Recently Hamiltonian systems on the plane defined in polar-like coordinates have been studied.^[1–4] A prototypical example of such a system is given in the polar-like coordinates by

$$ \begin{eqnarray}H=\displaystyle \frac{1}{2}{r}^{m-k}\left({p}_{r}^{2}+\displaystyle \frac{{p}_{\phi }^{2}}{{r}^{2}}\right)+{r}^{m}U(\phi ),\end{eqnarray}$$ (1)

$$ \begin{eqnarray}H=\displaystyle \frac{1}{2}{r}^{n}\left({p}_{r}^{2}+\displaystyle \frac{{p}_{\phi }^{2}}{{r}^{2}}\right)+{c}_{1}{r}^{1-n/2}\cos \displaystyle \frac{(n-2)\phi }{2}\end{eqnarray}$$ (2)

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The transformation between the original variables (r,ϕ) and flat Cartesian coordinates (Q₁,Q₂) is given by^[4]

$$ \begin{eqnarray}\begin{array}{lll} & & {Q}_{1}=b{r}^{1-\displaystyle \frac{n}{2}}\cos \displaystyle \frac{(n-2)\phi }{2},\\ & & {Q}_{2}=b{r}^{1-\displaystyle \frac{n}{2}}\sin \displaystyle \frac{(n-2)\phi }{2},\end{array}\end{eqnarray}$$ (3)

$$ \begin{eqnarray}\begin{array}{lll} & & u=\displaystyle \frac{1}{2}\left({Q}_{1}+\sqrt{{Q}_{1}^{2}+{Q}_{2}^{2}}\right),\\ & & v=\displaystyle \frac{1}{2}\left(-{Q}_{1}+\sqrt{{Q}_{1}^{2}+{Q}_{2}^{2}}\right),\end{array}\end{eqnarray}$$ (4)

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$$ \begin{eqnarray}H=\displaystyle \frac{1}{2}\displaystyle \frac{u{p}_{u}^{2}+v{p}_{v}^{2}}{u+v}+\displaystyle \frac{1}{2}k(u-v),\end{eqnarray}$$ (5)

In the theory of integrable systems there are various approaches to generalize a given integrable system. One of them is to consider the original system with a position-dependent mass (PDM). Hamiltonian systems with a PDM have been proposed and studied extensively,^[6–11] including classical integrable systems and quantum integrable ones. Many common models, e.g., harmonic oscillator and Kepler system, together with a PDM have been intensively studied, see e.g. Refs. [6,11].

In this paper we consider the Fordy system with a PDM. We show that it is separable in three coordinate systems and we deduce its integrals of motion in explicit forms. The nontrivial polynomial relation among integrals is worked out. We also propose a generalization of the Fordy system with a PDM by the Jacobi method.

This paper is organized as follows. In Section 2 we review first integrals and separability of the original Fordy system. In Section 3 the Fordy system with a PDM is proposed. We show the multi-separability of the system by explicitly writing its separation equations. The integrals of motion are computed and their algebraic relation established. In Section 4 we apply the Jacobi method and make a generalization of the Fordy system with a PDM. A condition for the flatness of underlying Riemannian space is also deduced. Finally, Section 5 is devoted to the conclusion and some discussions.

According to Ref. [4] the Fordy system (5) is superintegrable, endowed with two quadratic integrals of motion,

$$ \begin{eqnarray}\begin{array}{lll} & & {F}_{1}=\displaystyle \frac{uv({p}_{u}^{2}-{p}_{v}^{2})}{u+v}+kuv,\end{array}\end{eqnarray}$$ (6)

$$ \begin{eqnarray}\begin{array}{lll} & & {F}_{2}=\displaystyle \frac{\sqrt{uv}({p}_{u}+{p}_{v})(u{p}_{u}-v{p}_{v})}{{(u+v)}^{2}}+k\sqrt{uv},\end{array}\end{eqnarray}$$ (7)

