The two-dimensional inverse problem of dynamics consists in finding a potential $V$ which generates a family of curves

in the $xy$ Euclidean space. The roots of the problem are to be found in Newton’s Principia (1687) where a force law compatible with Kepler’s laws was deduced. Interest in this old problem increased after Szebehely [8] presented the partial differential equation

where the subscripts denote partial derivatives. This linear in $V$ partial differential equation is our tool to find the potential $V(x,y)$ which can produce as orbits a preassigned monoparametric family of curves (1), traced in the $xy$ plane, with adequate initial conditions, by a material point of unit mass, with energy dependence

given in advance.

In spite of its linear character, in practice, equation (2) cannot be solved analytically. Actually, its solvability is directly connected to the possibility of solving a (generally nonlinear) system of ordinary differential equations.

Additional assumptions regarding the form of either the known function $f(x,y)$ or the unknown potential $V(x,y)$ ease the solution of the problem. Thus, for example, for particular sets of functions $f$ (e.g., homogeneous in $x,y$ ) the compatibility with homogeneous (Bozis and Grigoriadou [1]) or nonhomogeneous (Bozis et al [4]) potentials was already studied.

In this framework, in the present paper a new special case is considered: that of the set of monoparametric families of curves with equation

where $h(x)$ is a nonlinear $\left(h^{\prime\prime}(x)\neq 0\right.$ ) function of $x$. We consider functions $h$ defined on an interval where $h^{\prime}$ has no zeros. For every function $h(x)$ equation (4) stands for a family of (equidistant) curves (not straight lines), shifted parallel to the $y$-axis. It is shown that all potentials which can generate families of the form (4) can be found by quadratures.

On the other hand, it is known that during the motion of a material point of unit mass along an orbit of the family (1), the inequality

must be observed, with

This means that the motion is allowed along those members (or part of the members) of the family (1) which are lying only inside some regions of the $xy$ plane, limited by the so-called family boundary curves (FBC) (Bozis and Ichtiaroglou [2]), which are given by the equation

The function $B(x,y)$ is the kinetic energy (expressed in terms of the position coordinates $x,y)$ of the material point of unit mass, as it moves on any of the orbits (4) in the presence of the potential $V(x,y)$. We shall refer to it here as the $B$-function and keep in mind that it is associated with the family (4), which, of course, can have infinitely many $B$-functions. The merit of such a function lies in that, by preassigning to (4) a certain $B$-function, we can manage to have conservative motion inside a preassigned region $T\subset\mathbb{R}^{2}$ of the $xy$ plane defined by inequality (5). We remind the reader that if force fields (not necessarily conservative) are demanded, in general, there exist such fields to create any preassigned family (1) inside any preassigned region $T$ (Bozis [3]).

The question raised and answered in this paper is the following: can any (positive in $T\subset\mathbb{R}^{2}$ ) function stand for a $B$-function? In other words: are there potentials $V(x,y)$ generating orbits of the form (4) traced with preassigned kinetic energy (6) and, as a consequence, lying inside a preassigned region (5)? Which is the pertinent family (4) and which is the corresponding energy dependence $E=E(f)$ ?

It turns out that, for every $h(x)$, there exists a simpler (positive in $T$ ) function $b(x,y)$, whose positiveness in $T$ implies the inequality (5). Due to its simplicity, we prefer to represent the FBC by $b(x,y)=0$ and we focus attention on the problem of obtaining compatible pairs of $b(x,y)$ and $h(x)$. We show that, if the given $b(x,y)$ satisfies two conditions, the function $h(x)$ (and, consequently, the family $f(x,y)=y-h(x)=c$ ) as well as the energy dependence function $E(f)$ and the potential $V(x,y)$ are determined.

