On the Hamiltonian formulation, integrability and algebraic structures of the Rajeev-Ranken model

The integrable 1+1-dimensional SU(2) principal chiral model (PCM) serves as a toy-model for 3+1-dimensional Yang-Mills theory as it is asymptotically free and displays a mass gap. Interestingly, the PCM is 'pseudodual' to a scalar field theory introduced by Zakharov and Mikhailov and Nappi that is strongly coupled in the ultraviolet and could serve as a toy-model for non-perturbative properties of theories with a Landau pole. Unlike the 'Euclidean' current algebra of the PCM, its pseudodual is based on a nilpotent current algebra. Recently, Rajeev and Ranken obtained a mechanical reduction by restricting the nilpotent scalar field theory to a class of constant energy-density classical waves expressible in terms of elliptic functions, whose quantization survives the passage to the strong-coupling limit. We study the Hamiltonian and Lagrangian formulations of this model and its classical integrability from an algebraic perspective, identifying Darboux coordinates, Lax pairs, classical r-matrices and a degenerate Poisson pencil. We identify Casimirs as well as a complete set of conserved quantities in involution and the canonical transformations they generate. They are related to Noether charges of the field theory and are shown to be generically independent, implying Liouville integrability. The singular submanifolds where this independence fails are identified and shown to be related to the static and circular submanifolds of the phase space. We also find an interesting relation between this model and the Neumann model allowing us to discover a new Hamiltonian formulation of the latter.


Introduction
It is well-known that the 1+1-dimensional SU (2) non-linear sigma model (NLSM) and the closely related principal chiral model (PCM) for the SU(2)-valued field g(x, t) are good toymodels for the physics of the strong interactions and 3+1-dimensional Yang-Mills theory. They have been shown to be asymptotically free and to possess a mass-gap [1]. Non-perturbative results concerning the S -matrix and the spectrum of the 1+1-dimensional NLSM and PCM have been obtained using the methods of integrable systems by Zamolodchikov and Zamolodchikov [2] (factorized S -matrices), by Polyakov and Wiegmann [3] (fermionization) and by Faddeev and Reshetikhin [4] (quantum inverse scattering method). Interestingly, a 'pseudo-dual' to the PCM introduced in the work of Zakharov and Mikhailov [5] and Nappi [6] is strongly coupled in the ultraviolet, displays particle production and has been shown by Curtright and Zachos [7] to possess infinitely many non-local conservation laws. Thus, this dual scalar field theory could serve as a toy-model for studying certain non-perturbative aspects of 3+1-dimensional λφ 4 theory which appears in the scalar sector of the standard model. This dual model is also of interest due to its connections to the Wess-Zumino-Witten model, the study of slow light and the theory of hypoelliptic operators as pointed out by Rajeev and Ranken in [8].
While the PCM is based on the semi-direct product of an su(2) current algebra and an abelian algebra [9], its dual is based on a step-3 nilpotent algebra of currents I = g −1 g ′ /λ 2 and J = g −1ġ /λ , where λ is a dimensionless coupling constant [8]. The equation of motion (EOM) of the PCM (J = λI ′ ) can be solved by expressing the currents I =φ/λ and J = φ ′ in terms of an su(2) -valued scalar field φ(x, t) . The zero-curvature consistency condition (İ − J ′ /λ = λ[I, J]) then becomes a non-linear wave equation: Recently, Rajeev and Ranken [8] studied a class of constant energy-density 'continuous wave' solutions to (1) obtained via the ansatz φ(x, t) = e Kx R(t)e −Kx + mKx where K = ikσ 3 2 (2) and R(t) is a traceless 2 × 2 anti-hermitian matrix. The continuous waves depend on two constants, a wavenumber k and a dimensionless parameter m . The reduction of the nilpotent scalar field theory to the manifold of these continuous waves is a mechanical system, the 'Rajeev-Ranken' (RR) model, with three degrees of freedom R a = Tr (Rσ a /2i) where Tr X = −2 tr X . Interestingly, the continuous wave solutions remain non-trivial even in the limit of strong coupling so that their quantization could play a role in understanding the microscopic degrees of freedom of the corresponding quantum theory. In [8], conserved quantities of the RR model were used to reduce the EOM for R(t) to a single non-linear ODE which was solved in terms of the Weierstrass ℘ function.
