Metrisability of Painleve equations and Hamiltonian systems of hydrodynamic type

Transcription

1 Metrisability of Painleve equations and Hamiltonian systems of hydrodynamic type Felipe Contatto Department of Applied Mathematics and Theoretical Physics University of Cambridge Joint work with Maciej Dunajski. arxiv: First integrals of affine connections and Hamiltonian systems of hydrodynamic type arxiv: Metrisability of Painlevé equations 28 July 2016 Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

2 Overview 1 Metrisability of projective structures 2 Deriving first integrals 3 Killing forms 4 Hydrodynamic-type systems Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

3 Overview 1 Metrisability of projective structures 2 Deriving first integrals 3 Killing forms 4 Hydrodynamic-type systems Problems: (i)consider the projective structures defined by the Painlevé equations, which of them are metrisable (if any)? Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

4 Overview 1 Metrisability of projective structures 2 Deriving first integrals 3 Killing forms 4 Hydrodynamic-type systems Problems: (i)consider the projective structures defined by the Painlevé equations, which of them are metrisable (if any)? (ii) How many first integrals linear in the momenta does a geodesic flow admit? Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

5 Overview 1 Metrisability of projective structures 2 Deriving first integrals 3 Killing forms 4 Hydrodynamic-type systems Problems: (i)consider the projective structures defined by the Painlevé equations, which of them are metrisable (if any)? (ii) How many first integrals linear in the momenta does a geodesic flow admit? (iii) Given a HT system, how many hamiltonian formulations (local sense) does it admit? Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

6 Metrisability problem (R. Liouville, 1889) Given a set of curves on R n, are they traces of geodesics of some metric? Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

7 Metrisability problem (R. Liouville, 1889) Given a set of curves on R n, are they traces of geodesics of some metric? On the (x, y)-plane, the necessary and sufficient condition for the curves to be the geodesics of an affine connection: to be the integral curves of an ODE y (x) = F(x, y, y ), where 4 y F(x, y, y ) = 0. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

8 Metrisability problem (R. Liouville, 1889) Given a set of curves on R n, are they traces of geodesics of some metric? On the (x, y)-plane, the necessary and sufficient condition for the curves to be the geodesics of an affine connection: to be the integral curves of an ODE y (x) = F(x, y, y ), where 4 y F(x, y, y ) = 0. Equivalence relation: two torsion-free connections are projectively equivalent (Γ ˆΓ) if they share the same unparametrised geodesics. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

9 Metrisability problem (R. Liouville, 1889) Given a set of curves on R n, are they traces of geodesics of some metric? On the (x, y)-plane, the necessary and sufficient condition for the curves to be the geodesics of an affine connection: to be the integral curves of an ODE y (x) = F(x, y, y ), where 4 y F(x, y, y ) = 0. Equivalence relation: two torsion-free connections are projectively equivalent (Γ ˆΓ) if they share the same unparametrised geodesics. Definition A projective structure [Γ] is the class of torsion-free connections that are projectively equivalent to Γ. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

10 Metrisability problem (R. Liouville, 1889) Given a set of curves on R n, are they traces of geodesics of some metric? On the (x, y)-plane, the necessary and sufficient condition for the curves to be the geodesics of an affine connection: to be the integral curves of an ODE y (x) = F(x, y, y ), where 4 y F(x, y, y ) = 0. Equivalence relation: two torsion-free connections are projectively equivalent (Γ ˆΓ) if they share the same unparametrised geodesics. Definition A projective structure [Γ] is the class of torsion-free connections that are projectively equivalent to Γ. Metrisability problem: given a projective structure, does it contain a metric connection? Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

11 The problem is (almost) solved in n = 2 dimensions: Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

12 The problem is (almost) solved in n = 2 dimensions: the necessary condition for the existence of a metric involves the vanishing of some invariants of differential order at least 5 in the connection [Bryant-Dunajski-Eastwood]. For the construction of a metric, the solution to the metrisability equations must satisfy the non-degeneracy condition. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

