Proceedings Ist International Meeting on Geometry and Topology Braga (Portugal)

Public. Centro de Matemática da Universidade do Minho p. 149–156, 1998

Bi-Hamiltonian manifolds – an application to the relativistic Toda lattice and a generalization to the Jacobi manifolds^{1}

J. M. Nunes da Costa^{2}

Abstract: An application of bi-Hamiltonian manifolds: using mastersymmetries, we generate a hierarchy of Poisson tensors that provides a multi-Hamiltonian formulation for the relativistic Toda lattice.

A generalization of bi-Hamiltonian manifolds: we extend the notion of compatible Poisson tensors to the Jacobi structures.

All manifolds, maps, vector and tensor fields are assumed to be differentiable of class C^{∞}.

I) Bi-Hamiltonian manifolds — an application

1 Recursion operators and Poisson-Nijenhuis manifolds

The bi-Hamiltonian manifolds were introduced by F. Magri [5] in 1978. In this section we review the basic facts concerning recursion operators and bi-Hamiltonian manifolds, namely Poisson-Nijenhuis manifolds [3, 6].

Definition 4 Let X be a vector field on a manifold M. A tensor field R of type (1, 1) (one time contravariant and one time covariant) is called a recursion operator of X if L_{X}R = 0.

As it is usual, we shall consider a tensor field of type (1, 1) as a vector bundle map of the tangent bundle T M into itself and we shall denote the transpose map by ^{t}R : T^{∗}M → T^{∗}M.

1AMS Classification: 58F05

2This work was partially supported by CMUC/FCT and PRAXIS/2/2.1/MAT (19/94 and 458/94).

150 J. M. Nunes da Costa

Definition 5 Let R be a tensor field of type (1, 1) on a manifold M. The Nijenhuis torsion of R is a tensor field τ(R) of type (1, 2) (one time contravariant and two times covariant) given, for any pair (X, Y) of vector fields on M, by

τ(R)(X, Y) = [RX, RY] − R([RX, Y] + [X, RY] − R[X, Y]). (1.1)

Definition 6 A tensor field R of type (1, 1) on the manifold M is called a Nijenhuis operator if its Nijenhuis torsion τ(R) vanishes.

If R is a Nijenhuis operator, then for any vector field X on M,

L_{RX}R = RL_{X}R. (1.2)

Proposition 2 Let R be a recursion operator of a vector field X on the manifold M. If R is a Nijenhuis operator, then R is recursion operator of each X_{k} = R^{k} X, k ∈ IN. Furthermore, [X_{k}, X_{l}] = 0, for all k, l ∈ IN.

Definition 7 A bi-Hamiltonian manifold is a triple (M, Λ_{0}, Λ_{1}) where Λ_{0} and Λ_{1} are compatible Poisson tensors on the smooth manifold M. The compatibility condition of Λ_{0} and Λ_{1} can be expressed by [Λ_{0}, Λ_{1}] = 0.

If Λ is a bivector on the manifold M, it defines a vector bundle map Λ^{#} : T^{∗}M → T M such that, for any x ∈ M and α, β ∈ T_{x}^{∗}M,

< β, Λ_{x}^{#}(α) >= Λ_{x}(α, β).

Definition 8 A Poisson-Nijenhuis manifold is a triple (M, Λ_{0}, R) where (M, Λ_{0}) is a Poisson manifold and R is a tensor field on M of type (1, 1) such that, if Λ_{1} is the bivector on M corresponding to the bundle map Λ_{1}^{#} = RΛ_{0}^{#}, then

Λ_{1}^{#} = Λ_{0}^{#} ∘ ^{t}R, [Λ_{0}, Λ_{1}] = 0 and [Λ_{1}, Λ_{1}] = 0.

From last definition, we may conclude that a Poisson-Nijenhuis manifold is a bi-Hamiltonian manifold and also that if (M, Λ_{0}, R) is a Poisson-Nijenhuis manifold, the Nijenhuis torsion of R vanishes on Λ_{0}^{#}(T^{∗}M) × Λ_{0}^{#}(T^{∗}M).