In this article, we study the classical dynamics of the RR model focussing on its Hamiltonian formulation and aspects of its integrability. We begin by reviewing the passage from the PCM to the nilpotent scalar field theory, followed by its reduction to the RR model in sections 2 and 3. Just as the canonical Poisson brackets (PBs) between I and its conjugate momentum in the Lagrangian of the PCM lead to the semi-direct product Poisson algebra among currents I and J [9], the canonical PBs between φ and its conjugate momentum are shown to imply a step-3 nilpotent Poisson algebra among these currents. In section 4.3, we identify canonical Darboux coordinates (R a , kP a ) on the six-dimensional phase space of the RR model and give it both Hamiltonian and Lagrangian formulations. Interestingly, since the evolution of R 3 decouples from that of the remaining variables, it is possible to give an alternative Hamiltonian formulation in terms of the variables L = [K, R] + mK and S =Ṙ + K/λ introduced by Rajeev and Ranken (see section 4.1). The latter include a non-dynamical constant L 3 = −mk but have the advantage of satisfying a step-3 nilpotent Poisson algebra which may be regarded as a finite dimensional version of the current algebra of the scalar field theory. Remarkably, the EOM in terms of the S and L variables admit another Hamiltonian formulation with the same Hamiltonian but PBs that are a finite dimensional analogue of the semi-direct product algebra of the PCM. Moreover, the nilpotent and semi-direct product Poisson structures are compatible and combine to form a Poisson pencil as shown in section 4.2. However, both Poisson structures are degenerate so that this Poisson pencil does not lead to a bi-Hamiltonian structure. In section 5.1, we find Lax pairs and classical r -matrices with respect to both Poisson structures. Interestingly, we also find a mapping of variables that allows us to relate the EOM and Lax pairs of the RR model to those of the Neumann model [10]. This is used to propose a new Hamiltonian formulation of the Neumann model with a nilpotent Poisson algebra. Despite these similarities there are differences: while P and J in the Neumann model are a projection and an anti-symmetric matrix, the corresponding S and L variables of the RR model are anti-hermitian, so that the Poisson structures as well as r matrices of the two models are distinct (see section 5.2). Returning to the RR model, we use its Lax pair and r -matrix formulation to identify a maximal set of four conserved quantities in involution ( C, m, s 2 and h ). These conserved quantities are quadratic polynomials in S and L . While C and m are Casimirs of the nilpotent S -L Poisson algebra, s 2 and h are Casimirs of the semidirect product Poisson algebra. While hk 2 = Tr SL is loosely like helicity, the Hamiltonian is proportional to s 2 k 2 = Tr S 2 upto the addition of terms involving C and m . In section 5.4, we find the canonical transformations generated by these conserved quantities and the associated symmetries. We also relate the conserved quantities to the reduction of Noether charges of the field theory. In section 5.6, we show that the four conserved quantities are generically functionally independent on the S -L phase space. The functional independence and involutive property of the conserved quantities imply Liouville integrability of the RR model [11]. We conclude with a brief discussion in section 6.
2 From the SU(2) PCM to the nilpotent scalar field theory The 1+1-dimensional principal chiral model for the SU(2)-valued field g(x, t) is defined by the action with primes and dots denoting x and t derivatives. Here, λ ≥ 0 is a dimensionless coupling constant and Tr = −2 tr . The corresponding equations of motion (EOM) are a non-linear wave equation for the components of g and may be written in terms of the su(2) Lie algebra-valued time and space components of the right current, r 0 = g −1ġ and r 1 = g −1 g ′ : An equivalent formulation is possible in terms of the left currents l µ = (∂ µ g)g −1 . By definition, r 0 and r 1 are components of a flat connection, so they satisfy the zero curvature 'consistency' conditionṙ Following Rajeev and Ranken [8], we define right current components rescaled by λ , which are especially useful in discussions of the strong coupling limit: In terms of these currents, the EOM and zero-curvature condition becomė These EOM may be derived from the Hamiltonian following from S PCM (upon dividing by λ > 0 ), and the PBs: Since both I and J are anti-hermitian, their squares are negative operators, but the minus sign in Tr ensures that H PCM ≥ 0 . The Poisson algebra (9) is a central extension of a semi-direct product of the abelian algebra generated by the I a and the su(2) current algebra generated by the J a . These PBs follow from the canonical PBs between I and its conjugate momentum in the action (3) [9]. The multiplicative constant in {J a , J b } is not fixed by the EOM. It has been chosen for convenience in identifying Casimirs of the reduced mechanical model in §4.2.
