Additional Constants of Motion for a Discretization of the Calogero--Moser Model

When one discretizes dynamical systems, it is hardly possible to avoid modifying the original systems. Controlling such modifications is thus a central problem in numerical analysis. 1 It would be ideal if a discretization conserves whole the structure of the original dynamical system such as orbits in the phase space, constants of motion, integrability and so on. As an example of such ideal discretizations, a discretization of the Kepler problem, which keeps all the constants of motion and the orbits in the phase space, was discovered. 2,3 The Kepler problem is an integrable system that has a set of mutually independent and Poisson commutative constants of motion, whose number is the same as the degrees of freedom of the system. A dynamical system of N -degrees of freedom which has mutually independent 2N -1 constants of motion in the form of single-valued functions is called maximally super-integrable and so is the Kepler problem. The above discretization conserves super-integrability of the Kepler problem.

Among the family of one-dimensional integrable systems with inverse-square interactions called the Calogero-Moser-Sutherland models, 4 the Calogero model, 5 which is the root of the family, the Calogero-Moser model 6 of the rational and hyperbolic types are known to be maximally super-integrable. [7][8][9] What we discuss here is the super-integrability of a discretization of the rational Calogero-Moser model, which is a classical dynamical system whose Hamiltonian is given by

where γ, N , p i := p i (t) and x i := x i (t) are the coupling parameter, the number of particles, the momentum and the coordinate of the i-th particle at the time t, respectively.

It will be no exaggeration to say that the Calogero-Moser model represents the models of Calogero-Moser-Sutherland type since the Lax formulation and a systematic construction of the constants of motion for the model was discovered earlier than those for any other models of the family. 6 Moser constructed a set of N constants of motion that are independent of each other. Later, mutual Poisson commutativity of the constants of motion of Moser-type was proved and the integrability of the model in Liouville’s sense was thus established. 10,11 Furthermore, it turned out that the model had N -1 additional constants of motion which are independent of the Moser-type ones and independent of each other as well. 8 This concludes the maximal super-integrability of the Calogero-Moser model.

A time-discretization of the Calogero-Moser model that conserves the Moser-type constants of motion was presented by Nijhoff and Pang, 12 which was reformulated into a more convenient form by Suris. 13 The aim of the paper is to show that the maximal super-integrability

Throughout the paper, we employ Suris’ formulation of the discrete Calogero-Moser model, which is given by the following discrete symplectic map

where ∆t, x i,n := x i (n∆t) and p i,n := p i (n∆t) denote the discrete time-step, the coordinate and the momentum of the i-th particle at the n-th discrete time n∆t. 13 The constant c 0 is defined by c 2 0 := -γ∆t. In terms of the Lax pair, which consists of two N × N matrices below,

the discrete symplectic map (2.1) is expressed by the discrete Lax equation,

which is equivalent to

(2.4)

The companion matrix M n thus plays a role of the time-evolution operator of the Lax matrix

The Moser-type quantities (2.5) are single-valued for they are rational functions of p i,n ’s and

x i,n ’s. In order to confirm the mutual independence of the Moser-type quantities, all one has to do is to check their explicit forms when γ = 0,

which is nothing but the power sums of p i,n ’s that are indeed independent of each other.

Note that the Hamiltonian (1.1) corresponds to the second constant of motion of Moser-type,

n /2. The companion matrix M n of the Lax pair (2.2) satisfies another Lax equation,

where I is the identity matrix and D n := diag(x 1,n , x 2,n , . . . , x N,n ). The above relation (2.7)

was the crucial key to the solution of the initial value problem of the discrete symplectic map (2.1). In the next section, we shall show how the relation (2.7) works in a systematic construction of N -1 additional constants of motion of the discrete Calogero-Moser model (2.1).

Our main purpose is to confirm that the N -1 quantities below

are conserved by the discrete time evolution of the discrete Calogero-Moser model (2.1) and that they are independent not only of the Moser-type quantities (2.5) but also of each other.

Note that the case m = 1 is omitted in eq. (3.1) because K

n = 0. The discrete symplectic map (2.1) is equivalent to the discrete Lax equations (2.3) and (2.7). From the discrete Lax equations (2.3) and (2.7), one obtains

which is rewritten as

The relation (3.3) gives the time-evolution of the matrix D n I -∆tc -1 0 L n . Using eqs. (2.4) and (3.3) as well as the trace identity, one can perform the calculation below,

which proves the conservati

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