An alternative point of view on the equations of the inverse problem of Dynamics

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Abstract

The version of the inverse problem of dynamics considered here is: given a family of planar curves \(f(x,y)=c\), find the potentials \(V(x,y)\) which give rise to this family. Its solution is based on two linear partial differential equations satisfied by \(V\): one of first order, containing the total energy function \(E(f)\), given by Szebehely in 1974, and the other one of second order, derived by Bozis in 1984 by eliminating the energy from Szebehely’s equation. In this paper, Bozis’ partial differential equation is obtained directly by eliminating the time derivatives of \(x(t)\) and \(y(t)\) up to the third order between seven differential relations based on the equations of motion and on the given family. Szebehely’s equation is then derived as a consequence. This shows the importance of Bozis’ equation, which is traditionally considered as following from Szebehely’s one. The connection with the nonconservative case is emphasized.

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Mira-Cristiana Anisiu
Tiberiu Popoviciu Institute of Numerical Analysis Cluj-Napoca, Romanian Academy, Romania

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M.-C. Anisiu, An alternative point of view on the equations of the inverse problem of Dynamics, Inverse Problems 20 (2004), 1865-1872, http://10.1088/0266-5611/20/6/011

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An alternative point of view on the equations of the inverse problem of Dynamics

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Inverse Problems

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http://10.1088/0266-5611/20/6/011

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 0266-5611

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0266-56112F6_2F011

An alternative point of view on the equations of the inverse problem of dynamics

Mira-Cristiana Anisiu'T Popoviciu' Institute of Numerical Analysis, Romanian Academy, PO Box 68, 400110 Cluj-Napoca, RomaniaE-mail: mira@math.ubbcluj.ro

Abstract

The version of the inverse problem of dynamics considered here is: given a family of planar curves f ( x , y ) = c f ( x , y ) = c f(x,y)=cf(x, y)=cf(x,y)=c, find the potentials V ( x , y ) V ( x , y ) V(x,y)V(x, y)V(x,y) which give rise to this family. Its solution is based on two linear partial differential equations satisfied by V V VVV : one of first order, containing the total energy function E ( f ) E ( f ) E(f)E(f)E(f), given by Szebehely in 1974, and the other one of second order, derived by Bozis in 1984 by eliminating the energy from Szebehely's equation. In this paper, Bozis' partial differential equation is obtained directly by eliminating the time derivatives of x ( t ) x ( t ) x(t)x(t)x(t) and y ( t ) y ( t ) y(t)y(t)y(t) up to the third order between seven differential relations based on the equations of motion and on the given family. Szebehely's equation is then derived as a consequence. This shows the importance of Bozis' equation, which is traditionally considered as following from Szebehely's one. The connection with the nonconservative case is emphasized.