$$ \begin{eqnarray}{K}_{2}=\displaystyle \frac{\sqrt{uv}({p}_{u}+{p}_{v})}{u+v}.\end{eqnarray}$$ (8)

$$ \begin{eqnarray}{F}_{2}^{2}=2H{K}_{2}^{2}-{K}_{2}^{4}+k{F}_{1}.\end{eqnarray}$$ (9)

$$ \begin{eqnarray}2H-{K}_{2}^{2}={P}_{1}^{2}+k{Q}_{1},\,{K}_{2}={P}_{2}.\end{eqnarray}$$ (10)

We can define other coordinates (Q^1,Q^2), which is a rotation by an angle of π/4 of flat coordinates (Q₁, Q₂),

$$ \begin{eqnarray}{Q}_{1}=\displaystyle \frac{1}{\sqrt{2}}({\hat{Q}}_{1}-{\hat{Q}}_{2}),\,{Q}_{2}=\displaystyle \frac{1}{\sqrt{2}}({\hat{Q}}_{1}+{\hat{Q}}_{2}).\end{eqnarray}$$ (11)

$$ \begin{eqnarray}H+{F}_{2}={\hat{P}}_{1}^{2}+\displaystyle \frac{k}{\sqrt{2}}{\hat{Q}}_{1},\,H-{F}_{2}={\hat{P}}_{2}^{2}-\displaystyle \frac{k}{\sqrt{2}}{\hat{Q}}_{2}.\end{eqnarray}$$ (12)

In the parabolic coordinates (u,v) defined by Eq. (4) the pair of quadratic first integrals H and F₁ satisfy the equalities

$$ \begin{eqnarray}2uH+{F}_{1}=u{p}_{u}^{2}+k{u}^{2},\,2vH-{F}_{1}=v{p}_{v}^{2}-k{v}^{2},\end{eqnarray}$$ (13)

The following proposition summarizes the above analysis.

For the Fordy system (2) the pairs of quadratic (or linear) integrals (H,K₂), (H,F₂), (H,F₁) are separable in flat coordinates (Q₁,Q₂), rotated coordinates (Q^1,Q^2) and parabolic coordinates (u,v), respectively. The corresponding separation equations are given by Eqs. (10), (12) and (13), respectively.

The Hamiltonian system with a position-dependent mass has been widely studied, which serves as a vital technique to generalize a given system. In this section we consider the Fordy system with a particular PDM as μ = εu – εv + 1, where ε is a suitable real constant.

The Hamiltonian of the new system is a scalar multiple of the original one (5) with the scalar factor λ = 1/μ,

$$ \begin{eqnarray}\begin{array}{ll} & \mathop{H}\limits^{\sim }=\lambda H=\displaystyle \frac{u{p}_{u}^{2}+v{p}_{v}^{2}}{2(1+\varepsilon u-\varepsilon v)(u+v)}+\displaystyle \frac{k(u-v)}{2(1+\varepsilon u-\varepsilon v)}.\end{array}\end{eqnarray}$$ (14)

$$ \begin{eqnarray}\begin{array}{lll}\mathop{H}\limits^{\sim } & = & \displaystyle \frac{(2-n){r}^{n}\left({p}_{r}^{2}+\displaystyle \frac{{p}_{\phi }^{2}}{{r}^{2}}\right)}{(4-2n)+4\varepsilon {r}^{1-\displaystyle \frac{n}{2}}\cos \displaystyle \frac{(n-2)\phi }{2}}\\ & & +\displaystyle \frac{k{r}^{1-\displaystyle \frac{n}{2}}\cos \displaystyle \frac{(n-2)\phi }{2}}{(2-n)+2\varepsilon {r}^{1-\displaystyle \frac{n}{2}}\cos \displaystyle \frac{(n-2)\phi }{2}}.\end{array}\end{eqnarray}$$ (15)

As the parameter ε approaches zero the mass tends to μ = 1 and the original system (2) is recovered. Therefore the system with a PDM, i.e., Eq. (15), can be regarded as a generalization or deformation of the Fordy system (2) with deformation parameter ε.