For the case of the family of curves given by (4), equation (2) has the simpler form

the prime denoting the derivative with respect to the $x$ variable, and, according to (1), $f(x,y)=y-h(x)$. The subsidiary system of ordinary differential equations is

The first of equations (9) gives

where

Equating the third to the first fraction in (9), we obtain a linear equation in $V$ which has to be solved after replacing $y$ appearing in the argument of $E$ by its expression in (10). So the argument of $E$ will be $y-h(x)=c_{1}-I(x)-h(x)$, and the equation in $V$ will have the form
$\frac{\mathrm{d}V}{\mathrm{~d}x}+\frac{2h^{\prime\prime}(x)}{h^{\prime}(x)\left(1+h^{\prime 2}(x)\right)}V-\frac{2h^{\prime\prime}(x)}{h^{\prime}(x)\left(1+h^{\prime 2}(x)\right)}E\left(c_{1}-I(x)-h(x)\right)=0$.

The solution of the ordinary differential equation (12) is

where

Integrating by parts the above integral $K$, we get

where

and where $E_{c}$ denotes the derivative of the one-variable function $E=E(c)$ with respect to its argument.

The general solution of the partial differential equation (8) is given by

with $A$ an arbitrary function of $c_{1}=y+I(x)$. So, for the family of curves (4) traced with a preassigned energy dependence $E=E(c)$, the potentials creating it are given by

where we adopt the notation

with the functions $h$ and $I$ depending merely on $x$.
Clearly, if we consider as given a function $h(x)$ and select arbitrarily a function $E(c)$, we can calculate the integrals $I$ and $J$ by quadratures from (11) and (15), respectively.

Real motion of the moving point takes place only on those members of the family of curves (1) or those parts of each member of the family which lie in the region of the $xy$ plane where the inequality (5) is observed.

We proceed to find the pertinent FBC (7) for the specific set of families (4). As the corresponding potentials are given by (17), the function in (6) is found to be given by

The meaning of (19) is the following: given a family (4), i.e. given a function $h(x)$, after selecting specific arbitrary functions $E(c)$ and $A\left(c_{1}\right)$, we can find the function $B(x,y)$ and draw conclusions regarding the FBC.

Naturally, wherever $B(x,y)\geqslant 0$, with $B(x,y)$ given by (19), it is also

with

and conversely. Thus, together with the $B$-function, we have a simpler function $b(x,y)$ which can serve to describe the FBC or, if given in advance, to create programmed motion inside a region. So, in what follows, we shall represent the preassigned allowed region by inequality (20). The same region, of course, could be represented by $\theta(x,y)b(x,y)\geqslant 0$, where $\theta(x,y)$ is any arbitrary non-negative function in $T$ (Bozis [3]).

On the grounds of the previous analysis we now pose the following question: consider a preassigned boundary function $b=b(x,y)$, positive inside a region $T\subset\mathbb{R}^{2}$ which we want to programme as an allowed region of some orbits (4). Can we find a function $h(x)$, leading to appropriate functions $E(c),A\left(c_{1}\right)$ and $J\left(x,c_{1}\right)$, such that equation (21) is satisfied?

If the answer to the above question is in the affirmative, inside $T$ there exist members of the family (4). These are orbits, traced by the moving point for adequate initial conditions, in the presence of the potential (17) with kinetic energy equal to $B(x,y)=\frac{1+h^{\prime 2}}{h^{\prime 2}}b(x,y)$ at each point of the orbit.

Consider a (positive in a region $T\subset\mathbb{R}^{2}$ ) boundary function $b(x,y)$ to account for our goal to obtain orbits of the form (4) inside the region $T$. For this version of the inverse problem, suppose that there is an affirmative answer, i.e. there exists an appropriate function $h(x)$ for which (21) can be satisfied.

