Bi-Hamiltonian flows and their geometric realizations Gloria Mar - - PowerPoint PPT Presentation

Bi-Hamiltonian flows and their geometric realizations

Gloria Mar´ ı Beffa February, 2010

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Assume u : J ⊂ R2 → RP1 is a solution of the Schwarzian KdV evolution ut = uxS(u) = uxxx − 3 2 u2

xx

ux where S(u) = uxxx

ux − 3 2

• uxx

ux

2 is the Schwarzian derivative of u.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Assume u : J ⊂ R2 → RP1 is a solution of the Schwarzian KdV evolution ut = uxS(u) = uxxx − 3 2 u2

xx

ux where S(u) = uxxx

ux − 3 2

• uxx

ux

2 is the Schwarzian derivative of u. Then S(u)t = S(u)xxx + 3S(u)xS(u) is a solution of the KdV equation.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Assume u : J ⊂ R2 → RP1 is a solution of the Schwarzian KdV evolution ut = uxS(u) = uxxx − 3 2 u2

xx

ux where S(u) = uxxx

ux − 3 2

• uxx

ux

2 is the Schwarzian derivative of u. Then S(u)t = S(u)xxx + 3S(u)xS(u) is a solution of the KdV equation. We say the Schwarzian KdV flow is a RP1 geometric realization of the KdV flow.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Assume u : J ⊂ R2 → R3 is a solution of the Vortex filament flow evolution ut = κB where κ is the curvature of the flow u and B is the binormal.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Assume u : J ⊂ R2 → R3 is a solution of the Vortex filament flow evolution ut = κB where κ is the curvature of the flow u and B is the binormal. Then, curvature and torsion of the flow satisfy an equation equivalent to the Nonlinear Schr¨

• dinger equation. If

φ = κei

• τdx,

φt = iφxx + i 2| |φ| |2φ (Hasimoto, 72)

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Assume u : J ⊂ R2 → R3 is a solution of the Vortex filament flow evolution ut = κB where κ is the curvature of the flow u and B is the binormal. Then, curvature and torsion of the flow satisfy an equation equivalent to the Nonlinear Schr¨

• dinger equation. If

φ = κei

• τdx,

φt = iφxx + i 2| |φ| |2φ (Hasimoto, 72) We say the Vortex Filament flow is a 3-dimensional Euclidean geometric realization of the NLS equation.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

(Loading movie...)

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Geometric realizations of soliton solutions have interesting properties and behavior

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

(Loading movie...)

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

(Loading movie...)

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Some other geometric realizations of completely integrable systems are:

◮ The Sine-Gordon and modified KdV equations have both an

Euclidean 3-dimensional realization.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Some other geometric realizations of completely integrable systems are:

◮ The Sine-Gordon and modified KdV equations have both an

Euclidean 3-dimensional realization.

◮ The Boussinesq equation can be realized as a flow of

projective curves in RP2.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Some other geometric realizations of completely integrable systems are:

◮ The Sine-Gordon and modified KdV equations have both an

Euclidean 3-dimensional realization.

◮ The Boussinesq equation can be realized as a flow of

projective curves in RP2.

◮ The Adler-Gelfand-Dikii flows, or generalized n-dimensional

KdV systems, have geometric realizations in RPn, for all n.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Some other geometric realizations of completely integrable systems are:

◮ The Sine-Gordon and modified KdV equations have both an

Euclidean 3-dimensional realization.

◮ The Boussinesq equation can be realized as a flow of

projective curves in RP2.

◮ The Adler-Gelfand-Dikii flows, or generalized n-dimensional

KdV systems, have geometric realizations in RPn, for all n.

◮ The Sawada Kotera equation can be realized as a flow in

equi-affine plane and also as a projective flow in RP1.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Some other geometric realizations of completely integrable systems are:

◮ The Sine-Gordon and modified KdV equations have both an

Euclidean 3-dimensional realization.

◮ The Boussinesq equation can be realized as a flow of

projective curves in RP2.

◮ The Adler-Gelfand-Dikii flows, or generalized n-dimensional

KdV systems, have geometric realizations in RPn, for all n.

◮ The Sawada Kotera equation can be realized as a flow in

equi-affine plane and also as a projective flow in RP1.