The EOM (J = λI ′ ) of the PCM is identically satisfied if we express the currents in terms of a Lie algebra-valued potential φ : The zero curvature condition (İ − J ′ /λ = λ [I, J] ) now becomes a 2 nd -order non-linear wave equation for the scalar φ : The field φ is an anti-hermitian traceless 2 × 2 matrix in the su(2) Lie algebra, which may be written as a linear combination of the generators t a = σ a /2i where σ a are the Pauli matrices: φ = φ a t a = 1 2i φ · σ with φ a = i tr (φσ a ) = Tr (φt a ) for a = 1, 2, 3.
The generators are normalized according to Tr (t a t b ) = δ ab and satisfy [t a , t b ] = ǫ abc t c . The wave equation (11) follows from the Lagrangian density The momentum conjugate to φ is π =φ The conserved energy and Hamiltonian coincide with H PCM of (8): If we postulate the canonical PBs then Hamilton's equationsφ = {φ, H} andπ = {π, H} reproduce (14). The canonical PBs between φ and π imply the following PBs among the currents I, J and φ : These PBs define a step-3 nilpotent Lie algebra in the sense that all triple PBs such as vanish. Note however that the currents I and J do not form a closed subalgebra of (17). Interestingly, the EOM (7) also follow from the same Hamiltonian (8) if we postulate the following closed Lie algebra among the currents Crudely, these PBs are related to (17) by 'integration by parts'. As with (17), this Poisson algebra of currents is a nilpotent Lie algebra of step-3. The scalar field model with EOM (11) and Hamiltonian (15) is classically related to the PCM through the change of variables r µ = λǫ µν ∂ ν φ . However, as emphasized in [7], this transformation is not canonical, leading to the moniker 'pseudo-dual'. Moreover, the corresponding quantum theories are quite different. While the PCM is asymptotically free, integrable and serves as a toy-model for 3+1-dimensional Yang-Mills theory, the quantized nilpotent scalar field theory displays particle production (it has a non-zero amplitude for 2 → 3 particle scattering), has a positive β function [6] and could serve as a toy-model for 3+1-dimensional λφ 4 theory [8].

Reduction of the nilpotent field theory and the RR model
Before attempting a non-perturbative study of the nilpotent scalar field theory, it is interesting to consider its reduction to finite dimensional mechanical systems obtained by considering special classes of solutions to the non-linear wave equation (11). The simplest such solutions are traveling waves φ(x, t) = f (x − vt) for constant v . However, for such φ , the commutator term −λ[vf ′ , f ′ ] vanishes so that traveling wave solutions of (11) are the same as those of the linear wave equation. Similarly, the non-linearities play no role in similarity solutions. Indeed, if we consider the scaling ansatzφ (ξ, τ ) = Λ −γ φ(x, t) where ξ = Λ −α x and τ = Λ −β t , then (11) takes the form: This equation is scale invariant when α = β and γ = 0 . Hence similarity solutions must be of the form φ(x, t) = ψ(η) where η = x/t and ψ satisfies the ODE It is evident that similarity solutions of (11) are insensitive to the non-linearities.