13 Hamiltonian description of geodesics Consider the metric g = g 11 dx 2 + 2g 12 dxdy + g 22 dy 2 and the geodesic Hamiltonian H = 1 2 g abp a p b. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

14 Hamiltonian description of geodesics Consider the metric g = g 11 dx 2 + 2g 12 dxdy + g 22 dy 2 and the geodesic Hamiltonian H = 1 2 g abp a p b. Hamilton s equations ẋ a (t) = H p a ṗ a (t) = H x a Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

15 Hamiltonian description of geodesics Consider the metric g = g 11 dx 2 + 2g 12 dxdy + g 22 dy 2 and the geodesic Hamiltonian H = 1 2 g abp a p b. Hamilton s equations ẋ a (t) = H p a Geodesic equations ṗ a (t) = H x a ẍ a + Γ a bcẋ b ẋ c = 0, a = 1, 2. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

16 Hamiltonian description of geodesics Consider the metric g = g 11 dx 2 + 2g 12 dxdy + g 22 dy 2 and the geodesic Hamiltonian H = 1 2 g abp a p b. Hamilton s equations ẋ a (t) = H p a Geodesic equations ṗ a (t) = H x a ẍ a + Γ a bcẋ b ẋ c = 0, a = 1, 2. Unparametrised geodesics (divide (a = 2)/(a = 1)) y = A 3 (x, y)y 3 + A 2 (x, y)y 2 + A 1 (x, y)y + A 0 (x, y) = F(x, y, y ), Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

17 Hamiltonian description of geodesics Consider the metric g = g 11 dx 2 + 2g 12 dxdy + g 22 dy 2 and the geodesic Hamiltonian H = 1 2 g abp a p b. Hamilton s equations ẋ a (t) = H p a Geodesic equations ṗ a (t) = H x a ẍ a + Γ a bcẋ b ẋ c = 0, a = 1, 2. Unparametrised geodesics (divide (a = 2)/(a = 1)) y = A 3 (x, y)y 3 + A 2 (x, y)y 2 + A 1 (x, y)y + A 0 (x, y) = F(x, y, y ), where A 0 = Γ 2 11, A 1 = Γ Γ2 12, A 2 = 2Γ 1 12 Γ2 22, A 3 = Γ Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

18 If K is a Killing vector conserved quantity: K a ẋ a = g ab K b ẋ a Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

19 If K is a Killing vector conserved quantity: K a ẋ a = g ab K b ẋ a Eliminate t using H = 1 2 (g 11ẋ 2 + 2g 12 ẋẏ + g 22 ẏ 2 ) and y (x) = ẏ/ẋ : I (x, y, y ) := 1 (K 1 + K 2 y ) 2 ( g11 + 2g 12 y + g 22 y 2) is a first integral of the unparametrised geodesic equation quadratic in y. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

20 If K is a Killing vector conserved quantity: K a ẋ a = g ab K b ẋ a Eliminate t using H = 1 2 (g 11ẋ 2 + 2g 12 ẋẏ + g 22 ẏ 2 ) and y (x) = ẏ/ẋ : I (x, y, y ) := 1 (K 1 + K 2 y ) 2 ( g11 + 2g 12 y + g 22 y 2) is a first integral of the unparametrised geodesic equation quadratic in y. The Painlevé equations define projective structures. (PI ) y = 6y 2 + x, (PII ) y = 2y 3 + xy + α,..., (PVI ). Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

21 If K is a Killing vector conserved quantity: K a ẋ a = g ab K b ẋ a Eliminate t using H = 1 2 (g 11ẋ 2 + 2g 12 ẋẏ + g 22 ẏ 2 ) and y (x) = ẏ/ẋ : I (x, y, y ) := 1 (K 1 + K 2 y ) 2 ( g11 + 2g 12 y + g 22 y 2) is a first integral of the unparametrised geodesic equation quadratic in y. The Painlevé equations define projective structures. (PI ) y = 6y 2 + x, (PII ) y = 2y 3 + xy + α,..., (PVI ). Satisfy the necessary conditions of [Bryant-Dunajski-Eastwood]. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

