HAMILTONIAN STRUCTURE OF

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HAMILTONIAN STRUCTURE OF THE PAINLEVE EQUATIONS
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HAMILTONIAN STRUCTURE OF THE PAINLEVE EQUATIONS

• Hamiltonian formulation

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HAMILTONIAN STRUCTURE OF THE PAINLEVE EQUATIONS

• Hamiltonian formulation

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HAMILTONIAN STRUCTURE OF THE PAINLEVE EQUATIONS

• Hamiltonian formulation

Example PII
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HAMILTONIAN STRUCTURE OF THE PAINLEVE EQUATIONS

• Hamiltonian formulation

Example PII

• Isomonodromic deformations method (IMD)

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HAMILTONIAN STRUCTURE OF THE PAINLEVE EQUATIONS

• Hamiltonian formulation

Example PII

• Isomonodromic deformations method (IMD)

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HAMILTONIAN STRUCTURE OF THE PAINLEVE EQUATIONS

• Hamiltonian formulation

Example PII

• Isomonodromic deformations method (IMD)

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HAMILTONIAN STRUCTURE OF THE PAINLEVE EQUATIONS

• Hamiltonian formulation

Example PII

• Isomonodromic deformations method (IMD)

Example PII
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In this course we shall see how to deduce the
Hamiltonian formulation from the IMD.
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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
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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?
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2
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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?
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1
2
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Literature Adler-Kostant-Symes,
Adams-Harnad-Hurtubise, Gehktman, Hitchin,
Krichever, Novikov-Veselov, Scott,
Sklyanin..
Recent books Adler-van Moerbeke-Vanhaeke
Babelon-Bernard-Talon

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Recap on Poisson and symplectic manifolds.
(Arnold, Classical Mechanics)
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Recap on Poisson and symplectic manifolds.
(Arnold, Classical Mechanics)

• M phase space

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Recap on Poisson and symplectic manifolds.
(Arnold, Classical Mechanics)

• M phase space

• F(M) algebra of differentiable functions

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Recap on Poisson and symplectic manifolds.
(Arnold, Classical Mechanics)

• M phase space

• F(M) algebra of differentiable functions

, F(M) x F(M) -gt 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
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Recap on Poisson and symplectic manifolds.
(Arnold, Classical Mechanics)

• M phase space

• F(M) algebra of differentiable functions

, F(M) x F(M) -gt 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
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Recap on Poisson and symplectic manifolds.
(Arnold, Classical Mechanics)

• M phase space

• F(M) algebra of differentiable functions

, F(M) x F(M) -gt 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 ,
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Recap on Lie groups and Lie algebras
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Recap on Lie groups and Lie algebras
Lie group G analytic manifold with a compatible
group structure

• multiplication G x G --gt G

• inversion G --gt G

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Recap on Lie groups and Lie algebras
Lie group G analytic manifold with a compatible
group structure

• multiplication G x G --gt G

• inversion G --gt G

Example
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Recap on Lie groups and Lie algebras
Lie group G analytic manifold with a compatible
group structure

• multiplication G x G --gt G

• inversion G --gt 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

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Recap on Lie groups and Lie algebras
Lie group G analytic manifold with a compatible
group structure

• multiplication G x G --gt G

• inversion G --gt 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
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Adjoint and coadjoint action.

• Given a Lie group G its Lie algebra g is Te G.

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Adjoint and coadjoint action.

• Given a Lie group G its Lie algebra g is Te G.

Example G SL(2,C). Then
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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

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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

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Example

• Symmetric non-degenerate bilinear form

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Example

• Symmetric non-degenerate bilinear form

• Coadjoint action

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Example

• Symmetric non-degenerate bilinear form

• Coadjoint action

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Example

• Symmetric non-degenerate bilinear form

• Coadjoint action

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Loop algebra
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Loop algebra

• Commutator

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Loop algebra

• Commutator

• Killing form

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Loop algebra

• Commutator

• Killing form

• Subalgebra

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Loop algebra

• Commutator

• Killing form

• Subalgebra

• Dual space

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Loop algebra

• Commutator

• Killing form

• Subalgebra

• Dual space

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Loop algebra

• Commutator

• Killing form

• Subalgebra

• Dual space

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Coadjoint orbits
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Coadjoint orbits
Integrable systems flows on coadjoint orbits
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Coadjoint orbits
Integrable systems flows on coadjoint orbits
Example PII
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Coadjoint orbits
Integrable systems flows on coadjoint orbits
Example PII
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Coadjoint orbits
Integrable systems flows on coadjoint orbits
Example PII
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Kostant - Kirillov Poisson bracket on the dual of
a Lie algebra
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Kostant - Kirillov Poisson bracket on the dual of
a Lie algebra

• Differential of a function

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

• Differential of a function

Example PII. Take
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Definition
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Definition
Example
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Definition
Example
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Definition
Example
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Definition
Example
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Hamiltonians
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Hamiltonians

• Fix a function

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Hamiltonians

• Fix a function

• For every define

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Hamiltonians

• Fix a function

• For every define

• Kostant Kirillov Poisson bracket

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Hamiltonians

• Fix a function

• For every define

• Kostant Kirillov Poisson bracket

• Define then we get the
  evolution equation