Let (M, Λ_{0}, R) be a Poisson-Nijenhuis manifold and define the following sequence of bivectors Λ_{k} = R^{k}Λ_{0}, k ∈ IN, on M. Then, (Λ_{k})_{k∈IN_{0}} is a hierarchy of Poisson tensors on M, that are pairwaise compatible.

Let us now consider the case of a bi-Hamiltonian manifold (M, Λ_{0}, Λ_{1}) where the Poisson tensor Λ_{0} is non-degenerate. In such a case, we may define the tensor field R = Λ_{1}^{#} ∘ (Λ_{0}^{#})^{−1} and (M, Λ_{0}, R) is a Poisson- Nijenhuis manifold. The tensor field R is a Nijenhuis operator and is also a recursion operator for all bi-Hamiltonian vector fields on M (that is, vector fields that are Hamiltonian with respect to both Poisson structures on M).

Bi-Hamiltonian manifolds 151

2 Mastersymmetries and bi-Hamiltonian systems

Mastersymmetries were introduced by B. Fuchssteiner [1] and mastersymmetries for bi-Hamiltonian systems were studied by W. Oevel [9]. First of all, we recall the notion of mastersymmetry.

Definition 9 Let Y be a bi-Hamiltonian vector field on a manifold M. The vector field X on M is a symmetry of the differential equation ẏ = Y (y) if [X, Y] = 0 and X is called a mastersymmetry if

[[X, Y], Y] = 0 and [X, Y] ≠ 0.

Let us take a bi-Hamiltonian manifold (M, Λ_{0}, Λ_{1}), with Λ_{0} non-degenerate. As we have already remarked, if we set R = Λ_{1}^{#} ∘ (Λ_{0}^{#})^{−1} then (M, Λ_{0}, R) is a Poisson-Nijenhuis manifold. Suppose that the first cohomology of M is trivial. Then, if Y_{1} = Λ_{0}^{#}(dH_{1}) = Λ_{1}^{#}(dH_{0}) is a bi-Hamiltonian vector field on M, there exists a sequence (H_{k})_{k∈IN} of first integrals of Y_{1} that are pairwaise in involution with respect to both Poisson brackets on M. The Hamiltonians H_{k} are defined (up to additive constants) by dH_{k} = (^{t}R)^{k}(dH_{0}). Under these conditions we have the following theorem due to Oevel.

Theorem 10 (Oevel) Let X_{0} be a vector field on the Poisson-Nijenhuis manifold (M, Λ_{0}, R) such that

L(X_{0})Λ_{0} = αΛ_{0}, L(X_{0})Λ_{1} = βΛ_{1} and L(X_{0})H_{0} = γH_{0},

for all α, β, γ ∈ IR.

Then, if Y_{k} = Λ_{k}^{#}(dH_{0}), the vector fields X_{k} = R^{k}X_{0} satisfy, for all k, l ∈ IN,

(i) [X_{k}, X_{l}] = (β − α)(l − k)X_{k+l};

(ii) [X_{k}, Y_{l}] = (β + γ + (β − α)(l − 1))Y_{k+l};

(iii) L(X_{k})Λ_{l} = (β + (β − α)(l − k − 1))Λ_{k+l};

(iv) L(X_{k})H_{l} = (γ + (β − α)(l + k))H_{k+l}.

Remark that for all k, l ∈ IN, [X_{k}, Y_{l}] ≠ 0 and [[X_{k}, Y_{l}], Y_{l}] = 0; that is, (X_{k})_{k∈IN} is a sequence of mastersymmetries for this bi-Hamiltonian system.