22 If K is a Killing vector conserved quantity: K a ẋ a = g ab K b ẋ a Eliminate t using H = 1 2 (g 11ẋ 2 + 2g 12 ẋẏ + g 22 ẏ 2 ) and y (x) = ẏ/ẋ : I (x, y, y ) := 1 (K 1 + K 2 y ) 2 ( g11 + 2g 12 y + g 22 y 2) is a first integral of the unparametrised geodesic equation quadratic in y. The Painlevé equations define projective structures. (PI ) y = 6y 2 + x, (PII ) y = 2y 3 + xy + α,..., (PVI ). Satisfy the necessary conditions of [Bryant-Dunajski-Eastwood]. Still need to check non-degeneracy. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

23 Metrisability of Painlevé equations Results: their projective structures are metrisable for (PIII), (PV) and (PVI) when they are projectively flat (equiv to Y (X ) = 0) or for PIII and PV when they admit an algebraic first integral. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

24 Metrisability of Painlevé equations Results: their projective structures are metrisable for (PIII), (PV) and (PVI) when they are projectively flat (equiv to Y (X ) = 0) or for PIII and PV when they admit an algebraic first integral. These first integrals are derivable from Killing vectors. E.g. (PV) ( 1 y = 2y + 1 ) y 2 1x ( y 1 y (y 1)2 + x 2 αy + β ) y First integral: I = 1 y ( xy y 1 ) 2 + 2β y 2αy. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

25 Killing 1-forms of affine connections Given an affine connection Γ on a surface Σ, its geodesics are the solutions to ẍ a + Γ a bcẋ b ẋ c = 0, a = 1, 2 Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

26 Killing 1-forms of affine connections Given an affine connection Γ on a surface Σ, its geodesics are the solutions to ẍ a + Γ a bcẋ b ẋ c = 0, a = 1, 2 I = K a ẋ a is a first integral (a K b) = 0 Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

27 Killing 1-forms of affine connections Given an affine connection Γ on a surface Σ, its geodesics are the solutions to ẍ a + Γ a bcẋ b ẋ c = 0, a = 1, 2 I = K a ẋ a is a first integral (a K b) = 0 First integrals linear in the momenta are equivalent to Killing 1-forms. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

28 Killing 1-forms of affine connections Given an affine connection Γ on a surface Σ, its geodesics are the solutions to ẍ a + Γ a bcẋ b ẋ c = 0, a = 1, 2 I = K a ẋ a is a first integral (a K b) = 0 First integrals linear in the momenta are equivalent to Killing 1-forms. Conditions for their existence will be established by prolongation. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

29 Killing 1-forms of affine connections Given an affine connection Γ on a surface Σ, its geodesics are the solutions to ẍ a + Γ a bcẋ b ẋ c = 0, a = 1, 2 I = K a ẋ a is a first integral (a K b) = 0 First integrals linear in the momenta are equivalent to Killing 1-forms. Conditions for their existence will be established by prolongation. Useful decomposition of the Riemann tensor: c R ab d = δ c a P bd δ c b P ad + B ab δ c d, where P ab = 2 3 R ab R ba and B ab = 2P [ab] = 2 3 R [ab]. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

30 Killing 1-forms of affine connections Given an affine connection Γ on a surface Σ, its geodesics are the solutions to ẍ a + Γ a bcẋ b ẋ c = 0, a = 1, 2 I = K a ẋ a is a first integral (a K b) = 0 First integrals linear in the momenta are equivalent to Killing 1-forms. Conditions for their existence will be established by prolongation. Useful decomposition of the Riemann tensor: c R ab d = δ c a P bd δ c b P ad + B ab δ c d, where P ab = 2 3 R ab R ba and B ab = 2P [ab] = 2 3 R [ab]. Introduce a volume form ɛ ab and its derivative c ɛ ab = θ c ɛ ab. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