When written in the flat coordinates (Q₁,Q₂) defined by Eq. (3), the Hamiltonian has the form of

$$ \begin{eqnarray}\mathop{H}\limits^{\sim }=\displaystyle \frac{{P}_{1}^{2}+{P}_{2}^{2}}{2(1+\varepsilon {Q}_{1})}+\displaystyle \frac{k{Q}_{1}}{2(1+\varepsilon {Q}_{1})}.\end{eqnarray}$$ (16)

The Fordy system with the PDM, Eq. (15), is separable in flat coordinates ( Q₁, Q₂) defined by Eq. (3). The separation equations are given by Eq. (17), and the additional integral of motion is also K₂.

In terms of rotated flat coordinates (Q^1,Q^2), the PDM Hamiltonian H∼ has the form as

$$ \begin{eqnarray}\mathop{H}\limits^{\sim }=\displaystyle \frac{{\hat{P}}_{1}^{2}+{\hat{P}}_{2}^{2}}{2+\sqrt{2}\varepsilon ({\hat{Q}}_{1}-{\hat{Q}}_{2})}+\displaystyle \frac{k}{2}\displaystyle \frac{{\hat{Q}}_{1}-{\hat{Q}}_{2}}{\sqrt{2}+\varepsilon ({\hat{Q}}_{1}-{\hat{Q}}_{2})}.\end{eqnarray}$$ (18)

turns out to be

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which is equivalent to

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both sides of which depend upon a canonical pair (Q^i,P^i) only. It follows that

$$ \begin{eqnarray}{\mathop{F}\limits^{\sim }}_{2}={\hat{P}}_{1}^{2}+\displaystyle \frac{k}{\sqrt{2}}{\hat{Q}}_{1}-\sqrt{2}\alpha \varepsilon {\hat{Q}}_{1}-\alpha \end{eqnarray}$$ (19)

is an additional first integral of the system with a PDM, i.e., Eq. (15).

The pair of first integrals (H∼,F∼2) satisfies the following separation equations:

$$ \begin{eqnarray}\begin{array}{lll} & & (1+\sqrt{2}\varepsilon {\hat{Q}}_{1})\mathop{H}\limits^{\sim }+{\mathop{F}\limits^{\sim }}_{2}={\hat{P}}_{1}^{2}+\displaystyle \frac{k}{\sqrt{2}}{\hat{Q}}_{1},\\ & & (1-\sqrt{2}\varepsilon {\hat{Q}}_{2})\mathop{H}\limits^{\sim }-{\mathop{F}\limits^{\sim }}_{2}={\hat{P}}_{2}^{2}-\displaystyle \frac{k}{\sqrt{2}}{\hat{Q}}_{2}.\end{array}\end{eqnarray}$$ (20)

as

$$ \begin{eqnarray}\begin{array}{lll}{\mathop{F}\limits^{\sim }}_{2} & = & \displaystyle \frac{(1-\sqrt{2}\varepsilon {\hat{Q}}_{2}){\hat{P}}_{1}^{2}-(1+\sqrt{2}\varepsilon {\hat{Q}}_{1}){\hat{P}}_{2}^{2}}{2(1+\varepsilon {Q}_{1})}\\ & & +\displaystyle \frac{k({\hat{Q}}_{1}+{\hat{Q}}_{2})}{2\sqrt{2}(1+\varepsilon {Q}_{1})}\end{array}\end{eqnarray}$$ (21)