3 The relativistic Toda lattice

Let us consider IR^{2n} with coordinates (q^{1}, …, q^{n}, p_{1}, …, p_{n}). The relativistic Toda lattice (RTL), introduced by S. Ruijsenaars [10], is a finite-dimensional completely integrable system that admits the following bi-Hamiltonian formulation [8]:

Λ_{0} = ∑_{i=1}^{n} exp(−p_{i})\frac{∂}{∂q^{i}} ∧ (\frac{∂}{∂p_{i}} + ∑_{j=i+1}^{n} \frac{∂}{∂q^{j}} )

+ ∑_{i=1}^{n−1} exp(q^{i} − q^{i+1} − p_{i+1}) ((\frac{∂}{∂p_{i}} + \frac{∂}{∂q^{i+1}} ) ∧ (\frac{∂}{∂p_{i+1}} + ∑_{j=i+2}^{n} \frac{∂}{∂q^{j}} )

−\frac{∂}{∂p_{i+1}} ∧ ∑_{j=i+2}^{n} \frac{∂}{∂q^{j}} ) (3.1)

Λ_{1} = ∑_{i=1}^{n} \frac{∂}{∂q^{i}} ∧ \frac{∂}{∂p_{i}}; (3.2)

with Hamiltonians

H_{0} = ∑_{i=1}^{n} (exp(q^{i} − q^{i+1} + p_{i}) + exp(p_{i})) (3.3)

and

(3.4) H_{1} = ∑_{i=1}^{n} (\frac{1}{2} (exp(q^{i} − q^{i+1} + p_{i}) + exp(p_{i}))^{2}

(3.5) + exp(q^{i−1} − q^{i} + p_{i−1}) (exp(q^{i} − q^{i+1} + p_{i}) + exp(p_{i}))) ,

where, by convention, q^{0} = −∞ and q^{n+1} = +∞.

Since Λ_{0} is a non-degenerate Poisson tensor on IR^{2n}, the triple (IR^{2n}, Λ_{0}, R) is a Poisson-Nijenhuis manifold, with R = Λ_{1}^{#} ∘ (Λ_{0}^{#})^{−1}. On the other hand, if we take

X_{0} = ∑_{i=1}^{n} \frac{∂}{∂p_{i}} , (3.6)

we compute

L(X_{0})Λ_{0} = −Λ_{0}, L(X_{0})Λ_{1} = 0 L(X_{0})H_{0} = H_{0}. (3.7)

So, we may apply Oevel’s theorem to generate a hierarchy (X_{k})_{k∈IN} of mastersymmetries, satisfying the following conditions, with Y_{k} = Λ_{k}^{#}(dH_{0}):

(3.8)  [X_{k}, X_{l}] = (l − k)X_{k+l};
(3.9)  [X_{k}, Y_{l}] = lY_{k+l};

(3.10)  L(X_{k})Λ_{l} = (l − k − 1))Λ_{k+l};
(3.11)  L(X_{k})H_{l} = (1 + l + k))H_{k+l}.

Thus we obtain a multi-Hamiltonian formulation for the RTL.

Let us consider the following map,

π : (q^{1}, …, q^{n}, p_{1}, …, p_{n}) ∈ IR^{2n} → (c_{1}, …, c_{n−1}, d_{1}, …, d_{n}) ∈ IR^{2n−1},

with c_{i} = exp(q^{i} − q^{i−1} + p_{i}) and d_{i} = exp(p_{i}).

The Poisson tensors Λ_{0} and Λ_{1} on IR^{2n} reduce by π onto Poisson tensors Λ̄_{0} and Λ̄_{1} on IR^{2n−1}, that are compatible. The Hamiltonians H_{0} and H_{1} can also be reduced by π. However, we may not reduce the recursion operator R and so we have no recursion operator on IR^{2n−1}. In spite of that, the mastersymmetries X_{k} = R^{k}X_{0} are projectable vector fields by π. In fact, relations (3.8)-(3.11) still hold for the reduced objects. This provides a way of generating multi-Hamiltonian formulations for the RTL on IR^{2n−1}.