31 Define the inverse volume form ɛ ab ɛ cb = δ a c. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

32 Define the inverse volume form ɛ ab ɛ cb = δ a c. Theorem There is a one to one correspondence between solutions to the Killing equations and parallel sections of the prolongation connection D on a rank three vector bundle E = Λ 1 (Σ) Λ 2 (Σ) Σ defined by D a ( Kb µ ) ( = ( a K b ɛ ab µ ) a µ P b a ɛef B ef δ b a K b + µθ a ). Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

33 Define the inverse volume form ɛ ab ɛ cb = δ a c. Theorem There is a one to one correspondence between solutions to the Killing equations and parallel sections of the prolongation connection D on a rank three vector bundle E = Λ 1 (Σ) Λ 2 (Σ) Σ defined by D a ( Kb µ ) ( = ( a K b ɛ ab µ ) a µ P b a ɛef B ef δ b a K b + µθ a ). The integrability conditions for the existence of parallel sections of this connection will lead to a set of invariants of the affine connection Γ. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

34 Theorem The necessary condition for a C 4 torsion free affine connection Γ on a surface Σ to admit a linear first integral is the vanishing, on Σ, of two scalars denoted by I N and I S of differential order 3 and 4 in Γ. Locally, I N = I S = 0 are necessary and sufficient for the existence of a Killing 1-form. there are precisely 2 Killing forms T a b = 0 and R [ab] 0, where T is a rank-2 tensor of differential order 3 in Γ. there are 3 independent Killing forms Γ is projectively flat and R [ab] = 0. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

35 Theorem The necessary condition for a C 4 torsion free affine connection Γ on a surface Σ to admit a linear first integral is the vanishing, on Σ, of two scalars denoted by I N and I S of differential order 3 and 4 in Γ. Locally, I N = I S = 0 are necessary and sufficient for the existence of a Killing 1-form. there are precisely 2 Killing forms T a b = 0 and R [ab] 0, where T is a rank-2 tensor of differential order 3 in Γ. there are 3 independent Killing forms Γ is projectively flat and R [ab] = 0. This does not hold globally. Counter example: the flat torus S 1 S 1 admits precisely 2 global Killing forms (or vectors). Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

36 Theorem The necessary condition for a C 4 torsion free affine connection Γ on a surface Σ to admit a linear first integral is the vanishing, on Σ, of two scalars denoted by I N and I S of differential order 3 and 4 in Γ. Locally, I N = I S = 0 are necessary and sufficient for the existence of a Killing 1-form. there are precisely 2 Killing forms T a b = 0 and R [ab] 0, where T is a rank-2 tensor of differential order 3 in Γ. there are 3 independent Killing forms Γ is projectively flat and R [ab] = 0. This does not hold globally. Counter example: the flat torus S 1 S 1 admits precisely 2 global Killing forms (or vectors). For special connections (R [ab] = 0), I N and I S become, essentially, Liouville s projective invariants ν 5 and w 1, respectively. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

37 Recall: R [ab] = [a Γ c b]c = 0 preserves a volume form. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

38 Recall: R [ab] = [a Γ c b]c = 0 preserves a volume form. [Γ] admits a traceless representative Π a bc, i.e., Πa ab = 0. It is given by Π a bc = Γa bc 1 3 δa b Γd dc 1 3 δa cγ d db. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

39 Recall: R [ab] = [a Γ c b]c = 0 preserves a volume form. [Γ] admits a traceless representative Π a bc, i.e., Πa ab = 0. It is given by Π a bc = Γa bc 1 3 δa b Γd dc 1 3 δa cγ d db. Π a bc are called Thomas symbols. It preserves ɛ 12 = 1. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

40 Recall: R [ab] = [a Γ c b]c = 0 preserves a volume form. [Γ] admits a traceless representative Π a bc, i.e., Πa ab = 0. It is given by Π a bc = Γa bc 1 3 δa b Γd dc 1 3 δa cγ d db. Π a bc are called Thomas symbols. It preserves ɛ 12 = 1. Question: what can we say about the unparametrised geodesics if its associated Thomas symbols admit a Killing form? Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

