

Nonlinear Evolution of a Vortex Ring

Nonlinear Evolution of a Vortex Ring Nonlinear Evolution of a Vortex Ring Yuji Hattori Kyushu Institute of Technology, JAPAN Yasuhide Fukumoto Kyushu University, JAPAN EUROMECH Colloquium 491 Vortex dynamics from quantum to geophysical scales

More information