Remarkably, Rajeev and Ranken [8] found a mechanical reduction of the nilpotent scalar field theory for which the non-linearities play a crucial role. They considered the wave ansatz: which leads to 'continuous wave' solutions of (11) with constant energy density. These configurations are obtained from a Lie algebra-valued dynamical matrix R(t) by a combination of an x -dependent internal rotation and a translation. For definiteness, the constant traceless anti-hermitian matrix K has been chosen in the third direction. The ansatz (22) depends on two parameters: a dimensionless real constant m and the constant K 3 = −k with dimensions of a wave number which could have either sign. When restricted to the submanifold of such propagating waves, the field equations (11) reduce to those of a mechanical system with three degrees of freedom which we refer to as the Rajeev-Ranken model. The currents (10) can be expressed in terms of R : These currents are periodic in x with period 2π/|k| . We work in units where c = 1 so that I and J have dimensions of a wave number. If we define the traceless anti-hermitian matrices then it is possible to express the EOM and consistency condition (7) as the paiṙ In components (L a = Tr (Lt a ) etc.), the equations becomė Here, L 3 = −mk is a constant but it will be convenient to treat it as a coordinate like the other components. The constancy of L 3 will be encoded in the Poisson structures to be introduced, where it will be seen to be either a conserved quantity or a Casimir. Sometimes it is convenient to express L 1,2 and S 1,2 in terms of polar coordinates: Here, r and ρ are dimensionless and positive. We may also express L and S in terms of coordinates and velocities: where u =Ṙ 3 /k − 1/λ or in components, It is clear from (24) that L and S do not depend on the coordinate R 3 . The EOM (25, 29) may be expressed as a system of three second order ODEs for the components of R(t) : (30) Rajeev and Ranken used conserved quantities to reduce these equations of motion to a single non-linear ODE for u which was solved in terms of the Weierstrass ℘ function. Here, we examine Hamiltonian and Lagrangian formulations of this model, certain aspects of its classical integrability and explore some properties of its conserved quantities. We also relate this model to the well-known Neumann model and thereby find a new Hamiltonian-Poisson bracket formulation for the latter.

Hamiltonian and PBs for the RR model
This mechanical system with 3 degrees of freedom and phase space M 6 S−L ( R 6 with coordinates L a , S a ) can be given a Hamiltonian-Poisson bracket formulation. A Hamiltonian is obtained by a simple reduction of that of the nilpotent field theory, H = (1/2λ) Tr dx φ 2 + φ ′2 . From (22), we have Trφ 2 = TrṘ 2 and Tr φ ′2 = Tr ([K, R] + mK) 2 . Thus the ansatz (22) has a constant energy density and it is reasonable to define the Hamiltonian of the mechanical model as the energy (15) per unit length, which has dimensions of 1/area: We have multiplied by λ for convenience. As in (8) H ≥ 0 since bothṘ and L are antihermitian. Poisson brackets among S and L which lead to the EOM (25) are given by We may view this Poisson algebra as a finite-dimensional version of the nilpotent Lie algebra of currents I and J in (19) Note that the PBs (32) have dimensions of a wave number. They may be expressed as {f, g} ν = r ab ∂ a f ∂ b g where the anti-symmetric Poisson tensor field is This Poisson algebra is degenerate: r ab has rank four and its kernel is spanned by the vector . The corresponding center of the algebra is generated by the Casimirs mk 2 ≡ Tr KL and Ck 2 ≡ Tr (L 2 /2) − (KS/λ) . Semi-direct product PBs: The L -S EOM (25) admit a second Hamiltonian formulation with a non-nilpotent Poisson algebra arising from the reduction of that of the PCM (9). It is straightforward to verify that the non-nilpotent PBs along with the Hamiltonian (31) lead to the EOM (25). This Poisson algebra is a semi-direct product of the simple su(2) Lie algebra generated by the L a and the abelian algebra of the S a . Furthermore, it is easily verified that s 2 k 2 ≡ Tr S 2 and hk 2 ≡ Tr SL are Casimirs of this Poisson algebra. It follows that the EOM (25) obtained from these PBs are unaltered if we remove the Tr S 2 term from the Hamiltonian (31). The factor λ in the {L a , S b } PB is fixed by the EOM while that in the {L a , L b } PB is necessary for h to be a Casimir. Formulation in terms of real antisymmetic matrices: It is sometimes convenient to re-express the 2 × 2 anti-hermitian su(2) Lie algebra elements L, S and K as 3 × 3 real anti-symmetric matrices: and similarly forS andK (more generally we would contract with the structure constants Moreover, the nilpotent (ν) (32) and semi-direct product (σ) (35) PBs become and The Hamiltonian (38) along with either of the PBs (39) or (40) give the EOM (37).