41 Recall: R [ab] = [a Γ c b]c = 0 preserves a volume form. [Γ] admits a traceless representative Π a bc, i.e., Πa ab = 0. It is given by Π a bc = Γa bc 1 3 δa b Γd dc 1 3 δa cγ d db. Π a bc are called Thomas symbols. It preserves ɛ 12 = 1. Question: what can we say about the unparametrised geodesics if its associated Thomas symbols admit a Killing form? The answer is given by the following theorem, which is partially due to [Liouville,1889]. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

42 Theorem The ODE y = A 0 (x, y) + A 1 (x, y)y + A 2 (x, y)(y ) 2 + A 3 (x, y)(y ) 3 defining a projective structure admits coordinates (X, Y ) such that Y XX = f (X, Y ) for some function f if and only if I N = I S = 0 for any special connection. Moreover, this is also equivalent to the fact that the connection with Thomas symbols admits a Killing 1-form given by dx. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

43 Theorem The ODE y = A 0 (x, y) + A 1 (x, y)y + A 2 (x, y)(y ) 2 + A 3 (x, y)(y ) 3 defining a projective structure admits coordinates (X, Y ) such that Y XX = f (X, Y ) for some function f if and only if I N = I S = 0 for any special connection. Moreover, this is also equivalent to the fact that the connection with Thomas symbols admits a Killing 1-form given by dx. Proof. Understand how Thomas symbols transform under coordinate transformations. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

44 Theorem The ODE y = A 0 (x, y) + A 1 (x, y)y + A 2 (x, y)(y ) 2 + A 3 (x, y)(y ) 3 defining a projective structure admits coordinates (X, Y ) such that Y XX = f (X, Y ) for some function f if and only if I N = I S = 0 for any special connection. Moreover, this is also equivalent to the fact that the connection with Thomas symbols admits a Killing 1-form given by dx. Proof. Understand how Thomas symbols transform under coordinate transformations. Understand how Killing tensors of Thomas symbols change under coordinate transformation. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

45 Theorem The ODE y = A 0 (x, y) + A 1 (x, y)y + A 2 (x, y)(y ) 2 + A 3 (x, y)(y ) 3 defining a projective structure admits coordinates (X, Y ) such that Y XX = f (X, Y ) for some function f if and only if I N = I S = 0 for any special connection. Moreover, this is also equivalent to the fact that the connection with Thomas symbols admits a Killing 1-form given by dx. Proof. Understand how Thomas symbols transform under coordinate transformations. Understand how Killing tensors of Thomas symbols change under coordinate transformation. Use these facts to show that one can choose coordinates (X, Y ) s.t. the Killing form is dx. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

46 Theorem The ODE y = A 0 (x, y) + A 1 (x, y)y + A 2 (x, y)(y ) 2 + A 3 (x, y)(y ) 3 defining a projective structure admits coordinates (X, Y ) such that Y XX = f (X, Y ) for some function f if and only if I N = I S = 0 for any special connection. Moreover, this is also equivalent to the fact that the connection with Thomas symbols admits a Killing 1-form given by dx. Proof. Understand how Thomas symbols transform under coordinate transformations. Understand how Killing tensors of Thomas symbols change under coordinate transformation. Use these facts to show that one can choose coordinates (X, Y ) s.t. the Killing form is dx. Check that this is equivalent to having Y XX = f (X, Y ). Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

47 Claim: Degenerate solutions to the metrisability equations correspond to Killing forms of special connections. Equivalently, I N = I S = 0 or ν 5 = w 1 = 0. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

48 Claim: Degenerate solutions to the metrisability equations correspond to Killing forms of special connections. Equivalently, I N = I S = 0 or ν 5 = w 1 = 0. Corollary (Babich,Bordag1999) The Painlevé equations can be put in the form y = f (x, y) under point transformation. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

49 Claim: Degenerate solutions to the metrisability equations correspond to Killing forms of special connections. Equivalently, I N = I S = 0 or ν 5 = w 1 = 0. Corollary (Babich,Bordag1999) The Painlevé equations can be put in the form y = f (x, y) under point transformation. Example: (PIII): y = αe x+y + βe x y + γe 2(x+y) + δe 2(x y) Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