◮ The complexly coupled system of KdV equations has

realizations as a conformal flow on the sphere S2, as a flow of star-shaped curves in the light cone of Lorentzian R4 and as a flow in the 2-Grassmannian in R4.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Some other geometric realizations of completely integrable systems are:

◮ The Sine-Gordon and modified KdV equations have both an

Euclidean 3-dimensional realization.

◮ The Boussinesq equation can be realized as a flow of

projective curves in RP2.

◮ The Adler-Gelfand-Dikii flows, or generalized n-dimensional

KdV systems, have geometric realizations in RPn, for all n.

◮ The Sawada Kotera equation can be realized as a flow in

equi-affine plane and also as a projective flow in RP1.

◮ The complexly coupled system of KdV equations has

realizations as a conformal flow on the sphere S2, as a flow of star-shaped curves in the light cone of Lorentzian R4 and as a flow in the 2-Grassmannian in R4.

◮ Both decoupled systems of KdV equations and matrix mKdV

equations have realizations as flows of Lagrangian planes, as flows of 2-Grassmannians in R4 and as flows of even dimensional spinors.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Anco, Calini & Ivey, Doliwa & Santini, Langer & Perline, Mar´ ı-Beffa, Olver, Pinkall, Qu, Sanders & Wang, Sym, Terng & Thorbergsson, Yasui & Sasaki.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Anco, Calini & Ivey, Doliwa & Santini, Langer & Perline, Mar´ ı-Beffa, Olver, Pinkall, Qu, Sanders & Wang, Sym, Terng & Thorbergsson, Yasui & Sasaki. What do these completely integrable systems have in common?

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Anco, Calini & Ivey, Doliwa & Santini, Langer & Perline, Mar´ ı-Beffa, Olver, Pinkall, Qu, Sanders & Wang, Sym, Terng & Thorbergsson, Yasui & Sasaki. What do these completely integrable systems have in common? First of all, all the geometric realizations live in homogeneous spaces, that is geometric manifolds of the form G/H with G a Lie group and H ⊂ G a closed subgroup.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Anco, Calini & Ivey, Doliwa & Santini, Langer & Perline, Mar´ ı-Beffa, Olver, Pinkall, Qu, Sanders & Wang, Sym, Terng & Thorbergsson, Yasui & Sasaki. What do these completely integrable systems have in common? First of all, all the geometric realizations live in homogeneous spaces, that is geometric manifolds of the form G/H with G a Lie group and H ⊂ G a closed subgroup. They are also all Bi-Hamiltonian systems. That is, they can be written as kt = PiδHi, i = 1, 2 for two choices of Hamiltonian structures Pi (described by differential operators), and two choices of Hamiltonian operators Hi.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Anco, Calini & Ivey, Doliwa & Santini, Langer & Perline, Mar´ ı-Beffa, Olver, Pinkall, Qu, Sanders & Wang, Sym, Terng & Thorbergsson, Yasui & Sasaki. What do these completely integrable systems have in common? First of all, all the geometric realizations live in homogeneous spaces, that is geometric manifolds of the form G/H with G a Lie group and H ⊂ G a closed subgroup. They are also all Bi-Hamiltonian systems. That is, they can be written as kt = PiδHi, i = 1, 2 for two choices of Hamiltonian structures Pi (described by differential operators), and two choices of Hamiltonian operators Hi.Furthermore, the Hamiltonian structures are compatible, that is, P1 + P2 is also a Hamiltonian structure.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Are the Hamiltonian structures linked to the background geometry

• f the realization?

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Are the Hamiltonian structures linked to the background geometry

• f the realization?

Not only they are linked, the background geometry generates them, determine many of their properties and guarantees geometric realizations.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Given u : I → M ∼ = G/H, what is a moving frame along u?

