1 / 56

Hamiltonian Formulation and Isomonodromic Deformations in Painleve Equations

This course focuses on deriving the Hamiltonian structure using the Isomonodromic Deformations Method (IMD) in Painleve equations, including generalizations. It covers the Hamiltonian formulation and offers examples, like PII. The course delves into complex equations and guides you through the process of deriving Hamiltonian formulation from IMD, providing insight into more intricate equations. Literature references such as Adler-Kostant-Symes and Hitchin are included, along with a recap on Poisson and symplectic manifolds and Lie groups. Learn about Poisson brackets, phase space, and Lie algebras, and explore examples like SL(2,C). Join this course to deepen your understanding of Hamiltonian structures in the context of Painleve equations.

calum
Download Presentation

Hamiltonian Formulation and Isomonodromic Deformations in Painleve Equations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. HAMILTONIAN STRUCTURE OF THE PAINLEVE’ EQUATIONS

  2. HAMILTONIAN STRUCTURE OF THE PAINLEVE’ EQUATIONS • Hamiltonian formulation:

  3. HAMILTONIAN STRUCTURE OF THE PAINLEVE’ EQUATIONS • Hamiltonian formulation:

  4. HAMILTONIAN STRUCTURE OF THE PAINLEVE’ EQUATIONS • Hamiltonian formulation: Example: PII

  5. HAMILTONIAN STRUCTURE OF THE PAINLEVE’ EQUATIONS • Hamiltonian formulation: Example: PII • Isomonodromic deformations method (IMD):

  6. HAMILTONIAN STRUCTURE OF THE PAINLEVE’ EQUATIONS • Hamiltonian formulation: Example: PII • Isomonodromic deformations method (IMD):

  7. HAMILTONIAN STRUCTURE OF THE PAINLEVE’ EQUATIONS • Hamiltonian formulation: Example: PII • Isomonodromic deformations method (IMD):

  8. HAMILTONIAN STRUCTURE OF THE PAINLEVE’ EQUATIONS • Hamiltonian formulation: Example: PII • Isomonodromic deformations method (IMD): Example: PII

  9. In this course we shall see how to deduce the Hamiltonian formulation from the IMD.

  10. In this course we shall see how to deduce the Hamiltonian formulation from the IMD. Motivation: find the Hamiltonian structure of more complicated generalizations of the Painleve’ equations

  11. In this course we shall see how to deduce the Hamiltonian formulation from the IMD. Motivation: find the Hamiltonian structure of more complicated generalizations of the Painleve’ equations (2) Example: PII What are p , p , q , q in this case? What is H? 1 2 1 2

  12. In this course we shall see how to deduce the Hamiltonian formulation from the IMD. Motivation: find the Hamiltonian structure of more complicated generalizations of the Painleve’ equations (2) Example: PII What are p , p , q , q in this case? What is H? 1 2 1 2 Literature: Adler-Kostant-Symes, Adams-Harnad-Hurtubise, Gehktman, Hitchin, Krichever, Novikov-Veselov, Scott, Sklyanin…………….. Recent books: Adler-van Moerbeke-Vanhaeke Babelon-Bernard-Talon

  13. Recap on Poisson and symplectic manifolds. (Arnol’d, Classical Mechanics)

  14. Recap on Poisson and symplectic manifolds. (Arnol’d, Classical Mechanics) • M = phase space

  15. Recap on Poisson and symplectic manifolds. (Arnol’d, Classical Mechanics) • M = phase space • F(M) = algebra of differentiable functions

  16. Recap on Poisson and symplectic manifolds. (Arnol’d, Classical Mechanics) • M = phase space • F(M) = algebra of differentiable functions { , }: F(M) x F(M) -> F(M) • Poisson bracket: {f,g} = -{g,f} skewsymmetry {f, a g+ b h} = a {f,g} + b {f,h} linearity {f, g h} = {f, g} h + {f, h} g Libenitz {f,{g,h}} + {h,{f,g}} + {g,{h,f}} = 0 Jacobi

  17. Recap on Poisson and symplectic manifolds. (Arnol’d, Classical Mechanics) • M = phase space • F(M) = algebra of differentiable functions { , }: F(M) x F(M) -> F(M) • Poisson bracket: {f,g} = -{g,f} skewsymmetry {f, a g+ b h} = a {f,g} + b {f,h} linearity {f, g h} = {f, g} h + {f, h} g Libenitz {f,{g,h}} + {h,{f,g}} + {g,{h,f}} = 0 Jacobi • Vector field XH associated to H eF(M): XH(f):= {H,f}