Poisson pencil from nilpotent and semi-direct product PBs
The defines a Poisson bracket for any real α . The linearity, skew-symmetry and derivation properties of the α -bracket follow from those of the individual PBs. Let us outline a proof of the Jacobi identity for the α -bracket. We first prove it for coordinate functions L a and S a . There are only four independent cases to be considered: The Jacobi identity for the α -bracket for linear functions of L and S follows from (42). For more general functions of L and S , it follows by applying the Leibnitz rule: Here, the coordinates ξ i are S 1,2,3 and L 1,2,3 .

Darboux coordinates and Lagrangian from Hamiltonian
Though they are convenient, the S and L variables are non-canonical generators of the nilpotent degenerate Poisson algebra (32). Moreover, they lack information about the coordinate R 3 . It is natural to seek canonical coordinates that contain information on all six generalized coordinates and velocities (R a ,Ṙ a ) (see (23)). Such Darboux coordinates will also facilitate a passage from Hamiltonian to Lagrangian. Unfortunately, as discussed below, the naive reduction of the field theoretic Lagrangian (13) does not result in one for the EOM (30).
It turns out that momenta canonically conjugate to the coordinates R 1 , R 2 and R 3 may be chosen as (see (29)) We obtained them from the nilpotent L -S Poisson algebra (32) by requiring the canonical PB relations These Darboux coordinates are associated to the nilpotent S -L PBs. On the other hand, R a cannot be treated as coordinates for the semi-direct product PBs (35), since {R 1 , R 2 } = (1/k 2 ){L 1 , L 2 } σ = 0 . Darboux coordinates associated to the semi-direct product PBs, may be analogously obtained from the coordinates Q in the wave ansatz for the mechanical reduction of the principal chiral field g = e λsKx Q(t)e −Kx given in Table I of [8].
Since R 3 does not appear in the Hamiltonian (31) (regarded as a function of (S, L) or (R,Ṙ) ), we have taken the momenta in (44) to be independent of R 3 so that it will be cyclic in the Lagrangian as well. However, the above formulae for P a are not uniquely determined. For instance, the PBs (45) are unaffected if we add to P a any function of the Casimirs (C, m) as also certain functions of the coordinates (see below for an example). In fact, we have used this freedom to pick P 3 to be a convenient function of the Casimirs. Moreover, {R 3 , kP 3 } = 1 is a new postulate, it is not a consequence of the S -L Poisson algebra.
The Hamiltonian (31) can be expressed in terms of the R 's and P 's: The EOM (25), (29) follow from this Hamiltonian and the canonical PBs (45). Thus R a and kP b are Darboux coordinates on the six-dimensional phase space of our mechanical system M 6 R−P ∼ = R 6 . Note that the previously introduced six-dimensional degenerate S -L phase space M 6 S−L is different from M 6 R−P though they share a five-dimensional submanifold in common parameterized by (L 1,2 , S 1,2,3 ) or (R 1,2 , P 1,2,3 ) . The former includes the constant parameter L 3 = −mk as its sixth coordinate but lacks information on R 3 which is the 'extra' coordinate in the latter.
A Lagrangian L mech (R,Ṙ) for our system may now be obtained via a Legendre transform by extremizing kP aṘa − H with respect to all the components of kP : R 3 is a cyclic coordinate leading to conservation of its conjugate momentum kP 3 = ∂L mech /∂Ṙ 3 . However L mech does not admit an invariant form as the trace of a polynomial in R andṘ .
