HAMILTONIAN STRUCTURE OF THE PAINLEVE EQUATIONS HAMILTONIAN STRUCTURE
- Slides: 56
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 p 1 , p 2, 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 p 1 , p 2, 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 e. F(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 e. F(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:
- Hamiltonian operator
- Hamiltonian operator
- Schrodinger wave equation
- Hamiltonian dynamics
- The hamiltonian operator is given by
- Hridis kumar pal
- функция гамильтона
- Lant elementar
- Chromatic number
- Canonical equations of motion
- A cyclic coordinate is in hamiltonian
- Shadow hamiltonian
- Unperturbed hamiltonian
- Cannon algorithm
- Particle on a ring boundary conditions
- Hamiltonian circuit examples
- Hamiltonian circuit
- Hamiltonian circuit
- Sorted edges algorithm steps
- Whats management science
- Hamiltonian circuit
- Hamiltonian graph
- Russell-saunders coupling
- Rectangular equations to polar equations
- Translate word equations to chemical equations
- Hình ảnh bộ gõ cơ thể búng tay
- Lp html
- Bổ thể
- Tỉ lệ cơ thể trẻ em
- Chó sói
- Tư thế worms-breton
- Chúa yêu trần thế alleluia
- Các môn thể thao bắt đầu bằng tiếng nhảy
- Thế nào là hệ số cao nhất
- Các châu lục và đại dương trên thế giới
- Công thức tiính động năng
- Trời xanh đây là của chúng ta thể thơ
- Cách giải mật thư tọa độ
- 101012 bằng
- độ dài liên kết
- Các châu lục và đại dương trên thế giới
- Thơ thất ngôn tứ tuyệt đường luật
- Quá trình desamine hóa có thể tạo ra
- Một số thể thơ truyền thống
- Cái miệng bé xinh thế chỉ nói điều hay thôi
- Vẽ hình chiếu vuông góc của vật thể sau
- Thế nào là sự mỏi cơ
- đặc điểm cơ thể của người tối cổ
- V cc cc
- Vẽ hình chiếu đứng bằng cạnh của vật thể
- Tia chieu sa te
- Thẻ vin
- đại từ thay thế
- điện thế nghỉ
- Tư thế ngồi viết
- Diễn thế sinh thái là
- Dot