Integrable Systems Related to Deformed $\mathfrak{so}(5)$

We investigate a family of integrable Hamiltonian systems on Lie-Poisson spaces $\mathcal{L}_+(5)$ dual to Lie algebras $\mathfrak{so}_{\lambda, \alpha}(5)$ being two-parameter deformations of $\mathfrak{so}(5)$. We integrate corresponding Hamiltonian equations on $\mathcal{L}_+(5)$ and $T^*\mathbb{R}^5$ by quadratures as well as discuss their possible physical interpretation.


Introduction
The notion of compatible Poisson structures on a manifold M , firstly introduced by Magri in [12], leads to one of the most productive methods of construction of functions on M being in involution. This method was used by many authors to integrate various Hamiltonian systems, see, e.g., monograph [19] for interesting examples as well as a huge number of references therein.
A pencil of Lie brackets on vector space g defines compatible Lie-Poisson structures on the dual g * to g. For the treatment of this case see [19,Chapter 7,Section 44]. One can find many examples of Hamiltonian systems on Lie-Poisson space g * obtained in this way in [2,3,10,12,13,17,18,20,21].
The main results of the paper are the following ones. In Section 2 we construct and integrate by quadratures a Hamiltonian system on Lie-Poisson space L + (5) with Poisson bracket {·, ·} λ,α defined by (2.2) and Hamiltonian defined by (2.11).
The lifting of the Hamiltonian system (2.12)-(2.15) on the symplectic manifold T * R 5 , see Hamiltonian (4.2) and Hamilton equations (4.3), is integrated in Section 4. We present some examples of the physical interpretation of the system given by (4.2) in Section 4 as well.
In the coordinates (a, x, y, µ) Lie-Poisson bracket for f, g ∈ C ∞ (L + (5)) is given by the formula where µ = (µ 1 , µ 2 , µ 3 ) . Let us note that this bracket belongs to the family of Lie-Poisson brackets investigated in [15]. According to Proposition 3 from [15] the global Casimirs for the bracket {·, ·} λ,α are as follows 3) Choosing a Hamiltonian H ∈ C ∞ (L + (5)) we obtain Hamilton equations on Lie-Poisson space L + (5). We will construct a family of Hamiltonians depending on two real parameters, which are completely integrable.
To this end we observe that Poisson brackets {·, ·} λ,α and {·, ·} ,β are compatible if α = β or λ = , see [15,Proposition 4] what means that the linear combination of these brackets is a Lie-Poisson bracket. In this paper we consider the case when λ = and α = β. Since the case α = 0 was considered in [6] we will not discuss it here. The bi-Hamiltonian systems given by the Lie-Poisson bracket {·, ·} 1,α and the constant Lie-Poisson bracket were studied in [7]. By Magri method [12] it can be shown that Casimir functions of the Poisson bracket {·, ·} ,α : One can verify functions h 1 , h 2 , µ 2 , µ 3 , a, h 1 −c 1 , h 2 −c 2 ∈ C ∞ (L + (5)) to be integrals of motion here which are in involution. Recall that c 1 and c 2 are Casimir functions defined in (2.3), (2.4). Since generic symplectic leaves of L + (5) have dimension eight then for the integrability of the above Hamiltonian system it is enough to possess four functionally independent integrals of motion being in involution with respect to the Poisson bracket {·, ·} λ,α . For example one of the possible choices of four integrals of motion is One easily verifies that the following proposition is valid.