Since the function $A$ depends on $x,y$ through $c_{1}$, it is

and, in view of (10) and (11), it must be

We want to replace into (22) the function $\bar{A}=\bar{E}+\bar{J}-b$ given by (21). To this end we prepare, in view of (3), (15) and (18),

In view of (23), the condition (22) is now written as

where $b_{10}$ stands for $b_{x}$ and $b_{01}$ for $b_{y}$. This notation is in agreement with the following more general convention which we adopt in what follows: we denote by $b_{mn}(m,n$ positive integers) the partial derivative of $b(x,y)m$ times with respect to $x$ and $n$ times with respect to $y$.

Since $h^{\prime}(x)\neq 0$, we denote $\frac{1}{h^{\prime}}$ by $z=z(x)$ and we write equation (24) as

The simpler case of isoenergetic families of orbits (4), for which $E_{c}=0$, will be treated separately in section 6 . In general, we know that $E_{c}$ depends on $x,y$ through $c$, given by equation (4): thus, $\frac{c_{y}}{c_{x}}=-\frac{1}{h^{\prime}}$ and we find

After some straightforward algebra, from (26) and (25) we obtain

where $z^{\prime}=\frac{\mathrm{d}z}{\mathrm{~d}x}\left(\neq 0\right.$ because $\left.h^{\prime\prime}(x)\neq 0\right)$.
Condition (27) is free from the energy and (as we restrict ourselves to choose $b$, and not $\theta b$, to represent the FBC) it is necessary and sufficient for our purpose to achieve programmed motion. As it stands, (27) relates the given boundary function $b(x,y)$ (and up to second-order derivatives of it) to the function

(and its derivative $z^{\prime}$ ).
To come to know $z$ and $z^{\prime}$, we differentiate twice both members of (27) with respect to $y$ and we obtain the two equations

and

Assuming that $b_{10}-2b_{01}z\neq 0$ (the special case will be treated in section 6) and dividing (29) by (27), we obtain the cubic in $z$ algebraic equation

with

Now dividing (30) by (27) we obtain a second cubic

with

It can be checked easily that $\beta_{i}$ can be obtained alternatively by differentiating each $\alpha_{i}$ ( $i=0,1,2,3$ ) with respect to $y$. A problem may arise when all the $\alpha_{i}$ are zero, so we cannot make use of equations (31) and (33). This case will be treated in section 6.

Equations (31) and (33) are necessary conditions for $z$ inherited from the unique necessary and sufficient condition (27) which we wish to satisfy, if this is possible, in the first place, with a $z=z(x)$. So, if, for a given $b$, such a $z$ does exist, this has to be the (at least one) common root of (31) and (33). Of course, for such a root to exist, a sixth-order determinant, called the resultant of the two polynomials in (31) and (33) (Mishina and Proskuryakov [7], p 164) must vanish identically. This requirement will lead to a (rather complicated) condition including derivatives of $b(x,y)$ up to the fourth order. Yet, it is understood that we need not write down this condition. Instead, taking for granted that an appropriate $z=z(x)$ to be put in (27) does exist and that this $z$ cannot be anything but the common root of (31) and (33), we proceed to find it as follows.

We multiply (31) by $\beta_{3}$ and (33) by $\alpha_{3}$ and subtract. Assuming that $\alpha_{2}\beta_{3}-\alpha_{3}\beta_{2}\neq 0$ (the special case will be treated in section 6), we obtain

with

In view of (35), we replace $z^{3}$ into (31) by $z^{3}=\left(\gamma_{1}^{2}+\gamma_{0}\right)z+\gamma_{0}\gamma_{1}$ and we find

with

We assume that $\delta_{1}\neq 0$ (the special case $\delta_{1}=0$ is examined in section 6) and we conclude by writing down the two conditions which the given function $b(x,y)$ must satisfy:
(i) The function $z$, found from (37), must depend only on the variable $x$, i.e.

(ii) The function $z$ must satisfy the equation (27), i.e.

Clearly, if we expressed the above conditions (39) and (40) in terms of the given function $b(x,y)$ only, there would appear partial derivatives of $b$ up to the fifth order.

