Classification of Homogeneous Third Order Differential ..._Pavlov_M.,_Potemin_G.,_Vitolo_… ·...
Transcript of Classification of Homogeneous Third Order Differential ..._Pavlov_M.,_Potemin_G.,_Vitolo_… ·...
Classification of Homogeneous Third Order
Differential-Geometric Poisson Brackets
E.V. Ferapontov, M.V. Pavlov, G.V. Potemin, R. Vitolo
UK-Russia-Italian Collaboration
17.10.2013
EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 1 / 17
First Order Differential-Geometric Poisson Brackets
First order differential-geometric Poisson brackets
{ui (x), uj (x ′)} = [g ij (u(x))∂x + bijk (u(x))ukx ]δ(x − x ′)
satisfy the skew-symmetry and Jacobi identity iff g ij (u) is a
symmetric nondegenerate tensor and bijk (u) = −g isΓjsk , where Γ
jsk is
a Levi-Civita connection, while the metric gij (u) is flat.
EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 2 / 17
First Order Differential-Geometric Poisson Brackets
First order differential-geometric Poisson brackets
{ui (x), uj (x ′)} = [g ij (u(x))∂x + bijk (u(x))ukx ]δ(x − x ′)
satisfy the skew-symmetry and Jacobi identity iff g ij (u) is a
symmetric nondegenerate tensor and bijk (u) = −g isΓjsk , where Γ
jsk is
a Levi-Civita connection, while the metric gij (u) is flat.
Corresponding Hamiltonian systems are
uit = [g ij∂x − g isΓjsku
kx ]
δH
δuj,
where the Hamiltonian functional H =∫
h(u,ux ,uxx ,uxxx , ...)dx .
EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 2 / 17
First Order Differential-Geometric Poisson Brackets
First order differential-geometric Poisson brackets
{ui (x), uj (x ′)} = [g ij (u(x))∂x + bijk (u(x))ukx ]δ(x − x ′)
satisfy the skew-symmetry and Jacobi identity iff g ij (u) is a
symmetric nondegenerate tensor and bijk (u) = −g isΓjsk , where Γ
jsk is
a Levi-Civita connection, while the metric gij (u) is flat.
Corresponding Hamiltonian systems are
uit = [g ij∂x − g isΓjsku
kx ]
δH
δuj,
where the Hamiltonian functional H =∫
h(u,ux ,uxx ,uxxx , ...)dx .
In a particular case H =∫
h(u)dx , we have N componenthydrodynamic type system
uit = (∇i∇jh)ujx .
EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 2 / 17
Third Differential-Geometric Poisson Brackets
{ui (x), uj (x ′)} = [g ij (u(x))∂3x + bijk (u(x))ukx ∂2x
+(c ijk (u(x))ukxx + c ijkm(u(x))u
kx u
mx )∂x
+d ijk (u(x))u
kxxx + d ij
km(u(x))ukx u
mxx + d ij
kmn(u(x))ukx u
mx u
nx ]δ(x − x ′).
EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 3 / 17
Third Differential-Geometric Poisson Brackets
{ui (x), uj (x ′)} = [g ij (u(x))∂3x + bijk (u(x))ukx ∂2x
+(c ijk (u(x))ukxx + c ijkm(u(x))u
kx u
mx )∂x
+d ijk (u(x))u
kxxx + d ij
km(u(x))ukx u
mxx + d ij
kmn(u(x))ukx u
mx u
nx ]δ(x − x ′).
In special coordinate system d ijk (a) = 0, d ij
km(a) = 0, d ijkmn(a) = 0. Then
EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 3 / 17
Third Differential-Geometric Poisson Brackets
{ui (x), uj (x ′)} = [g ij (u(x))∂3x + bijk (u(x))ukx ∂2x
+(c ijk (u(x))ukxx + c ijkm(u(x))u
kx u
mx )∂x
+d ijk (u(x))u
kxxx + d ij
km(u(x))ukx u
mxx + d ij
kmn(u(x))ukx u
mx u
nx ]δ(x − x ′).
In special coordinate system d ijk (a) = 0, d ij
km(a) = 0, d ijkmn(a) = 0. Then
{ai (x), aj (x ′)} = ∂x [gij (a(x))∂x + bijk (a(x))u
kx ]δ
′(x − x ′).
satisfy the skew-symmetry and Jacobi identity iff
EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 3 / 17
Third Differential-Geometric Poisson Brackets
{ui (x), uj (x ′)} = [g ij (u(x))∂3x + bijk (u(x))ukx ∂2x
+(c ijk (u(x))ukxx + c ijkm(u(x))u
kx u
mx )∂x
+d ijk (u(x))u
kxxx + d ij
km(u(x))ukx u
mxx + d ij
kmn(u(x))ukx u
mx u
nx ]δ(x − x ′).