II) Bi-Hamiltonian manifolds — a generalization

4 Compatible Jacobi manifolds

The notion of Jacobi manifold was introduced by A. Lichnerowicz [4]. Let us recall that a Jacobi manifold is a triple (M, C, E) where M is a smooth manifold, C is a bivector on M and E is a vector field on M such that

[E, C] = 0 and [C, C] = 2E ∧ C. (4.1)

If (M, C, E) is a Jacobi manifold, C^{∞}(M, IR) has the structure of a local Lie algebra [2] given by the Jacobi bracket:

{f, g} = C(df, dg) + f(E.g) − g(E.f), f, g ∈ C^{∞}(M, IR). (4.2)

With each function f ∈ C^{∞}(M, IR), we may associate a vector field on M,

X_{f} = C^{#}(df) + fE, (4.3)

which is called the Hamiltonian vector field associated with f.

Let M be a smooth manifold equipped with two Jacobi structures, (C_{1}, E_{1}) and (C_{2}, E_{2}).

Definition 11 The Jacobi manifolds (M, C_{1}, E_{1}) and (M, C_{2}, E_{2}) are said to be compatible if (C_{1} + C_{2}, E_{1} + E_{2}) is again a Jacobi structure on M.

All the results that follow can be found in [7]. Next theorem states the necessary and sufficient conditions for two Jacobi structures, defined on the same manifold, to be compatible.

Theorem 12 The Jacobi manifolds (M, C_{1}, E_{1}) and (M, C_{2}, E_{2}) are compatible if and only if

(i) [C_{1}, C_{2}] = E_{1} ∧ C_{2} + E_{2} ∧ C_{1};
(ii) [E_{1}, C_{2}] + [E_{2}, C_{1}] = 0.

The compatibility of two Jacobi structures defined on the same manifold can be stated using some equivalent properties.

Theorem 13 Let (M, C_{1}, E_{1}) and (M, C_{2}, E_{2}) be two Jacobi manifolds, whose Jacobi brackets are denoted respectively by {, }_{1} and {, }_{2}. Given a function f ∈ C^{∞}(M, IR), we denote by X_{f} and Y_{f} the Hamiltonian vector fields on M associated with f, with respect to the Jacobi structures (C_{1}, E_{1}) and (C_{2}, E_{2}), respectively.

The following properties 1, 2, 3 and 4 are equivalent:

(i) (a) [C_{1}, C_{2}] = E_{1} ∧ C_{2} + E_{2} ∧ C_{1}, (b) [E_{1}, C_{2}] + [E_{2}, C_{1}] = 0;

(ii) for all f, g, h ∈ C^{∞}(M, IR),

{f, {g, h}_{1}}_{2} + {g, {h, f}_{1}}_{2} + {h, {f, g}_{1}}_{2} +
+{f, {g, h}_{2}}_{1} + {g, {h, f}_{2}}_{1} + {h, {f, g}_{2}}_{1} = 0;

(iii) for all f, g ∈ C^{∞}(M, IR),

[X_{f}, Y_{g}] + [Y_{f}, X_{g}] − X_{{f,g}_{2}} − Y_{{f,g}_{1}} = 0;

(iv) for all f ∈ C^{∞}(M, IR), (a) L(Y_{f})C_{1} + L(X_{f})C_{2} + (E_{2}.f)C_{1} + (E_{1}.f)C_{2} = 0,

(b) [X_{f}, E_{2}] + [Y_{f}, E_{1}] + X_{(E_{2}.f)} + Y_{(E_{1}.f)} = 0.

Let (M, C_{1}, E_{1}) and (M, C_{2}, E_{2}) be two Jacobi manifolds. Let us take the corresponding associated homogeneous Poisson manifolds, (P, Λ_{1}) and (P, Λ_{2}), with P = IR × M and

Λ_{i} = e^{−t}(C_{i} + \frac{∂}{∂t} ∧ E_{i}), i = 1, 2. (4.4)

Proposition 3 The Jacobi manifolds (M, C_{1}, E_{1}) and (M, C_{2}, E_{2}) are compatible if and only if the Poisson tensors Λ_{1} and Λ_{2} on P are compatible.