in the rotated flat coordinates (Q^1,Q^2), or

$$ \begin{eqnarray}\begin{array}{lll}{\mathop{F}\limits^{\sim }}_{2} & = & \displaystyle \frac{{\hat{P}}_{1}^{2}-{\hat{P}}_{2}^{2}-\sqrt{2}\varepsilon ({\hat{Q}}_{2}{\hat{P}}_{1}^{2}+{\hat{Q}}_{1}{\hat{P}}_{2}^{2})}{2(1+\varepsilon {Q}_{1})}+\displaystyle \frac{k{Q}_{2}}{2(1+\varepsilon {Q}_{1})}\\ & = & \displaystyle \frac{-\varepsilon {Q}_{2}({P}_{1}^{2}+{P}_{2}^{2})+2{P}_{1}{P}_{2}(1+\varepsilon {Q}_{1})}{2(1+\varepsilon {Q}_{1})}+\displaystyle \frac{k{Q}_{2}}{2(1+\varepsilon {Q}_{1})}\,\,\,\,\,\,\,\,\,\end{array}\end{eqnarray}$$ (22)

in the flat coordinates (Q₁,Q₂), where we have, in the calculation, used the equality

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The Fordy system with a PDM, Eq. (15), is Stäckel separable in rotated flat coordinates (Q^1,Q^2). The separation equations are given by Eq. (20). The quadratic integral of motion F∼2 has the form of Eq. (21) or (22).

When written in the parabolic coordinates (u,v), the Hamiltonian H∼ has the form

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turns out to be

 1

where the parts of (u, p_u) and (v, p_v) are separated. It follows that

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is an additional first integral. The pair of first integrals (H∼,F∼1) satisfies the separation equations

$$ \begin{eqnarray}\begin{array}{lll} & & 2(u+\varepsilon {u}^{2})\mathop{H}\limits^{\sim }+{\mathop{F}\limits^{\sim }}_{1}=u{p}_{u}^{2}+k{u}^{2},\\ & & 2(v-\varepsilon {v}^{2})\mathop{H}\limits^{\sim }-{\mathop{F}\limits^{\sim }}_{1}=v{p}_{v}^{2}-k{v}^{2}.\end{array}\end{eqnarray}$$ (23)

The explicit form of first integral F∼1 is given by

$$ \begin{eqnarray}{\mathop{F}\limits^{\sim }}_{1}=\displaystyle \frac{uv(1-\varepsilon v){p}_{u}^{2}-uv(1+\varepsilon u){p}_{v}^{2}}{(u+v)(1+\varepsilon u-\varepsilon v)}+\displaystyle \frac{kuv}{1+\varepsilon u-\varepsilon v}\end{eqnarray}$$ (24)

in parabolic coordinates (u,v), or

$$ \begin{eqnarray}\begin{array}{lll}{\mathop{F}\limits^{\sim }}_{1} & = & \displaystyle \frac{{Q}_{2}^{2}}{4R(1+\varepsilon {Q}_{1})}(({p}_{u}^{2}-{p}_{v}^{2})-\varepsilon (v{p}_{u}^{2}+u{p}_{v}^{2}))+\displaystyle \frac{k}{1+\varepsilon {Q}_{1}}\displaystyle \frac{{Q}_{2}^{2}}{4}\\ & = & {Q}_{2}{P}_{1}{P}_{2}-{Q}_{1}{P}_{2}^{2}-\displaystyle \frac{\varepsilon {Q}_{2}^{2}}{4(1+\varepsilon {Q}_{1})}({P}_{1}^{2}+{P}_{2}^{2})\\ & & +\displaystyle \frac{k}{4(1+\varepsilon {Q}_{1})}{Q}_{2}^{2}\end{array}\end{eqnarray}$$ (25)

in the flat coordinates (Q₁,Q₂), where the quantities

 1

have been employed during the calculation.

The Fordy system with a PDM, Eq. (15), is Stäckel separable in parabolic coordinates (u,v). The separation equations are given by Eq. (23). The quadratic integral of motion F∼1 has the explicit form of Eq. (24) or (25).