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Given u : I → M ∼ = G/H, what is a moving frame along u? Classical definition A classical moving frame is a curve in the frame bundle over u invariant under the prolonged action of G.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Given u : I → M ∼ = G/H, what is a moving frame along u? Classical definition A classical moving frame is a curve in the frame bundle over u invariant under the prolonged action of G. Group-base definition (Fels and Olver, after Cartan, Green, Griffiths and others) A (left or right invariant) moving frame of order k is an equivariant map ρ : J(k)(R, M) → G with respect to the prolonged action of G on J(k)(R, M) and the (left or right) action of G on itself.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Fels & Olver, Hubert, Kogan, Mansfield, Mar´ ı Beffa, Pohjanpelto, Van der Kamp

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Fels & Olver, Hubert, Kogan, Mansfield, Mar´ ı Beffa, Pohjanpelto, Van der Kamp Are there effective ways to find group base moving frames?

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Fels & Olver, Hubert, Kogan, Mansfield, Mar´ ı Beffa, Pohjanpelto, Van der Kamp Are there effective ways to find group base moving frames?

Theorem

(FO 99) Assume we have normalization equations of the form g · u(k) = ck where ck are constants (they are called normalization constants). Here · denotes the prolonged action of the group on u(k). Assume we have enough equations so as to determine g as a function of u, ux, . . . . Then g = ρ−1 is a right moving frame and ρ is a left moving frame.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Definition

Let ρ be a left moving frame, and define K = ρ−1 (ρ)x ∈ g. We call K the left Maurer-Cartan matrix and (if g ⊂ gl(n)) ρx = ρK the left Serret-Frenet equations for the moving frame ρ.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Definition

Let ρ be a left moving frame, and define K = ρ−1 (ρ)x ∈ g. We call K the left Maurer-Cartan matrix and (if g ⊂ gl(n)) ρx = ρK the left Serret-Frenet equations for the moving frame ρ.

Theorem

(Hubert 07) Let ρ be a (left or right) moving frame for a curve u. Then, the coefficients of the (left or right) Serret-Frenet equations for ρ generate all differential invariants of the curve. That is, any

• ther differential invariant is a function of the entries of K and

their derivatives with respect to x.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Assume M = G/H is flat with G semisimple.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Assume M = G/H is flat with G semisimple.

Theorem

(MB 08) There exists U ⊂ C ∞(S1, g∗) open and H = C ∞(S1, H) ⊂ C ∞(S1, G) such that U/H ≡ K. Given a section S generated by Ik = ρ−1 · uk, K can be represented by Maurer-Cartan matrices associated to moving frames with the normalization associated to S, along curves in M with a monodromy.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Assume M = G/H is flat with G semisimple.

Theorem

(MB 08) There exists U ⊂ C ∞(S1, g∗) open and H = C ∞(S1, H) ⊂ C ∞(S1, G) such that U/H ≡ K. Given a section S generated by Ik = ρ−1 · uk, K can be represented by Maurer-Cartan matrices associated to moving frames with the normalization associated to S, along curves in M with a monodromy. The action of H on M is the gauge action on operators, that is a(h)(L) = h−1hx + h−1Lh.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Algebraically, one does not need to know the moving frame to find

• K. One only needs the normalization constant used to find the

moving frame.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Algebraically, one does not need to know the moving frame to find

• K. One only needs the normalization constant used to find the

moving frame.

Theorem

(Fels-Olver 99, M-B 08) Assume ρ is a left moving frame and assume it satisfies the normalization equations Ik = ρ−1 · uk = ck.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Algebraically, one does not need to know the moving frame to find

• K. One only needs the normalization constant used to find the

moving frame.

Theorem

(Fels-Olver 99, M-B 08) Assume ρ is a left moving frame and assume it satisfies the normalization equations Ik = ρ−1 · uk = ck. Then its left Maurer-Cartan matrix satisfies the equation K · Ir = (Ir)x − Ir+1 for all r and using the normalized values we can solve for K explicitly.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Example

(Joint work with Michael Eastwood) The 2-Grassmannian in R4 can be represented by the homogeneous space SL(4, R)/H, where H is H = { A C B

• , det A det B = 1}.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Example

(Joint work with Michael Eastwood) The 2-Grassmannian in R4 can be represented by the homogeneous space SL(4, R)/H, where H is H = { A C B

• , det A det B = 1}.