From (2.17) we conclude that a, µ 3 , h 1 − c 1 , h 2 − c 2 are integrals of motion functionally independent almost everywhere. There are the other choices of four integrals of motion for example a, µ 2 , h 1 , h 2 , which are also functionally independent almost everywhere. However, the proof of this property is technically more difficult than in the case (2.16). where O ∈ SO(3). The above motivates us to use the following SO(3)-invariant coordinates x := µ · x, y := µ · y, f := 2 x · y, (2.18) in order to solve (2.12)-(2.15). In these coordinates equations (2.14), (2.15) (for the case α = 0, a = 0) reduce to the following three equations d dt where the constants C, D and K are expressed is terms of Casimirs (2.3), (2.4) and integrals of motion a, h 1 , h 2 and µ 2 in the following way Introducing new variables ϕ, ψ and r by and substituting them into (2.19) and (2.20) we obtain Now from (2.21) and (2.22) we have Separating variables in (2.24) we find where constant R is defined by (2.24). Functions x(t), y(t), f (t) are expressed by means of elliptic function g(t) as follows Now, without loss of generality, we can assume µ = (0, 0, µ). Then we obtain that One obtains the other coordinate functions x 1 (t), x 2 (t), y 1 (t) and y 2 (t) from algebraic equations In order to integrate equations (2.12)-(2.15) in the case when a = 0 we note that functions are independent of the parameter t ∈ R. We note also that functions From (2.29) we find and thus the equation
So, I : T * R 5 → sl(2, R) is a momentum map. As usual the vector space isomorphism of Lie algebra sl(2, R) with its dual sl(2, R) * is defined by the trace. Let us recall that Lie-Poisson bracket for sl(2, R) is given by the formula Proposition 3.1. For both momentum maps mentioned above the following holds: (i) They prove to be Poisson maps, i.e. arrows in the diagram are morphisms of Poisson manifolds.
Proof . The property (3.10) follow from Leibniz rule and relations where d k and a, µ, x, y are given by (3.7) and (3.6), respectively.
From the above properties of I and J we conclude that diagram (3.9) realizes symplectic dual pair. For the definition of symplectic dual pair see [5, Chapter IV, Section 9.3].
We will consider T * R 5 as union of two complementary subsets where the subset T * sing R 5 consists of the pairs (q, p) ∈ T * sing R 5 such that q ∈ R 5 and η −1 λ,α p ∈ R 5 are linearly dependent while (q, p) ∈ T * reg R 5 iff q and η λ,α p are linearly independent. Note that T * sing R 5 = J −1 (0) and so, it is closed in T * R 5 . The function is a Casimir of the Poisson bracket (3.8) and the equality is valid, where c 1 is Casimir function defined in (2.3). See (3.4) for definition of ι : L + (5) −→ so (5). The function δ λ,α as well as the subsets T * sing R 5 and T * reg R 5 are invariant with respect to the action of the groups Φ(SO λ,α (5)) and Ψ (SL(2, R)). Let us also mention that for A ∈ SL(2, R) and g ∈ SO λ,α (5).
We will present other important facts in the following Proposition 3.2.
(iii) The fibres 14) are 7-dimensional submanifolds of T * reg R 5 . They are also invariant with respect to the action of SO λ,α (5) and the action of stabilizer subgroup SL(2, R) d .
Thus according to the theory of Grassmannians, see, e.g., [8, Chapter I, Section 5], we note that the momentum map (3.5) defines the Plücker embeding P : G(2, 5) → P( 2 R 5 ) ∼ = P(so (5)) of the Grassmannian G(2, 5) of the 2-dimensional vector subspaces of R 5 , spanned by vectors q, η −1 λ,α p ∈ R 5 . Thus the image J (T * reg R 5 ) of T * reg R 5 in so(5) is described by the Plücker relations which one obtains directly from (3.6). We also observe that T * reg R 5 has structure of the GL(2, R)principal bundle, i.e. it is the total space of Stiefel principal bundle over G (2,5), for definition of Stiefel bundle see [14]. Equations (3.17) define 7-dimensional submanifold J T * reg R 5 in so(5) which is invariant with respect to the multiplication so(5) → r ∈ so(5) of by r ∈ R \ {0}. So, one has the (R \ {0})-principal bundle over the Grassmannian G(2, 5). Let us note here that (3.19) is the determinant bundle of the bundle (3.18). Thus one has the surjective morphism of the principal bundles defined by the momentum map J : T * reg R 5 → so(5) and the determinant map det : GL(2, R) → R \ {0}.
Since, submanifold Γ s ⊂ T * reg R 5 is invariant with respect to the action of SL ± (2, R) it is a total space of the SL ± (2, R)-principal subbundle of the GL(2, R)-principal bundle (3.18). The structural groups morphism in this case is given by the inclusion SL ± (2, R) → GL(2, R).
