HAMILTONIAN STRUCTURE OF THE PAINLEVE’ EQUATIONS

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: