The mean field approach is important to the many-body systems, due to the fact that the $N$-body problem is practically impossible to handle for large values of $N$. The motion of a single particle may be studied in the field generated by many bodies, under the hypothesis that it does not perturb significantly the external field. This can be done, for example, in astrophysics (a star in a galaxy, a galaxy in a large galactic cluster) or in nuclear physics (nucleons in nuclei, atoms in metallic clusters).

In such models, the potential is not known in advance, but it is supposed to have some properties as spherical or axial symmetry. The knowledge of a family of orbits makes possible the determination of the potential, using the tools of the inverse problem of dynamics.

In this paper, we study the possibility of finding 3D potentials which give rise to a family of orbits known in advance. Among others, we construct a homogeneous potential which can produce a family of elliptic orbits (which are frequent in galaxy models), and an axially
symmetric potential compatible with a family of hyperbolae. A related 2D problem was first studied by Szebehely (1974), with the aim to use the results for determining the Earth’s potential on the basis of satellites’ orbits: find all the potentials $V(x,y)$ which can give rise to a given mono-parametric family of curves $f(x,y)=c$ traced in the $xy$-Cartesian plane by a material point of unit mass. These potentials satisfy a linear first-order equation in which the total energy appears, known as Szebehely’s equation. A linear second-order partial differential equation in the unknown function $V$, involving only the family of orbits and the potential-not the energy dependence, was produced by Bozis (1983). Anisiu (2004a) derived in a unified manner the two basic equations of the inverse problem of dynamics, and the region where real motion of the particle takes place (Bozis and Ichtiaroglou 1994).

Three-dimensional versions of the inverse problem were studied by Érdi (1982) for a monoparametric family of spatial orbits, and, for a two-parametric family, by Bozis (1983) (for general force fields) and by Váradi and Érdi (1983). Puel (1984) presented the intrinsic (independent of the coordinate frame) equations of the 3D inverse problem. Other results have been obtained by Bozis and Nakhla (1986), and Shorokov (1988). These results are summarized in the review paper of Bozis (1995). Recently, Bozis and Kotoulas (2004) have studied the case of two-parametric families of straight lines (FSL) produced by genuine three-dimensional potentials. Moreover, the same authors produced the free-ofenergy equations and derived 3D homogeneous potentials which give rise to two-parametric families of homogeneous orbits in space (Bozis and Kotoulas 2005). At the same time, Anisiu (2004b, 2005) obtained in a direct way the two free-of-energy PDEs of the three-dimensional inverse problem and the region where real motion is allowed, presenting also several families of orbits compatible with 3D potentials.

The free-of-energy equation for the planar inverse problem was used to find the richest set of potentials which can generate a given family of ellipses (Bozis and Caranicolas 1997). The same equation enabled Caranicolas (1998) to produce potentials which allow for figure-eight families of orbits; such families appear in astrophysics, being detected in the $N$-body problem simulation in barred galaxies. More recently, Bozis and Anisiu (2005) have dealt with a solvable version of the planar inverse problem and found potentials of special type $V=v(\gamma)$ where $\gamma=f_{y}/f_{x}$. This approach was motivated by the fact that potentials of the form $v(\gamma)$ had already appeared as solutions of the planar inverse problem. Another reason was that, when potentials $v(\gamma)$ compatible with the given family $\gamma$ do exist, they can be calculated by quadrature.

The present work extends to three dimensions the results concerning the planar potentials of the form $V=v(\gamma)$ compatible with a pre-assigned family of orbits $f(x,y)=c$. We deal with the following version of the 3D inverse problem of dynamics: given a two-parametric family of regular curves $f(x,y,z)=c_{1},g(x,y,z)=c_{2}$ (with orbital functions $\alpha,\beta$ defined in (6)), find the potentials of the form $V=v(\alpha)$ or $V=F(\alpha,\beta)$ producing these families of curves as trajectories. For example, it is proved that the family (73) of elliptic orbits can be traced by a material point under the action of the potential $V(\alpha)=-\alpha^{2}\left(2+\alpha^{2}\right)/4$. Working with the orbital functions $\alpha$ and $\beta$ from (62), we obtain the potential $V=F(\alpha,\beta)$ from (64), which is an axially symmetric one. As mentioned by Boccaletti and Puccaco (1996, vol 1, p 349), axially symmetric potentials are important for astrophysical applications, since they are generated by mass distributions provided with rotational motions around the symmetry axis, typical of solid celestial bodies, stars, disc galaxies and elliptical galaxies. Besides the applications in galactic dynamics (Contopoulos 1960, Binney and Tremaine 1987, p 121; Caranicolas 2004), they are used in models in thermodynamics (e.g. Olaya-Castro and Quiroga (2000)) and in quantum mechanics (e.g. Heiss et al (1994)).

The problem under consideration does not have solutions for arbitrary families of curves. We determine the necessary and sufficient conditions on $\alpha$ and $\beta$ which guarantee the existence of such potentials. A promising feature is that, when the conditions are fulfilled, we can construct the potentials by quadrature. We focus on genuine three-dimensional potentials, i.e. potentials whose analytical expression involves all the variables $x,y,z$. It follows that we take into account only potentials for which no partial derivative is identically zero. All over the paper we shall regard $V$ as the potential energy function, hence the total energy is given by $\mathcal{E}=\left(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2}\right)/2+V(x,y,z)$. In section 2 we present the basic equations of the 3D inverse problem of dynamics. In section 3 we study the case $V=v(\alpha)$. The case $V=F(\alpha,\beta)$ is examined in section 4. In both sections 3 and 4 we give step-by-step procedures, which lead to the expression of the potential when the orbital functions satisfy certain differential conditions. The mathematics are used as a tool, and the method can be applied whenever a family of orbits will be observed, or generated by a simulation. We refer to the case of straight lines in section 5 and conclude in section 6.