The Hamiltonian flow {σ λ,α t } t∈R on T * reg R 5 defined by the Hamiltonian δ λ,α = 1 αλ (c 1 •ι −1 •J ) is described explicitly by expressions (4.12), (4.13) established in Section 4. It preserves fibres J −1 ( ) and I −1 ( d) of both momentum maps and on I −1 ( d) it is identical to the action of the stabilizer subgroup SL(2, R) d ⊂ SL(2, R). From (3.12) one sees that δ λ,α is the pull-back of the Casimirs c and 1 αλ (c 1 • ι −1 ). Thus the groups SL(2, R) and SO λ,α (5) act also on the reduced symplectic manifold Γ s := Γ s /{σ λ,α t } by symplectomorphisms. Summarizing the above facts we can formulate  Since Ω s ⊂ so(5) is invariant with respect to the coadjoint action of SO λ,α (5) we will investigate the decomposition of Ω s into the orbits of this action. For this reason we note that ( I) −1 ( d) ⊂ T * reg R 5 , where ( I)(q, p) := I(q, p), is invariant with respect to the action (3.1).
Completing this section let us shortly discuss the case T * sing R 5 = J −1 (0). If (q, p) ∈ J −1 (0) then one has b 1 q + b 2 η −1 λ,α p = 0 for some 0 = b 1 b 2 ∈ R 2 . Thus, we find that I(q, p) b 1 b 2 = 0 and hence I J −1 (0) ⊂ ∆ 0 . Summing up the above facts we obtain bundle I : The canonical form γ after restriction to J −1 (0) \ {0} is given by So, dγ J −1 (0) is equal to the lifting I * ω 0 of the SL(2, R)-invariant symplectic form ω 0 of the symplectic leaf ∆ 0 ⊂ SL(2, R). We will not consider this case in what follows. The reason is that the Hamiltonian H •J after restriction to J −1 (0) vanishes, so it generates trivial dynamics.
In the next section we will use fibration (3.20) to integrate Hamiltonian equations defined by Hamiltonian H • J for regular case I −1 ( d) ∩ T * r R 5 .

Solutions and their physical interpretations
Our goal is to use results of two previous section for solving Hamilton equations dq dt = ∂h ∂p and dp dt = − ∂h ∂q (4.1) on T * R 5 with Hamiltonian where H is defined in (2.11). After substituting (4.2) into (4.1) we obtain Using (3.6) and (3.14)-(3.16) we transform above system of equations to the following one where Integrating linear system given in (4.4) we obtain where Further, substituting solutions (4.6) into (4.5), see also (3.14)-(3.16), we come to a non-autonomous linear system of equations for functions q(t) and p(t). In order to solve this system let us consider separately two subcases a = 0 and a = 0. If a = 0 then from (3.6) we get where q −1 (t), q 0 (t), p −1 (t) and p 0 (t) are given by (4.6) and ( x(t), y(t)) were found in Section 2, see (2.25), (2.26).
Finally let us discuss a few possible physical interpretations of the above integrated Hamiltonian systems.
Firstly let us note that if γ = 1 and = λ then h = 1 αλ c 1 • ι −1 • J = δ λ,α . In this case equations (4.1) take the form solution of (4.11) is given by where is a one-parameter subgroup of SL(2, R) d . This allows us to restrict the Hamiltonian δ λ,α and the flow Ψ(A λ,α (t)) to symplectic submanifold of T * R 5 defined by the equations d 1 = const and d 3 = 0. Such a submanifold is the bundle T * Q λ,α cotangent to the quadric Q λ,α := {q ∈ R 5 : αλq 2 −1 + λq 2 0 + q 2 = d 1 = const}. The Hamiltonian δ λ,α after restriction to T * Q λ,α represents kinetic energy (4.14) of the free particle localized on the quadric Q λ,α . In (4.14) we identify 2d 1 with the mass of the particle and express momentum p by velocity dq dt by means of metric tensor p = η λ,α dq dt .
Therefore (4.12) is the geodesic flow on the four-dimensional hypersurface Q λ,α which is for example: iii) anti-de Sitter spaces AdS 4 if α = 1 and λ = −1.