HAMILTONIAN STRUCTURE OF
THE PAINLEVE’ EQUATIONS
HAMILTONIAN STRUCTURE OF
THE PAINLEVE’ EQUATIONS
• Hamiltonian formulation:
HAMILTONIAN STRUCTURE OF
THE PAINLEVE’ EQUATIONS
• Hamiltonian formulation:
HAMILTONIAN STRUCTURE OF
THE PAINLEVE’ EQUATIONS
• Hamiltonian formulation:
Example: PII
HAMILTONIAN STRUCTURE OF
THE PAINLEVE’ EQUATIONS
• Hamiltonian formulation:
Example: PII
• Isomonodromic deformations method (IMD):
HAMILTONIAN STRUCTURE OF
THE PAINLEVE’ EQUATIONS
• Hamiltonian formulation:
Example: PII
• Isomonodromic deformations method (IMD):
HAMILTONIAN STRUCTURE OF
THE PAINLEVE’ EQUATIONS
• Hamiltonian formulation:
Example: PII
• Isomonodromic deformations method (IMD):
HAMILTONIAN STRUCTURE OF
THE PAINLEVE’ EQUATIONS
• Hamiltonian formulation:
Example: PII
• Isomonodromic deformations method (IMD):
Example: PII
In this course we shall see how to deduce the Hamiltonian formulation from the IMD.
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
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
Example: PII (2)
What are p1 , p2, q 1, q 2 in this case? What is H?
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
Example: PII (2)
What are p1 , p2, q 1, q 2 in this case? What is H?
Literature: Adler-Kostant-Symes, Adams-Harnad-Hurtubise, Gehktman, Hitchin,
Krichever, Novikov-Veselov, Scott, Sklyanin……………..
Recent books: Adler-van Moerbeke-Vanhaeke
Babelon-Bernard-Talon
Recap on Poisson and symplectic manifolds.
(Arnol’d, Classical Mechanics)
Recap on Poisson and symplectic manifolds.
(Arnol’d, Classical Mechanics)
• M = phase space
Recap on Poisson and symplectic manifolds.
(Arnol’d, Classical Mechanics)
• M = phase space
• F(M) = algebra of differentiable functions
Recap on Poisson and symplectic manifolds.
(Arnol’d, Classical Mechanics)
• M = phase space
• F(M) = algebra of differentiable functions
• Poisson bracket:
{f,g} = -{g,f}
{ , }: F(M) x F(M) -> F(M)
skewsymmetry
{f, a g+ b h} = a {f,g} + b {f,h}
{f, g h} = {f, g} h + {f, h} g
linearity
Libenitz
{f,{g,h}} + {h,{f,g}} + {g,{h,f}} = 0
Jacobi
Recap on Poisson and symplectic manifolds.
(Arnol’d, Classical Mechanics)
• M = phase space
• F(M) = algebra of differentiable functions
• Poisson bracket:
{f,g} = -{g,f}
{ , }: F(M) x F(M) -> F(M)
skewsymmetry
{f, a g+ b h} = a {f,g} + b {f,h}
{f, g h} = {f, g} h + {f, h} g
linearity
Libenitz
{f,{g,h}} + {h,{f,g}} + {g,{h,f}} = 0
• Vector field XH associated to H eF(M):
Jacobi
XH(f):= {H,f}
Recap on Poisson and symplectic manifolds.
(Arnol’d, Classical Mechanics)
• M = phase space
• F(M) = algebra of differentiable functions
• Poisson bracket:
{f,g} = -{g,f}
{ , }: F(M) x F(M) -> F(M)
skewsymmetry
{f, a g+ b h} = a {f,g} + b {f,h}
{f, g h} = {f, g} h + {f, h} g
linearity
Libenitz
{f,{g,h}} + {h,{f,g}} + {g,{h,f}} = 0
• Vector field XH associated to H eF(M):
Jacobi
XH(f):= {H,f}
A Posson manifold is a differentiable manifold M with a Poisson bracket { , }
Recap on Lie groups and Lie algebras
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
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:
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 algebra g: 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
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 algebra g: 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:
Adjoint and coadjoint action.
• Given a Lie group G its Lie algebra g is Te G.
Adjoint and coadjoint action.
• Given a Lie group G its Lie algebra g is Te G.
Example: G = SL(2,C). Then
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:
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:
Example:
• Symmetric non-degenerate bilinear form:
Example:
• Symmetric non-degenerate bilinear form:
• Coadjoint action:
Example:
• Symmetric non-degenerate bilinear form:
• Coadjoint action:
Example:
• Symmetric non-degenerate bilinear form:
• Coadjoint action:
Loop algebra
Loop algebra
• Commutator:
Loop algebra
• Commutator:
• Killing form:
Loop algebra
• Commutator:
• Killing form:
• Subalgebra:
Loop algebra
• Commutator:
• Killing form:
• Subalgebra:
• Dual space:
Loop algebra
• Commutator:
• Killing form:
• Subalgebra:
• Dual space:
Loop algebra
• Commutator:
• Killing form:
• Subalgebra:
• Dual space:
Coadjoint orbits
Coadjoint orbits
Integrable systems = flows on coadjoint orbits:
Coadjoint orbits
Integrable systems = flows on coadjoint orbits:
Example: PII
Coadjoint orbits
Integrable systems = flows on coadjoint orbits:
Example: PII
Coadjoint orbits
Integrable systems = flows on coadjoint orbits:
Example: PII
Kostant - Kirillov Poisson bracket on the dual of a Lie algebra
Kostant - Kirillov Poisson bracket on the dual of a Lie algebra
• Differential of a function
Kostant - Kirillov Poisson bracket on the dual of a Lie algebra
• Differential of a function
Example: PII. Take
Definition:
Definition:
Example:
Definition:
Example:
Definition:
Example:
Definition:
Example:
Hamiltonians
Hamiltonians
• Fix a function
Hamiltonians
• Fix a function
• For every
define:
Hamiltonians
• Fix a function
• For every
define:
• Kostant Kirillov Poisson bracket:
Hamiltonians
• Fix a function
• For every
define:
• Kostant Kirillov Poisson bracket:
• Define
then we get the evolution equation: