Classical multiseparable Hamiltonian systems, superintegrability and Haantjes geometry

Highlights

• A new geometric approach to the classical theory of separation of variables based on the theory of $\omega \mathcal{H}$ manifolds is proposed.

• A class of deformed Drach-Holt-type systems is analyzed and their separability properties established.

• A detailed study of Hamiltonian multiseparable systems in terms of their semisimple abelian $\omega \mathcal{H}$ structures is presented.

• Several examples of superintegrable models in two and three space dimensions are studied; their multiseparability properties are formulated in terms of the existence of Darboux-Haantjes coordinates.

Abstract

We show that the theory of classical Hamiltonian systems admitting separating variables can be formulated in the context of ($\omega ,\mathcal{H}$) structures. They are symplectic manifolds endowed with a compatible Haantjes algebra $\mathcal{H}$, namely an algebra of (1,1)tensor fields with vanishing Haantjes torsion. A special class of coordinates, called Darboux-Haantjes coordinates, will be constructed from the Haantjes algebras associated with a separable system. These coordinates enable the additive separation of variables of the corresponding Hamilton-Jacobi equation.

We shall prove that a multiseparable system admits as many $\omega \mathcal{H}$ structures as separation coordinate systems. In particular, we will show that a large class of multiseparable, superintegrable systems, including the Smorodinsky-Winternitz systems and some physically relevant systems with three degrees of freedom, possesses multiple Haantjes structures.

Introduction

The prominence of integrable models in many areas of pure and applied mathematics and theoretical physics has motivated, in the last decades, a resurgence of interest in the algebraic and geometric structures underlying the notion of integrability. The study of the geometry of Hamiltonian integrable systems has a long history, dating back to the classical works by Liouville, Jacobi, Stäckel, Eisenhart, Arnold, etc. The approaches proposed in the literature are intimately related with the problem of the determination of suitable coordinate systems guaranteeing the additive separation of the Hamilton-Jacobi (HJ) equation.

Recently, many new ideas coming from differential and algebraic geometry, topology and tensor analysis, have contributed to the formulation of important approaches such as the theory of bi-Hamiltonian systems [23], the Lenard-Nijenhuis geometry [30] and the theory of Dubrovin-Frobenius manifolds [10]. These theoretical developments shed new light on the multiple connections among integrability, topological field theories, singularity theory, co-isotropic deformations of associative algebras, etc. Besides, several integrable models, both classical and quantum ones, have recently been discovered, in particular in the domain of superintegrability.

Superintegrable systems are a special class of integrable systems which possess surprisingly rich algebraic and geometric properties [32], [33], [38], [42], [52], [53]. Essentially, they possess more independent integrals of motion than degrees of freedom. If a system with $n$ degrees of freedom admits $2n-1$ functionally independent integrals of motion (the maximal number allowed), it is said to be maximally superintegrable; if it admits $n+1$ integrals, then is minimally superintegrable.

Among the most famous examples of superintegrable models we mention the classical harmonic oscillator, the Kepler potential, the Calogero-Moser potential, the Smorodinsky-Winternitz systems and the Euler top. The presence of “hidden” symmetries, expressed by integrals which are second (or higher) degree polynomials in the momenta, usually allows us to determine the dynamical behaviour of superintegrable models. In the maximal case, the bounded orbits are closed and periodic [38]. As is well known, the phase space topology is also very rich: it can be described in terms of a symplectic bifoliation, determined by the standard Liouville-Arnold invariant fibration [1] of Lagrangian tori, and complemented by a co-isotropic polar foliation [14], [38].

Although the present work focuses on classical systems, we point out that quantum superintegrable systems also possess many relevant properties. Exact quantum solvability of a Hamiltonian system, related to the existence of suitable Lie algebras of raising and lowering operators, could be regarded as the quantum analogue of the classical notion of maximal superintegrability [52].

A fundamental class of integrable models is the separable one: these models are characterized by the fact that one can find at least a system of canonical coordinates in which the corresponding Hamilton-Jacobi equation takes the additively separated form

$$W=\sum _{k=1}^n{W}_k=\sum _{k=1}^n\int p_k{(q_k^{\prime };H_1,\dots ,H_n)}_{{\mid }_{H_i=a_i}}d{q}_k^{\prime }.$$

The problem of finding separating variables for integrable Hamiltonian systems has been extensively investigated. In 1904, Levi-Civita proposed a test which permits to establish whether a given Hamiltonian separates in an assigned coordinate system [22]. Another important result, due to Benenti [3], states that a family of Hamiltonian functions ${\left\{H_i\right\}}_{1\le i\le n}$ are separable in a set of canonical coordinates $(\mathit{q},\mathit{p})$ if and only if they are in separable involution, i.e. they satisfy the relations

$${\{H_i,H_j\}}_{|k}=\frac{\partial H_i}{\partial q^k}\frac{\partial H_j}{\partial p_k}-\frac{\partial H_i}{\partial p_k}\frac{\partial H_j}{\partial q^k}=0,1\le k\le n$$

where no summation over $k$ is understood. However, such theorem as well as the Levi–Civita test are not constructive, and do not allow us to determine a complete integral of the Hamilton–Jacobi equation.

A constructive approach for the determination of separating variables was given by Sklyanin [47] within the framework of Lax systems. The Hamiltonian functions ${\left\{H_i\right\}}_{1\le i\le n}$ are separable in a set of canonical coordinates $(\mathit{q},\mathit{p})$ if there exist $n$ suitable equations, called the Jacobi-Sklyanin separation equations for ${\left\{H_i\right\}}_{1\le i\le n}$, having the form

$${\Phi }_i(q^i,p_i;H_1,\dots ,H_n)=0\text{det}\left[\frac{\partial {\Phi }_i}{\partial H_j}\right]\ne 0,i=1,\dots ,n.$$

These equations allow us to construct a solution of the HJ equation. In fact, by solving Eq. (3)with respect to $p_k=\frac{\partial W_k}{\partial q_k}$, we get the additively separated form (1).

Nevertheless, the three above-mentioned criteria of separability are not intrinsic: in order to be applied, they require the explicit knowledge of the local chart $(\mathit{q},\mathit{p})$. To overcome such a drawback, in the last decades the modern theory of separation of variables (SoV) has been conceived in the context of symplectic and Poisson geometry; in particular, the bi-Hamiltonian theory has offered a fundamental geometric insight into the theory of integrable systems [13], [30].

The main purpose of the present work is to establish a novel relationship between the theory of separable Hamiltonian systems and the geometry of an important class of tensor fields, the Haantjes tensors, introduced in [19] as a relevant, natural generalization of the notion of Nijenhuis tensors [40], [41]. The class of Nijenhuis tensors plays a significant role in differential geometry and the theory of almost-complex structures, due to the celebrated Newlander-Nirenberg theorem [39].

Our approach is based on the notion of $\omega \mathcal{H}$ manifolds, introduced in [51] by analogy with the theory of $\omega N$ manifolds [13], [30] for finite-dimensional Hamiltonian systems (see also [15], [27] and [28] for a treatment of integrable hierarchies of PDEs). Essentially, an $\omega \mathcal{H}$ manifold is a symplectic manifold endowed with an algebra $\mathcal{H}$ of (1,1) tensor fields with vanishing Haantjes torsion, which are compatible with the symplectic structure. Under the hypotheses of the Liouville-Haantjes (LH) theorem proved in [51], a non-degenerate Hamiltonian system is completely integrable in the Liouville-Arnold sense if and only if it admits a $\omega \mathcal{H}$ structure.

In our context, Haantjes chains represent in the Haantjes framework the generalization of the notion of Lenard-Magri chain [24] and of generalized Lenard chain [12], [26] defined previously for quasi-bi-Hamiltonian systems [36], [37]. By means of these structures, one obtains a complete description of the integrals of the motion of a system in terms of the associated Haantjes operators.

The problem of SoV can also be recast and studied in our approach. Precisely, as stated in Theorem 2 below, if an integrable system admits a semisimple $\omega \mathcal{H}$ structure, one can derive a set of coordinates, that we shall call the Darboux-Haantjes (DH) coordinates, representing separation coordinates for the Hamilton-Jacobi equation associated with the system. In these coordinates, the symplectic form takes a Darboux form, and the operators of the Haantjes algebra take all simultaneously a diagonal form. As we will show in Theorem 3, and in the examples of Sections 8 and 10, multiseparable systems possess different Haantjes structures associated in a nontrivial way with their separation coordinates.

In the study of separable Hamiltonian systems, the theory of $\omega N$ manifolds has proved to be a powerful tool. Since any semisimple Haantjes algebra admits a Nijenhuis generator, we deduce that the $DH$ coordinates in the Haantjes scenario are also Darboux-Nijenhuis (DN) coordinates of the $\omega N$ theory.

In the semisimple case, we shall prove that there is a one-to-one correspondence between a given $\omega \mathcal{H}$ manifold and an equivalence class of $\omega N$ structures (see Section 3.7). Interestingly enough, given a semisimple $\omega \mathcal{H}$ manifold, one can find Nijenhuis generators which fulfill both the algebraic and the differential compatibility conditions, required by the $\omega N$ theory.

However, from a general, theoretical point of view, the two theories are not equivalent. Indeed, in the non-semisimple case, there are integrable systems, as the Post-Winternitz system (discussed in [51]), which admit a non-Abelian $\omega \mathcal{H}$ structure endowed with three Haantjes generators. In other words, there exist $\omega \mathcal{H}$ manifolds with several generators. In these cases, an alternative description in terms of a standard $\omega N$ structure is not available, since in the $\omega N$ approach the existence of a unique Nijenhuis tensor $N$ is assumed.

This article is organized into two parts. In the first one, including Sections 2 and 3, for the sake of self-consistency we briefly summarize the notions necessary for the study of multiseparable systems. Precisely, in Section 2, the basic definitions concerning Nijenhuis and Haantjes tensors are proposed; in Section 3, the Haantjes geometry is reviewed. In particular, the notions of Haantjes algebras, $\omega \mathcal{H}$ manifolds and Darboux-Haantjes coordinates are revised. In the second part, starting from Section 4, we shall propose the original results of our work. In Section 4, we propose the main theorem concerning the existence of Haantjes structures for separable systems. In Section 5, as a direct application of the theory previously developed, we solve the problem of SoV for a family of Drach-Holt type systems, that were previously considered to be non-separable. Interestingly enough, the new separating variables we found are defined in the full phase space. In Section 6, this theorem is extended to the case of multiseparable (and superintegrable) models. We propose in Section 7 a novel geometric construction: a lift of operators from the configuration space $Q$ of dimension two to $T^{*}Q$, which generalizes the standard Yano lift [58]. By means of our procedure, a Haantjes operator can be lifted into another Haantjes operator (unlike the Yano lift, which only preserves Nijenhuis operators, but not the Haantjes ones). Section 8 is devoted to the study of the Haantjes structures for the Smorodinsky-Winternitz systems in the plane, whereas Section 9 deals with the study of the anisotropic oscillator. In Section 10, the $\omega \mathcal{H}$ manifolds associated with certain important multiseparable systems in three dimensions are determined. Future research perspectives are discussed in the final Section 11.