The three-dimensional inverse problem of dynamics, as formulated by Bozis and Nakhla (1986), seeks all the potentials $V=V(x,y,z)$ of $C^{2}$-class (continuous with continuous derivatives up to second order) which are compatible with a pre-assigned family of regular orbits

By regular orbits we mean that the functions $f,g$ are of $C^{3}$-class on a domain $D\subset\mathbb{R}^{3}$ and such that $\nabla f\neq 0$ and $\nabla g\neq 0$, and, in addition $\nabla f\times\nabla g\neq 0$ on $D$, so that $\delta_{0}\neq 0$ in (3). The total energy is constant on each curve of the family, namely

Differentiating equations (1) with respect to time $t$ and taking into account the conservation of the energy one can express the velocity vector; differentiating again (1) and using the equations of motion one obtains the two linear PDEs satisfied by the potential $V$. These equations, in which the energy dependence function $\mathcal{E}=\mathcal{E}(f,g)$ appears, are

where

and

The vectorial and scalar products in $\mathbb{R}^{3}$ are denoted by ’ $\times$ ’, respectively ’ ’, and the subscripts $x,y,z$ denote the corresponding partial derivatives. Equations (3) in a vectorial form have been derived by Puel (1984) using Frenet’s reference frame.

The orbital functions

have been defined by Bozis and Kotoulas (2004), in a domain where $\delta_{1}\neq 0$. With the use of notations

equations (3) take the simpler form (Bozis and Kotoulas 2005)

The region where real motion exists (Shorokhov 1988, Anisiu 2004b) is

Furthermore, Bozis and Kotoulas (2005) eliminated the energy dependence function from equations (8) and obtained two energy-free PDEs for the unknown potential function $V=V(x,y,z)$, one of first order and the other of second order. These equations relate the potential to the given family of curves, supposing that the last one does not consist of straight lines. It was proved by Bozis and Kotoulas (2004) that $\alpha_{0}=0$ and $\beta_{0}=0$ only when the family (1) consists of straight lines; we shall treat this special case in section 5.

Remark 1. As was mentioned by Bozis and Kotoulas (2005), the transformation $x\rightarrow x,y\rightarrow z,z\rightarrow y$ brings $\alpha\rightarrow\beta$ and $\beta\rightarrow\alpha$ as well as $\alpha_{0}\rightarrow\beta_{0}$ and $\beta_{0}\rightarrow\alpha_{0}$.

From now on we shall consider $\alpha_{0}\neq 0$. (The case $\beta_{0}\neq 0$ can be treated directly using instead of equation (12) the equation obtained from the second-order equation (25) in the paper of Anisiu (2004b).) The first-order equation is

where

The second-order PDE of the 3D inverse problem reads
$k_{11}V_{xx}+k_{12}V_{xy}+k_{13}V_{xz}+k_{22}V_{yy}+k_{23}V_{yz}+k_{01}V_{x}+k_{02}V_{y}+k_{03}V_{z}=0$,
where

We note here that the previous equations (10) and (12) have also been produced on first grounds by Anisiu (2004b). These equations are equivalent to system (3), as was proved by Anisiu (2005), and have the advantage that it is not necessary to give the energy in advance. From (10) and (12) it is easy to check that if $V$ is a solution, then $\tilde{V}=c_{1}V+c_{2}$ is a solution too ( $c_{1},c_{2}$ are constants). So, without loss of generality, we shall omit these constants below.

In the present paper we extend the 2D results of Bozis and Anisiu (2005) and study the following version of the 3D inverse problem of dynamics: given a two-parametric family of regular orbits (1), we try to find the three-dimensional potentials of the form $V=v(\alpha)$ or $V=F(\alpha,\beta)$ which are compatible with that family. Having no a priori information on the energy, we deal with the two linear PDEs (10) and (12).

In this section we shall try to find solutions of the form $V=v(\alpha)$ for the above two equations (10) and (12). In view of remark 1, we shall not consider here the case of $V=v(\beta)$.

(1) We start with the functions $\alpha$ and $\beta$ (or with $f$ and $g$, and then obtain $\alpha,\beta$ from (6)).
(2) We calculate the expression in (15); if it is different from zero, the problem has no solution.
(3) If (15) is satisfied, we estimate $\mathcal{A}$ from (18); if $\mathcal{A}=0$, we have no solution.
(4) If $\mathcal{A}\neq 0$, with $\mathcal{B}$ from (18) we calculate $\mathcal{C}$ from (21) and verify conditions (24i, ii). If at least one of them is not satisfied, there is no solution.
(5) If (24i, ii) are verified, we obtain the potential $V=v(\alpha)$ from (25).

In this section we shall find potentials of the form $V=F(\alpha,\beta)$, with $\alpha,\beta$ functionally independent and $F$ of $C^{2}$-class. The condition $\alpha_{0}\neq 0$ is maintained. We impose necessary conditions on the orbital functions $\alpha,\beta$ and then we shall determine, when possible, the potential function $V=F(\alpha,\beta)$ so that it satisfies the two linear PDEs (10) and (12).