Let us recall that if (M, C, E) is a Jacobi manifold with Jacobi bracket denoted by {, } and if h ∈ C^{∞}(M, IR) is a smooth function that never vanishes on M , we may define a new Jacobi bracket {, }_{h} on M:

\{f, g\}_{h} = \frac{1}{h}\{hf, hg\}, f, g ∈ C^{∞}(M, IR). (4.5)

The new Jacobi structure (C_{h}, E_{h}) on M is given by

C_{h} = hC, E_{h} = C^{#}(dh) + hE. (4.6)

In such a case, the two Jacobi structures (C, E) and (C_{h}, E_{h}) on M are said to be conformally equivalent.

The next two Propositions show the relationship between the compatibility of Jacobi manifolds and the conformal equivalence of their Jacobi structures.

Proposition 4 Let (M, C, E) be a Jacobi manifold and let h, k ∈ C^{∞}(M, IR) be two distinct functions that never vanish on M. If (C_{h}, E_{h}) and (C_{k}, E_{k}) are two Jacobi structures on M, that are conformally equivalent to (C, E), then (C_{h}, E_{h}) and (C_{k}, E_{k}) are compatible, where C_{h} = hC, C_{k} = kC, E_{h} = hE + [C, h] and E_{k} = kE + [C, k].

In particular, any Jacobi structure on M which is conformally equivalent to (C, E), is compatible with (C, E).

Proposition 5 Let (M, C_{1}, E_{1}) and (M, C_{2}, E_{2}) be two compatible Jacobi manifolds. Then, for any function h ∈ C^{∞}(M, IR) that never vanishes on M, the Jacobi manifolds (M, C_{1h}, E_{1h}) and (M, C_{2h}, E_{2h}), with C_{ih} = hC_{i} and E_{ih} = hE_{i} + [C_{i}, h], i = 1, 2, are compatible.

References

[1] B. Fuchssteiner, Mastersymmetries and higher order time-dependent symmetries and conserved densities of nonlinear evolution equations, Progr. Theor. Phys. 70, 1508 (1983).

[2] Kirillov A., Local Lie algebras, Russian Math. Surveys 31 (4), 55 (1976).

[3] Kosmann-Schwarzbach Y. and Magri F., Poisson-Nijenhuis structures, Ann. Inst. Henry Poincaré 53 (1), 35 (1990).

[4] Lichnerowicz A., Les variétés de Jacobi et leurs algèbres de Lie associées, J. Math. pures et appl. 57, 453 (1978).

[5] Magri F., A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (5), 1156 (1978).

[6] Magri F. and Morosi C., A Geometrical characterization of integrable Hamiltonian systems through the theory of Poisson-Nijenhuis manifolds, Quaderno S19, Univ. of Milan, 1984.

[7] Nunes da Costa J. M., Compatible Jacobi manifolds: geometry and reduction, to appear in J. Phys. A: Math. Gen.

[8] Nunes da Costa J. M. and Marle C.-M., Master symmetries and bi-Hamiltonian structures for the relativistic Toda lattice, J. Phys. A: Math. Gen. 30, 7551 (1997).

[9] Oevel W., A geometrical approach to integrable systems admitting time dependent invariants, in Topics in Soliton Theory and Exactly Solvable Nonlinear Equations, ed. M. Ablowitz, B. Fuchssteiner and M. Kruskal (World Scientific, Singapore, 1987).

[10] Ruijsenaars S. N. M., Relativistic Toda Systems, Commun. Math. Phys., 133, 217 (1990).

[11] Suris Yu. B., On the bi-Hamiltonian structure of Toda and relativistic Toda lattices, Phys. Lett. A 180, 419 (1993).

Joana Margarida Nunes da Costa Departamento de Matematica Universidade de Coimbra Apartado 3008 3000 COIMBRA Portugal

Email: jmcosta@mat.uc.pt