The system with a PDM, Eq. (14), has four first integrals, H∼,K2,F∼1,F∼2. In this subsection we will determine the algebraic relation and Poisson algebra among them. A straightforward computation gives the nonvanishing Poisson brackets between them as

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Since the number of integrals exceeds the number of degrees of freedom of the system there exists at most three functionally independent integrals of motion and the four integrals of motion must be functionally dependant.^[5] However, it is a nontrivial task to find the suitable algebraic equation for the integrals especially when the integrals have complicated forms.

An instructive principle is that, in order to establish such an algebraic relation, it is necessary to make the potential parts of the integrals canceled out. We denote by h∼, f∼1, f∼2 the potential parts of integrals H∼, F∼1, F∼2, respectively. It is crucial to rewrite h∼ as

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$$ \begin{eqnarray}2\left(\varepsilon \mathop{h}\limits^{\sim }-\displaystyle \frac{k}{2}\right){\mathop{f}\limits^{\sim }}_{1}+{\mathop{f}\limits^{\sim }}_{2}^{2}=0.\end{eqnarray}$$ (26)

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The classical Jacobi method can be used to construct new integrable systems from known ones. It originated in Ref. [12] where the author proposed elliptic coordinates and employed them to integrate some vital mechanical systems. This method reverses the conventional path from a given system to its separation variables and formulate new systems from known separation variables and arbitrary separation relations. For more discussions and applications of it, one can see, e.g., Refs. [13–16].

Note that the separation Eq. (23) of the Fordy system with a PDM can be viewed as a deformation of Eq. (13) for the Fordy system. Alternatively to the PDM approach, we can also obtain the new system by applying the classical Jacobi method to the separation equations of the Fordy system, i.e., by adding quadratic terms εu² and –εv².

According to this idea, we can further generalize the Fordy system with a PDM, i.e., Eq. (15). By applying the Jacobi method to separation equations (23) we can get a family of generalized integrable systems with the following separation equations:

$$ \begin{eqnarray}\begin{array}{lll} & & 2(u+\varepsilon {u}^{2})\mathop{H}\limits^{\sim }+{\mathop{F}\limits^{\sim }}_{1}=(u+f(u)){p}_{u}^{2}+{k}_{1}{u}^{2}+{k}_{2}{u}^{3},\\ & & 2(v-\varepsilon {v}^{2})\mathop{H}\limits^{\sim }-{\mathop{F}\limits^{\sim }}_{1}=(v+h(v)){p}_{v}^{2}+{k}_{3}{v}^{2}+{k}_{4}{v}^{3},\end{array}\end{eqnarray}$$ (28)

$$ \begin{eqnarray}\begin{array}{lll}\mathop{H}\limits^{\sim } & = & \displaystyle \frac{(u+f(u)){p}_{u}^{2}+(v+h(v)){p}_{v}^{2}}{2(u+v+\varepsilon {u}^{2}-\varepsilon {v}^{2})}\\ & & +\displaystyle \frac{{k}_{1}{u}^{2}+{k}_{2}{u}^{3}+{k}_{3}{v}^{2}+{k}_{4}{v}^{3}}{2(u+v+\varepsilon {u}^{2}-\varepsilon {v}^{2})}.\end{array}\end{eqnarray}$$ (29)

From the generalized separation equations (28) we obtain

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$$ \begin{eqnarray}W={W}_{1}(u,\mathop{\alpha }\limits^{\sim },{\mathop{\alpha }\limits^{\sim }}_{1})+{W}_{2}(v,\mathop{\alpha }\limits^{\sim },{\mathop{\alpha }\limits^{\sim }}_{1}).\end{eqnarray}$$ (30)

$$ \begin{eqnarray}\mathop{\beta }\limits^{\sim }=-t+\displaystyle \frac{\partial W}{\partial \mathop{\alpha }\limits^{\sim }},\,{\mathop{\beta }\limits^{\sim }}_{1}=\displaystyle \frac{\partial W}{\partial {\mathop{\alpha }\limits^{\sim }}_{1}},\end{eqnarray}$$ (31)

In general the generalized system (29) is defined on cotangent bundle of non-flat Riemannian manifold. In the following, we analyze the conditions which imply the flatness of the underlying space.