50 Claim: Degenerate solutions to the metrisability equations correspond to Killing forms of special connections. Equivalently, I N = I S = 0 or ν 5 = w 1 = 0. Corollary (Babich,Bordag1999) The Painlevé equations can be put in the form y = f (x, y) under point transformation. Example: (PIII): y = αe x+y + βe x y + γe 2(x+y) + δe 2(x y) Remark: Scalars I N and I S, along with [B-D-E], answer the question about degeneracy in metrisability Metrisability problem itself is completely solved in 2D. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

51 Hydrodynamic-type (HT) systems Definition (HT system (our case)) A system of PDEs is of HT if it has the form t u a = v a b(u) x u b, a, b = 1, 2 where u a = u a (x, t) and v is a diagonalisable matrix with distinct real eigenvalues λ 1 (u) and λ 2 (u). Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

52 Hydrodynamic-type (HT) systems Definition (HT system (our case)) A system of PDEs is of HT if it has the form t u a = v a b(u) x u b, a, b = 1, 2 where u a = u a (x, t) and v is a diagonalisable matrix with distinct real eigenvalues λ 1 (u) and λ 2 (u). Theorem (Riemann invariants) A HT system admits coordinates R i (u) (called Riemann invariants) such that t R i = λ i (u(r)) x R i, i = 1, 2 (no summation). Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

53 Question: does my HT system admit a Hamiltonian formulation under a Poisson bracket of Dubrovin-Novikov type? δf ( {F, G} = δu a g ab (u) x + bab c (u) uc ) δg x δu b dx. R Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

54 Question: does my HT system admit a Hamiltonian formulation under a Poisson bracket of Dubrovin-Novikov type? δf ( {F, G} = δu a g ab (u) x + bab c (u) uc ) δg x δu b dx. Or u a t R ab δh = Ω δu b = g ab u c b c H }{{} x, v a c where is the Levi-Civita connection of g, H[u 1, u 2 ] = H(u 1, u 2 )dx and Ω ab = g ab x + u c bab c x Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

55 Answer [Ferapontov91]: It does iff there exists a flat diagonal metric k 1 d ( R 1) 2 + f 1 d ( R 2) 2 satisfying the following system of PDEs 2 k + 2Ak = 0, 1 f + 2Bf = 0, where A = 2λ 1 λ 2 λ 1, B = 1λ 2 λ 1 λ 2, and i = / R i Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

56 Answer [Ferapontov91]: It does iff there exists a flat diagonal metric k 1 d ( R 1) 2 + f 1 d ( R 2) 2 satisfying the following system of PDEs where And, by flatness, 2 k + 2Ak = 0, 1 f + 2Bf = 0, A = 2λ 1 λ 2 λ 1, B = 1λ 2 λ 1 λ 2, and i = / R i ( 2 A + A 2 )f + ( 1 B + B 2 )k A 2f B 1k = 0. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

57 Answer [Ferapontov91]: It does iff there exists a flat diagonal metric k 1 d ( R 1) 2 + f 1 d ( R 2) 2 satisfying the following system of PDEs where And, by flatness, 2 k + 2Ak = 0, 1 f + 2Bf = 0, A = 2λ 1 λ 2 λ 1, B = 1λ 2 λ 1 λ 2, and i = / R i ( 2 A + A 2 )f + ( 1 B + B 2 )k A 2f B 1k = 0. These are the compatibility conditions of the overdetermined system for H g ab b c H = v a c Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

58 Claim: The above overdetermined system of PDEs is equivalent to the Killing equations (a K b) = 0, where Γ 1 11 = 1 ln A 2B, Γ 2 22 = 2 ln B 2A, 1 ) 1 ) Γ 1 12 = ( 2 2 ln A + A, Γ 2 12 = ( 2 1 ln B + B, and K 1 = Af, K 2 = Bk. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