If we factor elements in SL(4, R) as g = I Z I A B I Y I

• The action of SL(4, R) on Grassmannians is

g · U = A(U + Y )(B + ZA(U + Y ))−1.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Using the action we can find the prolonged infinitesimal action and we obtain the following equations for the Maurer-Cartan matrix. If K = KA KY KZ KB

• then, with the normalization choices I0 = 0 = I2, I1 = I

K (0) · I0 = KY = I1 − (I0)x = I K (1) · I1 = KA − KB = I2 − (I1)x = 0 K (2) · I2 = −2KZ = I3 We need to further normalize I3 and I4.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

If we normalize I3 = a b

• and I4 =

∗ b − a ∗ ∗

• , then we have

KZ to be a diagonal matrix and K (3) · I3 = KAI3 − I3KA = I4 − (I3)x which completely determined K to be K =     k3 1 1 k4 −k3 1 k1 k3 1 k2 k4 −k3    

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

On the other hand, we can gauge this matrix K by an element of H with invariant entries to obtain a matrix of the form K =     ^ k3 1 ^ k4 1 ^ k1 ^ k3 ^ k2 ^ k4    

• r K =

    1 1 κ1 κ3 κ4 κ2     . According to Hubert’s results, these entries will also be generators

• f the Grassmannian differential invariants.

They are all equivalent, so why does it matter?

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

How do we generate Hamiltonian structures and geometric realizations using moving frames and differential invariants?

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

How do we generate Hamiltonian structures and geometric realizations using moving frames and differential invariants? There are two well known Hamiltonian structures on C ∞(S1, g∗).

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

How do we generate Hamiltonian structures and geometric realizations using moving frames and differential invariants? There are two well known Hamiltonian structures on C ∞(S1, g∗). Let F, G : C ∞(S1, g∗) → R, and let δF

δL , δG δL ∈ C ∞(S1, g) be their

variational derivatives.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

How do we generate Hamiltonian structures and geometric realizations using moving frames and differential invariants? There are two well known Hamiltonian structures on C ∞(S1, g∗). Let F, G : C ∞(S1, g∗) → R, and let δF

δL , δG δL ∈ C ∞(S1, g) be their

variational derivatives. {F, G}1(L) =

• S1δG

δL , δF δL

• x

+ [L, δF δL ]dx (1)

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

How do we generate Hamiltonian structures and geometric realizations using moving frames and differential invariants? There are two well known Hamiltonian structures on C ∞(S1, g∗). Let F, G : C ∞(S1, g∗) → R, and let δF

δL , δG δL ∈ C ∞(S1, g) be their

variational derivatives. {F, G}1(L) =

• S1δG

δL , δF δL

• x

+ [L, δF δL ]dx (1) {F, G}2(L) =

• S1δG

δL , [L0, δF δL ]dx (2) with L0 ∈ g∗ constant.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Theorem

(MB 08) Poisson bracket (1) reduces to K ≡ U/H to produce a Hamiltonian structure on the space of differential invariants.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Theorem

(MB 08) Poisson bracket (1) reduces to K ≡ U/H to produce a Hamiltonian structure on the space of differential invariants. Furthermore, if K can be represented as an affine subspace of C ∞(S1, g∗) any reduced Hamiltonian equation possesses a geometric realization as a flow of curves in G/H.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Theorem

(MB 08) Poisson bracket (1) reduces to K ≡ U/H to produce a Hamiltonian structure on the space of differential invariants. Furthermore, if K can be represented as an affine subspace of C ∞(S1, g∗) any reduced Hamiltonian equation possesses a geometric realization as a flow of curves in G/H. The second bracket (2) reduces only in some cases signally the existence of an associated integrable system. It needs to be checked in each individual case.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Theorem

(MB 08) Poisson bracket (1) reduces to K ≡ U/H to produce a Hamiltonian structure on the space of differential invariants. Furthermore, if K can be represented as an affine subspace of C ∞(S1, g∗) any reduced Hamiltonian equation possesses a geometric realization as a flow of curves in G/H. The second bracket (2) reduces only in some cases signally the existence of an associated integrable system. It needs to be checked in each individual case. Can we find the Poisson tensor algebraically from the normalization equations?