Such a form may be obtained by adding a time derivative to L mech : The price to pay for this invariant form is that R 3 is no longer cyclic, so that the conservation of P 3 is not manifest. The Lagrangian L ′ mech may also be obtained directly from the Hamiltonian (46) if we choose as conjugate momenta kΠ a instead of the kP a of (44): Interestingly, while L mech and L ′ mech give the correct EOM (30), unlike with the Hamiltonian, the naive reduction of the field theoretic Lagrangian (13) does not. This discrepancy was unfortunately overlooked in Eq. (3.7) of [8]. Indeed, the naive reduction L naive of the field theoretic Lagrangian differs from L ′ mech by a term which is not a time derivative: To see this, we put the ansatz (22) for φ in the nilpotent field theory Lagrangian (13) and use to get the naively reduced Lagrangain In obtaining L naive we have ignored an x -dependent term as it is a total time derivative, a factor of the length of space and multiplied through by λ . As mentioned earlier, L naive does not give the correct EOM for R 1 and R 2 nor does it lead to the PBs among L and S (32) if we postulate canonical PBs among R a and their conjugate momenta. However the Legendre transforms of L mech , L ′ mech and L naive all give the same Hamiltonian (31). It would be interesting to understand why the naive reduction of the scalar field gives a Hamiltonian but not a suitable Lagrangian for the mechanical system.
Here, σ ± = 1 2 (σ 1 ± iσ 2 ) . These FPBs can be expressed as a commutator where the classical r -matrix is proportional to the permutation matrix P : To obtain this r -matrix we used the following identities among Pauli matrices: We may now motivate the particular choice of Lax matrix A (53). The nilpotent S -L PBs (32) do not involve S , so the PBs between matrix elements of A are also independent of S .
provided we define a new Lax matrix A σ = A/ζ 2 . The EOM for S and L are then equivalent to the Lax equationȦ σ = [B, A σ ] at order ζ −2 and ζ −1 . In this case, the analogue of the FPBs (54) is

Similarities and differences with the Neumann model
The EOM (25) and Lax pair (53) of the RR model have a formal structural similarity with those of the Neumann model. The latter describes the motion of a particle on S N −1 subject to harmonic forces with frequencies a 1 , · · · , a N [10]. In other words, a particle moves on S N −1 ⊂ R N and is connected by N springs, the other ends of which are free to move on the N coordinate hyperplanes. The EOM of the Neumann model follow from a symplectic reduction of dynamics on a 2N dimensional phase space with coordinates x 1 , · · · , x N and y 1 , · · · , y N . The canonical PBs {x k , y l } = δ kl and Hamiltonian lead to Hamilton's equationṡ Here, J kl = x k y l − x l y k is the antisymmetric angular momentum tensor. Introducing the column vectors X k = x k and Y k = y k and the diagonal frequency matrix Ω with diagonal entries a 1 , · · · , a N , Hamilton's equations take the forṁ It is easily seen that X t X is a constant of motion. Moreover, the Hamiltonian and PBs are invariant under the 'gauge' transformation (X, Y ) → (X, Y + ǫX) for ǫ ∈ R . Imposing the gauge condition X t (Y + ǫ(t)X) = 0 along with X t X = 1 allows us to reduce the dynamics to a phase space of dimension 2(N − 1) . If we define the rank 1 projection P = XX t then J = XY t − Y X t and P are seen to be gauge-invariant and satisfy the evolution equationṡ The Hamiltonian (60) in terms of J, P and Ω becomes The PBs following from the canonical x -y PBs is related to our Lax pair A σ (ζ) = −K + L/ζ + S/(λζ 2 ) and B(ζ) = S/ζ (53) via Despite these similarities, there are significance differences. (a) While L and S are Lie algebra-valued traceless anti-hermitian matrices, J and P are a real anti-symmetric and a real symmetric rank-one projection matrix. Furthermore, while K is a constant traceless anti-hermitian matrix ( (ik/2)σ 3 for the su(2) case), the frequency matrix Ω is diagonal with positive entries.
(b) The Hamiltonian (64) of the Neumann model also differs from that of our model (31) as it does not contain a quadratic term in P . However, the addition of (1/4) tr P 2 to the Hamiltonian (64) would not alter the EOM (63) as tr P 2 is a Casimir of the algebra (65).
(c) The PBs (65) of the Neumann model bear a close resemblance to those of the RR model (40) expressed in terms of the real anti-symmetric matricesS andL of §4.1. Under the map L → J ,S → −P and λ → 1 , the semi-direct product PBs (40) go over to the PBs (65) up to an overall factor of −1/2 and a couple of signs in the {P, J} PB. These sign changes are necessary to ensure that the latter PBs respect the symmetry of P kl as opposed to the anti-symmetry ofS kl .