59 Claim: The above overdetermined system of PDEs is equivalent to the Killing equations (a K b) = 0, where Γ 1 11 = 1 ln A 2B, Γ 2 22 = 2 ln B 2A, 1 ) 1 ) Γ 1 12 = ( 2 2 ln A + A, Γ 2 12 = ( 2 1 ln B + B, and K 1 = Af, K 2 = Bk. Recall: A and B are determined by your HT system. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

60 Killing forms of the above connection are in 1 1 correspondence with Hamiltonians of the HT system. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

61 Killing forms of the above connection are in 1 1 correspondence with Hamiltonians of the HT system. Theorem A HT system admits 1, 2 or 3 Hamiltonian formulations iff its associated connection defined above admits 1, 2 or 3 independent linear first integrals respectively. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

62 Killing forms of the above connection are in 1 1 correspondence with Hamiltonians of the HT system. Theorem A HT system admits 1, 2 or 3 Hamiltonian formulations iff its associated connection defined above admits 1, 2 or 3 independent linear first integrals respectively. This connection defines the following projective structure Y = ( X ln (AB))Y ( Y ln (AB))(Y ) 2. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

63 Killing forms of the above connection are in 1 1 correspondence with Hamiltonians of the HT system. Theorem A HT system admits 1, 2 or 3 Hamiltonian formulations iff its associated connection defined above admits 1, 2 or 3 independent linear first integrals respectively. This connection defines the following projective structure Y = ( X ln (AB))Y ( Y ln (AB))(Y ) 2. Theorem This projective structure is metrisable by the Lorentzian metric A B d ( R 1) d ( R 2). Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

64 Corollary A HT sytem is trihamiltonian iff its associated connection has symmetric Ricci tensor and (AB) ln(ab) = const (metric of constant curvature). Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

65 Corollary A HT sytem is trihamiltonian iff its associated connection has symmetric Ricci tensor and (AB) ln(ab) = const (metric of constant curvature). A two-dimensional Frobenius manifold (M, η, e, E, ) (η=flat metric, e=identity, E=Euler vector field) with flat coordinates (t 1, t 2 ) defines a HT system t i t = ej cjl i t l x, where i j = c k ij k and e j = j. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

66 Corollary A HT sytem is trihamiltonian iff its associated connection has symmetric Ricci tensor and (AB) ln(ab) = const (metric of constant curvature). A two-dimensional Frobenius manifold (M, η, e, E, ) (η=flat metric, e=identity, E=Euler vector field) with flat coordinates (t 1, t 2 ) defines a HT system t i t = ej cjl i t l x, where i j = c k ij k and e j = j. Theorem HT systems arising from Frobenius manifolds are trihamiltonian. Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

67 Corollary A HT sytem is trihamiltonian iff its associated connection has symmetric Ricci tensor and (AB) ln(ab) = const (metric of constant curvature). A two-dimensional Frobenius manifold (M, η, e, E, ) (η=flat metric, e=identity, E=Euler vector field) with flat coordinates (t 1, t 2 ) defines a HT system t i t = ej cjl i t l x, where i j = c k ij k and e j = j. Theorem HT systems arising from Frobenius manifolds are trihamiltonian. The flat metrics determining the Poisson brackets are η ij (metric), h ij = E k η ik c j kl (intersection form) and h ik h jl η kl (whatever). Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

68 Outlook Statement of the problem of R. Liouville Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

69 Outlook Statement of the problem of R. Liouville Metrisability of the Painlevé equations and derivation of their first integrals Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

70 Outlook Statement of the problem of R. Liouville Metrisability of the Painlevé equations and derivation of their first integrals Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

71 Outlook Statement of the problem of R. Liouville Metrisability of the Painlevé equations and derivation of their first integrals Killing forms of affine connections Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

72 Outlook Statement of the problem of R. Liouville Metrisability of the Painlevé equations and derivation of their first integrals Killing forms of affine connections Hamiltonian structures of HT systems Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20

73 Outlook Statement of the problem of R. Liouville Metrisability of the Painlevé equations and derivation of their first integrals Killing forms of affine connections Hamiltonian structures of HT systems Thank you! Felipe Contatto (DAMTP, Cambridge) LMS Durham Symposium July / 20