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Theorem

(MB 08) Poisson bracket (1) reduces to K ≡ U/H to produce a Hamiltonian structure on the space of differential invariants. Furthermore, if K can be represented as an affine subspace of C ∞(S1, g∗) any reduced Hamiltonian equation possesses a geometric realization as a flow of curves in G/H. The second bracket (2) reduces only in some cases signally the existence of an associated integrable system. It needs to be checked in each individual case. Can we find the Poisson tensor algebraically from the normalization equations? Yes! with one assumption. We need K ≡ U/H to be represented by an affine subspace of C ∞(S1, g∗).

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

The Poisson reduction process can be algebraically defined the following way:

◮ We start with two functionals defined on K ≡ U/N,

f , g : K → R, and with variational derivatives δf , δg.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

The Poisson reduction process can be algebraically defined the following way:

◮ We start with two functionals defined on K ≡ U/N,

f , g : K → R, and with variational derivatives δf , δg.

◮ We extend them to functionals F, G : C ∞(S1, g∗) → R with

variational derivatives δF, δG ∈ C ∞(S1, g).

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

The Poisson reduction process can be algebraically defined the following way:

◮ We start with two functionals defined on K ≡ U/N,

f , g : K → R, and with variational derivatives δf , δg.

◮ We extend them to functionals F, G : C ∞(S1, g∗) → R with

variational derivatives δF, δG ∈ C ∞(S1, g). We do not need to know the extensions, we only need to know δF, δG along K.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

The Poisson reduction process can be algebraically defined the following way:

◮ We start with two functionals defined on K ≡ U/N,

f , g : K → R, and with variational derivatives δf , δg.

◮ We extend them to functionals F, G : C ∞(S1, g∗) → R with

variational derivatives δF, δG ∈ C ∞(S1, g). We do not need to know the extensions, we only need to know δF, δG along K.

◮ Assuming K is an affine subspace, the element δF can be

found algebraically from the equation (δF)x + [K, δF] ∈ h0 (3) where h is the Lie algebra of H and h0 its annihilator.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

The Poisson reduction process can be algebraically defined the following way:

◮ We start with two functionals defined on K ≡ U/N,

f , g : K → R, and with variational derivatives δf , δg.

◮ We extend them to functionals F, G : C ∞(S1, g∗) → R with

variational derivatives δF, δG ∈ C ∞(S1, g). We do not need to know the extensions, we only need to know δF, δG along K.

◮ Assuming K is an affine subspace, the element δF can be

found algebraically from the equation (δF)x + [K, δF] ∈ h0 (3) where h is the Lie algebra of H and h0 its annihilator.

◮ The entries of δF dual to K will be given by δf and the rest

will be determined algebraically by equation (3) in terms δf and K.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

If we know δF, δG, then the reduction of the first bracket is {f , g}1(K) =

• S1 trace (((δF)x + [K, δF]) δG) dx =
• S1 δf TP1δgdx

where P1 is the Poisson tensor, guaranteed to be Poisson.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

If we know δF, δG, then the reduction of the first bracket is {f , g}1(K) =

• S1 trace (((δF)x + [K, δF]) δG) dx =
• S1 δf TP1δgdx

where P1 is the Poisson tensor, guaranteed to be Poisson. The reduction of the second one will be {f , g}2(K) =

• S1 trace ([K0, δF]δG) dx =
• S1 δf TP2δgdx

for some properly chosen K0. This bracket is not guaranteed to be Poisson.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

If we know δF, δG, then the reduction of the first bracket is {f , g}1(K) =

• S1 trace (((δF)x + [K, δF]) δG) dx =
• S1 δf TP1δgdx

where P1 is the Poisson tensor, guaranteed to be Poisson. The reduction of the second one will be {f , g}2(K) =

• S1 trace ([K0, δF]δG) dx =
• S1 δf TP2δgdx

for some properly chosen K0. This bracket is not guaranteed to be

• Poisson. Both P1 and P2 can be algebraically found from the

expressions of δF and δG.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

If we know δF, δG, then the reduction of the first bracket is {f , g}1(K) =

• S1 trace (((δF)x + [K, δF]) δG) dx =
• S1 δf TP1δgdx

where P1 is the Poisson tensor, guaranteed to be Poisson. The reduction of the second one will be {f , g}2(K) =

• S1 trace ([K0, δF]δG) dx =
• S1 δf TP2δgdx

for some properly chosen K0. This bracket is not guaranteed to be

• Poisson. Both P1 and P2 can be algebraically found from the

expressions of δF and δG. Sometimes they can be non-local.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Example