In special coordinate system d ijk (a) = 0, d ij
km(a) = 0, d ijkmn(a) = 0. Then
{ai (x), aj (x ′)} = ∂x [gij (a(x))∂x + bijk (a(x))u
kx ]δ
′(x − x ′).
satisfy the skew-symmetry and Jacobi identity iff
gmk,n + gkn,m + gmn,k = 0,
gmn,kl =1
9gpq(gqk,ngpl ,m − gqk,ngpm,l + gqk,mgpl ,n − gqk,mgpn,l
+gqn,kgpm,l − gqn,kgpl ,m + gqm,kgpn,l − gqm,kgpl ,n),
EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 3 / 17
Third Differential-Geometric Poisson Brackets
{ui (x), uj (x ′)} = [g ij (u(x))∂3x + bijk (u(x))ukx ∂2x
+(c ijk (u(x))ukxx + c ijkm(u(x))u
kx u
mx )∂x
+d ijk (u(x))u
kxxx + d ij
km(u(x))ukx u
mxx + d ij
kmn(u(x))ukx u
mx u
nx ]δ(x − x ′).
In special coordinate system d ijk (a) = 0, d ij
km(a) = 0, d ijkmn(a) = 0. Then
{ai (x), aj (x ′)} = ∂x [gij (a(x))∂x + bijk (a(x))u
kx ]δ
′(x − x ′).
satisfy the skew-symmetry and Jacobi identity iff
gmk,n + gkn,m + gmn,k = 0,
gmn,kl =1
9gpq(gqk,ngpl ,m − gqk,ngpm,l + gqk,mgpl ,n − gqk,mgpn,l
+gqn,kgpm,l − gqn,kgpl ,m + gqm,kgpn,l − gqm,kgpl ,n),
where cnkm = 13 (gmn,k − gkn,m) and cijk = giqgjpb
pqk .
EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 3 / 17
Metric. Nonlinear System
gmk,n + gkn,m + gmn,k = 0,
gmn,kl =1
9gpq(gqk,ngpl ,m − gqk,ngpm,l + gqk,mgpl ,n − gqk,mgpn,l
+gqn,kgpm,l − gqn,kgpl ,m + gqm,kgpn,l − gqm,kgpl ,n).
This system has a solution in the form
gik = g(0)ik + g
(1)ikma
m + g(2)ikmna
man.
The nonlinear part
gmn,kl =1
9gpq(gqk,ngpl ,m − gqk,ngpm,l + gqk,mgpl ,n − gqk,mgpn,l
+gqn,kgpm,l − gqn,kgpl ,m + gqm,kgpn,l − gqm,kgpl ,n)
can be parameterized in the form
gik = φβγψiβψkγ,
where ψiβ = ψiβmam + ξ iβ and ψiβk = −ψkβi .
EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 4 / 17
Metric. Nonlinear System
gmk,n + gkn,m + gmn,k = 0,
gmn,kl =1
9gpq(gqk,ngpl ,m − gqk,ngpm,l + gqk,mgpl ,n − gqk,mgpn,l
+gqn,kgpm,l − gqn,kgpl ,m + gqm,kgpn,l − gqm,kgpl ,n).
This system has a solution in the form
gik = g(0)ik + g
(1)ikma
m + g(2)ikmna
man.
The nonlinear part
gmn,kl =1
9gpq(gqk,ngpl ,m − gqk,ngpm,l + gqk,mgpl ,n − gqk,mgpn,l
+gqn,kgpm,l − gqn,kgpl ,m + gqm,kgpn,l − gqm,kgpl ,n)
can be parameterized in the form
gik = φβγψiβψkγ,
where ψiβ = ψiβmam + ξ iβ and ψiβk = −ψkβi .Lie Algebras.
EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 4 / 17
Metric. Nonlinear System
The linear partgmk,n + gkn,m + gmn,k = 0
can be solved by virtue of the Monge parameterization:
ds2 = gik(a)daidak = ~dT Q̂~d ,
where ~d = (da1, da2, ..., daN , a1da2 − a2da1, a2da3 − a3da2, ...) andQ̂ is a constant nondegenerate symmetric matrix.
EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 5 / 17
Metric. Nonlinear System
The linear partgmk,n + gkn,m + gmn,k = 0
can be solved by virtue of the Monge parameterization:
ds2 = gik(a)daidak = ~dT Q̂~d ,
where ~d = (da1, da2, ..., daN , a1da2 − a2da1, a2da3 − a3da2, ...) andQ̂ is a constant nondegenerate symmetric matrix.
Passive Form.gmk,n + gkn,m + gmn,k = 0,
gmn,kl =1
9gpq(gqk,ngpl ,m − gqk,ngpm,l + gqk,mgpl ,n − gqk,mgpn,l
+gqn,kgpm,l − gqn,kgpl ,m + gqm,kgpn,l − gqm,kgpl ,n).
EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 5 / 17
Particular Case
Theorem: In the particular case cijk = 0 (three distinct indices), a generalsolution of the above nonlinear system is parameterized by a solepolynomial function G of degree 4 such that
gkk = −2G,kk , gkm = G,km, ckkm = −ckmk = G,kkm, k 6= m,
while all other connection coefficients cijk = 0.
EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 6 / 17
Particular Case
Theorem: In the particular case cijk = 0 (three distinct indices), a generalsolution of the above nonlinear system is parameterized by a solepolynomial function G of degree 4 such that
gkk = −2G,kk , gkm = G,km, ckkm = −ckmk = G,kkm, k 6= m,
while all other connection coefficients cijk = 0.