We want to have monoparametric families of orbits of the form (4) ’inside’ a preassigned region $T\subset\mathbb{R}^{2}$ given by inequality (20) with the given function $b(x,y)$ to be defined as in (21).

In view of the analysis in section 4 and aided by a symbolic algebra program (e.g. MATHEMATICA) we proceed as follows:
(i) For the given $b(x,y)$ we prepare its partial derivatives $b_{ij}$ up to the fourth order and, from equations (32), (34), (36) and (38), we find all the functions $\alpha_{i},\beta_{i}(i=0,1,2,3)$ and $\gamma_{k},\delta_{k}(k=0,1)$. We make sure that

(Otherwise we act as in section 6.)
(ii) We prepare the partial derivatives $\delta_{0,x},\delta_{0,y},\delta_{1,x},\delta_{1,y}$ of the two functions $\delta_{k}(x,y)$ ( $k=$ 0,1 ) and we check the conditions (39) and (40).

After the remark which follows we shall continue with our synthesis by assuming that we do have an appropriate $b$-function at our disposal.

Remark. As a rule, of course, one does not expect the conditions (39) and (40) to be satisfied. Consequently, one cannot programme motion inside $T\subset\mathbb{R}^{2}$, described by the $b$-function at hand. Yet, such an appropriate $b$-function may be hidden! It may correspond to another selection of the function $h(x)$ in equation (4) and, most likely, to different selections of the arbitrary functions $E$ and $A$, as given by the equations (3) and (16).

One then may try to find a good $b$-function. Thus, for example:

• •

  If $b$ is given with some free constants, one may try to determine these constants so that the conditions (39) and (40) are satisfied.

  Report issue for preceding element
• •

  If this cannot be achieved, we may write (39) and (40) for $b\rightarrow\theta b(x,y)$ and try to find even one (positive in $T$ ) particular solution $\theta_{0}(x,y)$ satisfying these conditions. This assignment is far from being a simple task, of course.

  Report issue for preceding element

Assuming that the given $b$ is appropriate, we now proceed as follows:
(iii) From (37) and (28) we find $z$ and $h^{\prime}$ and then, in turn, $h(x)$ up to an additive constant $h_{0}$. Then, out of the set of orbits (4), we obtain the specific monoparametric family $f(x,y)=y-h(x)=c$. In general, the two equations (31) and (33) have one common root, so we obtain one family (4) for some members of which we can manage to have them trapped inside the preassigned region (20). To this end we must determine the appropriate potential.
(iv) From equation (25) we now determine $E_{c}$ and, as we already know the expression for $c=y-h(x)$, we determine uniquely (apart from an additive constant $E_{0}$ ) the energy dependence function (3) $E=E(c)$ and, from the first of equations (18), we come to know $\bar{E}$, into which the constants $h_{0}$ and $E_{0}$ will enter.
(v) Finally, since the kinetic energy is

we write down the potential

which gives rise to orbits with equation (4) lying inside the region (20).

Comment. In spite of the fact that the two integration constants, $h_{0}$ and $E_{0}$ enter into (43) through $\bar{E}$, the potential $V(x,y)$ is essentially unique, at least as far as the study of the present family is concerned. Indeed, $h_{0}$ and $E_{0}$ appear in (43) either purely as an additive constant or through a combination which amounts to a (constant, again) additive arbitrary function $V_{0}(f)$, where $f=y-h(x)=c$ is the family under consideration. For these reasons we can set both $h_{0}$ and $E_{0}$ equal to zero and this is actually what we do in the example of section 7 .

For $x>0$, let us start with the $b$-function

with

We want to check if families of the form (4) exist inside the (nonshaded) region of figure 2, given by $b(x,y)\geqslant 0$, which we engage as the allowed region for such families.

According to the procedure exposed in section 5, we write down the equations (31) and (33) and see first if they possess any $y$-independent common roots. The coefficients $\alpha_{i},\beta_{i}$, found from (32) and (34), are (apart from unimportant multiplicative factors)