(d) Though both models possess non-dynamical r -matrices, they are somewhat different as are the forms of the fundamental PBs among Lax matrices. Recall that the fundamental Poisson brackets (FPBs) and r -matrix (58) of the RR model say, for the semi-direct product PBs are: . (68) Here, α, β, γ and δ take the values 1, 2 . This r -matrix has a single simple pole at ζ = ζ ′ . On the other hand, the FPBs of the Neumann model may be expressed as a sum of two commutators The corresponding r -matrices have a pair of simple poles at ζ = ±ζ ′ : Note that the anti-symmetry of (69) is guaranteed by the relation r 12 (ζ, ζ ′ ) klpq = r 21 (ζ, ζ ′ ) lkqp .
New Hamiltonian formulation for the Neumann model: An interesting consequence of our analogy between these two models is a new Hamiltonian formulation for the Neumann model inspired by the nilpotent PBs (39) of the RR model. Indeed, suppose we take the Hamiltonian for the Neumann model as and postulate the step-3 nilpotent PBs, then Hamilton's equations reduce to the EOM (63). These PBs differ from those obtained from (39) via the mapS → −P,L → J andK → Ω by a factor of 1/2 and a couple of signs in the {P, P } ν PB. As before, these sign changes are necessary since P is symmetric whileS is anti-symmetric. It is straightforward to verify that the Jacobi identity is satisfied: the only non-trivial case being {{P, P }, P } + cyclic = 0 where cancellations occur among the cyclically permuted terms. In all other cases the individual PBs such as {{P, J}, J} are identically zero. Though inspired by the su(2) case of the RR model, the PBs (72) are applicable to the Neumann model for all values of N .

Conserved quantities in involution for the RR model
The expression (55) for the fundamental PBs implies that the conserved quantities Tr A n (ζ) are in involution. This follows from the identity for m, n = 1, 2, 3 . . . . Now Tr A n (ζ) is a polynomial in ζ of degree 2n . Each of its coefficients furnishes a conserved quantity in involution with the others. However, they cannot all be independent as the model has only 3 degrees of freedom. For instance Tr A(ζ) ≡ 0 but In this case, the coefficients of various powers of ζ give four functionally independent conserved quantities in involution. Let us denote them by s 2 k 2 = Tr S 2 , hk 2 = Tr SL, mk 2 = Tr KL = −kL 3 and Ck 2 = Tr Factors of k 2 have been introduced so that C , m , h and s 2 (whose positive square-root we denote by s ) are dimensionless. In [8], h and C were named C 1 and C 2 . Of the four conserved quantities in ( Higher powers of A do not lead to new conserved quantities: Tr A 3 is identically zero by the identity Tr (t a t b t c ) = 1 2 ǫ abc for t a = σ a /2i . The same applies to other odd powers of A . On the other hand, the expression for A 4 (ζ) given in Appendix A, along with the identity Evidently, the coefficients of various powers of ζ are functions of the four known conserved quantities (75). It is possible to show that the higher powers Tr A 6 , Tr A 8 , . . . also cannot yield additional conserved quantities by examining the dynamics on the common level sets of the known conserved quantities.
Canonical vector fields on M 6 : On the phase space M 6 S−L , the canonical vector fields ( V a f = r ab ∂ b f ) associated to conserved quantities, follow from the Poisson tensor (34). They vanish for the Casimirs ( V C = V m = 0 ) while for helicity and the Hamiltonian (H = Ek 2 ) , Note that the coefficient of each of the coordinate vector fields in V E gives the time derivative of the corresponding coordinate (upto a factor of k 2 ) and leads to the EOM (26). These vector fields commute, since Conserved quantities for semi-direct product Poisson algebra: Let us briefly consider the dual semi-direct product Poisson brackets {·, ·} σ (35). As noted, the same Hamiltonian (31) leads to the S -L EOM (25) with these PBs. Moreover, it can be shown that C, m, s and h (75) continue to be in involution with respect to {·, ·} σ and to commute with H . Interestingly, the Casimirs ( C, m ) and non-Casimir conserved quantities (s 2 , h) exchange roles in going from the nilpotent to the semi-direct product Poisson algebras.