In the Grassmannian case we had three choices: K =     k3 1 1 k4 −k3 1 k1 k3 1 k2 k4 −k3     , K =     ^ k3 1 ^ k4 1 ^ k1 ^ k3 ^ k2 ^ k4    

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Example

In the Grassmannian case we had three choices: K =     k3 1 1 k4 −k3 1 k1 k3 1 k2 k4 −k3     , K =     ^ k3 1 ^ k4 1 ^ k1 ^ k3 ^ k2 ^ k4     and K =     1 1 κ1 κ3 κ4 κ2     .

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Example

In the Grassmannian case we had three choices: K =     k3 1 1 k4 −k3 1 k1 k3 1 k2 k4 −k3     , K =     ^ k3 1 ^ k4 1 ^ k1 ^ k3 ^ k2 ^ k4     and K =     1 1 κ1 κ3 κ4 κ2     . Assume we choose the last one.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Here H = { A C B

• ,

det A det B = 1};

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Here H = { A C B

• ,

det A det B = 1}; h are traceless matrices

• f the form

∗ ∗ ∗

• Gloria Mar´

ı Beffa Bi-Hamiltonian flows and their geometric realizations

Here H = { A C B

• ,

det A det B = 1}; h are traceless matrices

• f the form

∗ ∗ ∗

• and h0 are matrices of the form

∗

• .

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Here H = { A C B

• ,

det A det B = 1}; h are traceless matrices

• f the form

∗ ∗ ∗

• and h0 are matrices of the form

∗

• .

The variational derivative of an extension δF along K looks like δF = F1 δf F3 F2

• .

If K = I K0

• , the determining equation looks like

(δF)x + [K, δF] = (F1)x + F3 − δfK0 (δf )x + F2 − F1 (F3)x + K0F1 − F2K0 (F2)x + K0δf − F3

• =

∗

• .

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

This solves for F1 = 1

2(δf )x, F2 = − 1 2(δf )x and

F3 = 1

2 (−(δf )xx + Kδf + δfK).

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

This solves for F1 = 1

2(δf )x, F2 = − 1 2(δf )x and

F3 = 1

2 (−(δf )xx + Kδf + δfK).

The reduced bracket is {h, g}1(K0) =

• S1 tr
• (F3)x + K0F1 − F2K0

G1 δg G3 G2

• dx

= 1 2

• S1 tr
• δg(−D3δf + D(K0δf + δfK0) + 2K0Dδf + 2D(δf )K0)
• dx.

This is the matrix KdV Hamiltonian structure.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

This solves for F1 = 1

2(δf )x, F2 = − 1 2(δf )x and

F3 = 1

2 (−(δf )xx + Kδf + δfK).

The reduced bracket is {h, g}1(K0) =

• S1 tr
• (F3)x + K0F1 − F2K0

G1 δg G3 G2

• dx

= 1 2

• S1 tr
• δg(−D3δf + D(K0δf + δfK0) + 2K0Dδf + 2D(δf )K0)
• dx.

This is the matrix KdV Hamiltonian structure.The second bracket reduces as {h, g}2(K0) =

• S1 tr

I

• ,

F1 δf F3 F2 G1 δg G3 G2

• dx

= 2

• S1 tr (δgDδh) dx

which is the companion Hamiltonian structure for the matrix KdV equation.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Finally, the equation Ut = κ1T1 + κ2T2 + κ3T3 + κ4T4 is a Grassmannian geometric realization of the matrix KdV equation, (K0)t = −(K0)xxx + 6K0(K0)x + 6(K0)xK0 for a properly chosen classical frame {T1, T2, T3, T4}.

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Interesting open problems and questions:

• 1. When does {, }2 reduce? What are the obstacles?
• 2. Can we always find a moving frame for which K is an affine

subspace of C ∞(S1, g)?

• 3. What are the different shapes/normal forms that K can have

and how do they link to different integrable systems?

• 4. How do different types of differential invariants relate to

different types of completely integrable systems?

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations

Thanks!

Gloria Mar´ ı Beffa Bi-Hamiltonian flows and their geometric realizations