  18. Recap on Poisson and symplectic manifolds. (Arnol’d, Classical Mechanics) • M = phase space • F(M) = algebra of differentiable functions { , }: F(M) x F(M) -> F(M) • Poisson bracket: {f,g} = -{g,f} skewsymmetry {f, a g+ b h} = a {f,g} + b {f,h} linearity {f, g h} = {f, g} h + {f, h} g Libenitz {f,{g,h}} + {h,{f,g}} + {g,{h,f}} = 0 Jacobi • Vector field XH associated to H eF(M): XH(f):= {H,f} A Posson manifold is a differentiable manifold M with a Poisson bracket { , }

  19. Recap on Lie groups and Lie algebras

  20. Recap on Lie groups and Lie algebras Lie group G: analytic manifold with a compatible group structure • multiplication: G x G --> G • inversion: G --> G

  21. Recap on Lie groups and Lie algebras Lie group G: analytic manifold with a compatible group structure • multiplication: G x G --> G • inversion: G --> G Example:

  22. Recap on Lie groups and Lie algebras Lie group G: analytic manifold with a compatible group structure • multiplication: G x G --> G • inversion: G --> G Example: Lie algebrag: vector space with Lie bracket • [x, y] = -[y,x] antisymmetry • [a x + b y,z] = a [x, z] + b [y, z] linearity • [x, [y, z]] + [z, [x, y]] + [y, [z, x]] = 0 Jacobi

  23. Recap on Lie groups and Lie algebras Lie group G: analytic manifold with a compatible group structure • multiplication: G x G --> G • inversion: G --> G Example: Lie algebrag: vector space with Lie bracket • [x, y] = -[y,x] antisymmetry • [a x + b y,z] = a [x, z] + b [y, z] linearity • [x, [y, z]] + [z, [x, y]] + [y, [z, x]] = 0 Jacobi Example:

  24. Adjoint and coadjoint action. • Given a Lie group G its Lie algebra g is Te G.

  25. Adjoint and coadjoint action. • Given a Lie group G its Lie algebra g is Te G. Example: G = SL(2,C). Then

  26. Adjoint and coadjoint action. • Given a Lie group G its Lie algebra g is Te G. Example: G = SL(2,C). Then • g acts on itself by the adjoint action:

  27. Adjoint and coadjoint action. • Given a Lie group G its Lie algebra g is Te G. Example: G = SL(2,C). Then • g acts on itself by the adjoint action: • g acts on g* by the coadjoint action:

  28. Example: • Symmetric non-degenerate bilinear form:

  29. Example: • Symmetric non-degenerate bilinear form: • Coadjoint action:

  30. Example: • Symmetric non-degenerate bilinear form: • Coadjoint action:

  31. Example: • Symmetric non-degenerate bilinear form: • Coadjoint action:

  32. Loop algebra

  33. Loop algebra • Commutator:

  34. Loop algebra • Commutator: • Killing form:

  35. Loop algebra • Commutator: • Killing form: • Subalgebra:

  36. Loop algebra • Commutator: • Killing form: • Subalgebra: • Dual space:

  37. Loop algebra • Commutator: • Killing form: • Subalgebra: • Dual space:

  38. Loop algebra • Commutator: • Killing form: • Subalgebra: • Dual space:

  39. Coadjoint orbits

  40. Coadjoint orbits Integrable systems = flows on coadjoint orbits:

  41. Coadjoint orbits Integrable systems = flows on coadjoint orbits: Example: PII

  42. Coadjoint orbits Integrable systems = flows on coadjoint orbits: Example: PII

  43. Coadjoint orbits Integrable systems = flows on coadjoint orbits: Example: PII

  44. Kostant - Kirillov Poisson bracket on the dual of a Lie algebra

  45. Kostant - Kirillov Poisson bracket on the dual of a Lie algebra • Differential of a function

  46. Kostant - Kirillov Poisson bracket on the dual of a Lie algebra • Differential of a function Example: PII. Take

  47. Definition:

  48. Definition: Example:

  49. Definition: Example:

  50. Definition: Example:

More Related