Symmetries and associated canonical transformations
Here, we identify the Noether symmetries and canonical transformations (CT) generated by the conserved quantities. The constant m = −L 3 /k commutes (relative to {·, ·} ν ) with all observables and acts trivially on the coordinates R a and momenta P b of the mechanical system. As noted in §4.3, the Lagrangian L mech (47) admits R 3 as a cyclic coordinate with conserved momentum kP 3 = (λk/2)(2C − m 2 ) + k/λ (44). In other words, (ελk/2)(2C − m 2 ) generates the infinitesimal CT R 3 → R 3 + ε . L mech is also invariant under infinitesimal rotations in the R 1 -R 2 plane. This corresponds to the infinitesimal CT δR a = εǫ ab R b , δP a = εǫ ab P b for a, b = 1, 2 and δR 3 = δP 3 = 0, with generator (Noether charge) εk h + (λm/2)(2C − m 2 ) . Of course, the additive constants involving m may be dropped from these generators. Thus, while P 3 (or equivalently C ) generates translations in R 3 , h (up to addition of a multiple of P 3 ) generates rotations in the R 1 -R 2 plane. In addition to these two point-symmetries, the Hamiltonian (46) is also invariant under an infinitesimal CT that mixes coordinates and momenta: This CT is generated by the conserved quantity which differs from s 2 by terms involving h and C which serve to simplify the CT by removing an infinitesimal rotation in the R 1 -R 2 plane as well as a constant shift in R 3 . Here, upto Casimirs, (81) is related to the Hamiltonian (76) of the system, indeed s 2 + 2C = (1/k 2 )(2H − k 2 /λ 2 ) . The above assertions follow from using the canonical PBs, {R a , kP b } = δ ab to compute the changes δR a = {R a , Q} etc., generated by the three conserved quantities Q expressed as:

Relation of conserved quantities to Noether charges of the field theory
We show below that three of the Noether charges of the field theory corresponding to symmetries under translations of φ , x and t reduce to the conserved quantities P 3 , (k 2 /λ)(h − m/λ) and H in the mechanical model. The charge corresponding to rotations of φ in the internal space however does not admit a consistent reduction to a conserved quantity of the mechanical model.
The corresponding Noether conserved density and current are (84) The local conservation law ∂ t j t +∂ x j x = 0 is equivalent to the non linear wave equation (11) [7]. Taking the shift η ∝ λ , all elements of the matrix Q s = φ − (λ/2)[φ, φ ′ ] dx are conserved.
Interestingly, the conserved quantity P 3 (44) may be viewed as the reduction of Q s . In fact, inserting the ansatz (22)  The first two integrals vanish while we find thatQ s 3 = TrQ s t 3 = P 3 so that Q s = lP 3 t 3 , where l is the length of the spatial domain.
Noether charge for rotations in internal space: The adjoint action φ → U † φU by a unitary matrix U and its infinitesimal version φ → φ + θ[n, φ] (where n ∈ su(2) and θ is a small angle) are symmetries of the field theory (13) leading to the Noether density and current: Note that the conservation law ∂ t j t + ∂ x j x = 0 or Tr n φ,φ −φ ′′ λ − [φ, φ ′ ] = 0 may also be obtained from the EOM (11) by taking a commutator with φ . However, the conserved charge does not reduce to a conserved quantity in the mechanical model. This is because the space of mechanical states is not invariant under the above adjoint action. In fact, inserting the ansatz involution. Though the conserved quantities are generically independent, there are singular submanifolds of the phase space where they fail to be independent. It would be interesting to find these singular submanifolds and the corresponding relations among conserved quantities on them. Though we have argued that this model is Liouville integrable, it remains to find a canonical transformation to action-angle variables. It is also of interest to identify common level sets of conserved quantities and describe the foliation of the phase space by invariant tori of various dimensions. The possible extension of the integrability of this mechanical reduction to its parent nilpotent scalar field theory is of course of much interest. We hope to address these issues in future work.