This leads to
gmm = − ∑p 6=m
Rmp(ap)2 − 2 ∑
p 6=m
Hmpap +Dm,
gkm = Rkmakam +Hkma
k +Hmkam + Fkm, k 6= m,
where Rkm = Rmk ,Fkm = Fmk , Dk are constants, and
cmmk = −cmkm = Rkmak +Hmk , k 6= m.
EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 6 / 17
Two Component Case
Theorem: only two metrics in 2-component case:
g(1)ik =
(
1− (a2)2 1+ a1a2
1+ a1a2 1− (a1)2
)
, g(2)ik =
(
−2a2 a1
a1 0
)
.
EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 7 / 17
Three Component Case
Using the ansatz cijk = 0 we are able to reduce the nonlinear system to:
R12F12 = H12H21, F13R13 = H13H31, F23R23 = H23H32,
D1R12R13 +H212R13 +H2
13R12 = 0,
D2R12R23 +H221R23 +H2
23R12 = 0,
D3R13R23 +H231R23 +H2
32R13 = 0.
In the generic case Rij 6= 0 we obtain the three-parameter family of metrics
gik =
−(a2 + β2)2 − (a3)2 a1(a2 + β2) a3(a1 + β1)
a1(a2 + β2) −(a1)2 − (a3 + β3)2 a2(a3 + β3)
a3(a1 + β1) a2(a3 + β3) −(a1 + β1)2 − (a2)2
EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 8 / 17
Three Component Case
Using the ansatz cijk = 0 we are able to reduce the nonlinear system to:
R12F12 = H12H21, F13R13 = H13H31, F23R23 = H23H32,
D1R12R13 +H212R13 +H2
13R12 = 0,
D2R12R23 +H221R23 +H2
23R12 = 0,
D3R13R23 +H231R23 +H2
32R13 = 0.
In the generic case Rij 6= 0 we obtain the three-parameter family of metrics
gik =
−(a2 + β2)2 − (a3)2 a1(a2 + β2) a3(a1 + β1)
a1(a2 + β2) −(a1)2 − (a3 + β3)2 a2(a3 + β3)
a3(a1 + β1) a2(a3 + β3) −(a1 + β1)2 − (a2)2
The particular cases
1 R12 = 0, R13 6= 0, R23 6= 0,2 R12 = 0, R13 = 0, R23 6= 0,3 R12 = 0, R13 = 0, R23 = 0,
were solved separately, obtaining a complete classification of 53 metrics.EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 8 / 17
WDVV Associativity Equations
In three component case, WDVV associativity equations reduce to thesingle equation
fttt = f 2xxt − fxxx fxtt ,
EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 9 / 17
WDVV Associativity Equations
In three component case, WDVV associativity equations reduce to thesingle equation
fttt = f 2xxt − fxxx fxtt ,
which can be written as the hydrodynamic type system
at = bx , bt = cx , ct = (b2 − ac)x ,
where a = fxxx , b = fxxt , c = fxtt .
EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 9 / 17
WDVV Associativity Equations
In three component case, WDVV associativity equations reduce to thesingle equation
fttt = f 2xxt − fxxx fxtt ,
which can be written as the hydrodynamic type system
at = bx , bt = cx , ct = (b2 − ac)x ,
where a = fxxx , b = fxxt , c = fxtt .
This system is bi-Hamiltonian, i.e.
abc
t
= J0δH1 = J1δH0,
where
J0 =
− 32D
12Da Db
12aD
12 (bD +Db) 3
2cD + cxbD 3
2Dc − cx (b2 − ac)D +D(b2 − ac)
EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 9 / 17
WDVV Associativity Equations
and
J1 =
0 0 D3
0 D3 −D2aDD3 −DaD2 D2bD +DbD2 +DaDaD
.
EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 10 / 17
WDVV Associativity Equations
and
J1 =
0 0 D3
0 D3 −D2aDD3 −DaD2 D2bD +DbD2 +DaDaD
.
Here H1 =∫
cdx ,H0 = − 12a(D
−1b)2 − (D−1b)(D−1c).
EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 10 / 17
Lorentzian Metric
The third order Hamiltonian structure is associated with the action
S =1
2
∫
(f 2xt fxxx + fxt ftt)dxdt.
EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 11 / 17
Riemann Curvature Tensor
The Riemann tensor of curvature has the form
Rjikl = gjsRsikl = gjs [∂kΓs
il − ∂lΓsik + Γs
knΓnil − Γs
lnΓnik ],
where
Γijk =
1
2g im(gmj ,k + gmk,j − gjk,m).
However, taking into account
gmk,n + gkn,m + gmn,k = 0,
we obtainΓijk = −g imgjk,m.
ThenRjikl = gik,jl − gil ,jk + g sm(gik,mgjl ,s − gil ,mgjk,s).
EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 12 / 17
References
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References
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References
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EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 15 / 17
References
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EVF, MVP, GVP, RV (Italy, Russia, UK) Poisson Brackets 17.10.2013 16 / 17
References
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