The underlying Riemannian space has the metric

$$ \begin{eqnarray}{\rm{d}}{s}^{2}=(u+v+\varepsilon {u}^{2}-\varepsilon {v}^{2})\left(\displaystyle \frac{{\rm{d}}{u}^{2}}{u+f(u)}+\displaystyle \frac{{\rm{d}}{v}^{2}}{v+h(v)}\right).\end{eqnarray}$$ (32)

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$$ \begin{eqnarray}{\rm{d}}{s}^{2}=\displaystyle \frac{J{\rm{d}}{u}^{2}}{U(u)}+\displaystyle \frac{J{\rm{d}}{v}^{2}}{V(v)}=E{\rm{d}}{u}^{2}+G{\rm{d}}{v}^{2},\end{eqnarray}$$ (33)

$$ \begin{eqnarray}K=-\displaystyle \frac{1}{2\sqrt{EG}}\left(\displaystyle \frac{\partial }{\partial u}\displaystyle \frac{{G}_{u}}{\sqrt{EG}}+\displaystyle \frac{\partial }{\partial v}\displaystyle \frac{{E}_{v}}{\sqrt{EG}}\right).\end{eqnarray}$$ (34)

A detailed calculation gives

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$$ \begin{eqnarray}\begin{array}{lll}K & = & \displaystyle \frac{1}{2{J}^{3}}\,\left[-\left(\varepsilon u+\displaystyle \frac{1}{2}\right)J{f}^{^{\prime} }(u)+\left(\varepsilon v-\displaystyle \frac{1}{2}\right)J{h}^{^{\prime} }(v)\right.\\ & & \left.+(1+2({u}^{2}+{v}^{2}){\varepsilon }^{2}+2(u-v)\varepsilon )\right.\\ & & \left.\times (f(u)+h(v))+{\varepsilon }^{2}{(u+v)}^{3}\,\right].\end{array}\end{eqnarray}$$ (35)

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$$ \begin{eqnarray}f(u)={\ell }_{1}{u}^{2}+{\ell }_{2}u,\,h(v)=-{\ell }_{1}{v}^{2}+{\ell }_{2}v,\end{eqnarray}$$ (36)

For the condition of ε = 0 and f(u), g(v) given by Eq. (36) and arbitrary parameters k_i (i = 1,…,4), the Hamiltonian system (29) is defined on the cotangent bundle of a flat Riemannian space.

In summary, we have made a deformation for the superintegrable system (2) proposed by Fordy, by a position-dependent mass. The new system with a PDM is shown separable in three distinct coordinate systems with separation equations and integrals of motion explicitly demonstrated. By applying the Jacobi method to the Fordy system with a PDM a family of integrable systems containing arbitrary constants and functions are generated.

It will be an interesting direction of study to explore the connection between the new systems proposed in this paper and those known in the literature. In particular, a broad class of the finite-gap systems with position-depend mass has been considered by Bravo and Plyushchay in Ref. [8]. They obtained elliptic finite-gap systems of Lamé and Darboux–Treibich–Verdier types by reduction to Seiffert’s spherical spiral and Bernoulli lemniscate with Calogero-like or harmonic oscillator potentials. Our proposed systems have intimate relations with them. While they studied quantum mechanical systems with PDM, we consider classical ones. It will be interesting to develop the quantum counterparts of our proposed systems and explore their applications in areas of theoretical physics such as supersymmetric quantum mechanics or quantum field theory.^[18,19]

Another line of research is to investigate the newly proposed systems by employing some common approaches in theory of integrable systems, such as the Lax representation, classical r-matrix formulation, (bi-)Hamiltonian structure, and action-angle variables. Their symmetry properties^[20,21] can also be investigated.