1. Introduction

We consider the following version of the inverse problem for one material point of unit mass, moving in the x y x y xyx yxy inertial Cartesian plane. Given a family of curves
(1) f ( x , y ) = c (1) f ( x , y ) = c {:(1)f(x","y)=c:}\begin{equation*} f(x, y)=c \tag{1} \end{equation*}(1)f(x,y)=c
with f f fff of C 3 C 3 C^(3)C^{3}C3-class (continuous and with continuous derivatives up to third order on a domain of the plane), find the potentials V ( x , y ) V ( x , y ) V(x,y)V(x, y)V(x,y) under whose action, for appropriate initial conditions, the particle will describe the curves of that family. The equations of the motion are
(2) x ¨ = − V x , y ¨ = − V y , (2) x ¨ = − V x , y ¨ = − V y , {:(2)x^(¨)=-V_(x)","quady^(¨)=-V_(y)",":}\begin{equation*} \ddot{x}=-V_{x}, \quad \ddot{y}=-V_{y}, \tag{2} \end{equation*}(2)x¨=−Vx,y¨=−Vy,
where the dots denote derivatives with respect to the time t t ttt, and the subscripts partial derivatives.
We emphasize that in this version of the inverse problem a family of curves (1) is given, which is in fact determined by the ratio f y / f x f y / f x f_(y)//f_(x)f_{y} / f_{x}fy/fx. Up to now, in the research connected to the inverse problem of dynamics, the families of curves were selected on the grounds of theoretical reasons: families of conic sections, of homogeneous functions or of other special analytic forms. It would be important to consider the inverse problem from the numerical viewpoint. An orbit will be obtained as a result of a curve-fitting process applied to some observed data. As Bozis and Blaga (2004) have shown, this single orbit can be classified in different monoparametric families of curves (1). A practical application would be to find the Newtonian potential of the nonspherical Earth from observed satellite orbits.
Therefore a family of curves (1) can be obtained either from theory or from measured data.
By making use of the energy integral, Szebehely (1974) proved that the desired potentials satisfy the first-order partial differential equation
(3) f x V x + f y V y + 2 ( V − E ( f ) ) f x 2 + f y 2 ( f x x f y 2 − 2 f x y f x f y + f y y f x 2 ) = 0 (3) f x V x + f y V y + 2 ( V − E ( f ) ) f x 2 + f y 2 f x x f y 2 − 2 f x y f x f y + f y y f x 2 = 0 {:(3)f_(x)V_(x)+f_(y)V_(y)+(2(V-E(f)))/(f_(x)^(2)+f_(y)^(2))(f_(xx)f_(y)^(2)-2f_(xy)f_(x)f_(y)+f_(yy)f_(x)^(2))=0:}\begin{equation*} f_{x} V_{x}+f_{y} V_{y}+\frac{2(V-E(f))}{f_{x}^{2}+f_{y}^{2}}\left(f_{x x} f_{y}^{2}-2 f_{x y} f_{x} f_{y}+f_{y y} f_{x}^{2}\right)=0 \tag{3} \end{equation*}(3)fxVx+fyVy+2(V−E(f))fx2+fy2(fxxfy2−2fxyfxfy+fyyfx2)=0
where E ( f ) E ( f ) E(f)E(f)E(f) denotes the total energy, which is constant on each curve of the family (1). Using the functions
(4) γ = f y f x and Γ = γ γ x − γ y (4) γ = f y f x  and  Γ = γ γ x − γ y {:(4)gamma=(f_(y))/(f_(x))quad" and "quad Gamma=gammagamma_(x)-gamma_(y):}\begin{equation*} \gamma=\frac{f_{y}}{f_{x}} \quad \text { and } \quad \Gamma=\gamma \gamma_{x}-\gamma_{y} \tag{4} \end{equation*}(4)γ=fyfx and Γ=γγx−γy
Bozis (1983) wrote Szebehely's equation in the simpler form
(5) V x + γ V y + 2 Γ ( E ( f ) − V ) 1 + γ 2 = 0 (5) V x + γ V y + 2 Γ ( E ( f ) − V ) 1 + γ 2 = 0 {:(5)V_(x)+gammaV_(y)+(2Gamma(E(f)-V))/(1+gamma^(2))=0:}\begin{equation*} V_{x}+\gamma V_{y}+\frac{2 \Gamma(E(f)-V)}{1+\gamma^{2}}=0 \tag{5} \end{equation*}(5)Vx+γVy+2Γ(E(f)−V)1+γ2=0
Remark 1. The function γ γ gamma\gammaγ is related to the slope of the curves in family (1); more precisely, it represents the slope at each point of a family f ∗ ( x , y ) = c ∗ f ∗ ( x , y ) = c ∗ f^(**)(x,y)=c^(**)f^{*}(x, y)=c^{*}f∗(x,y)=c∗ which is orthogonal to family (1). The function Γ Γ Gamma\GammaΓ has also a geometrical meaning, the curvature K K KKK of the members of family (1) being given by K = | Γ | / ( 1 + γ 2 ) 3 / 2 K = | Γ | / 1 + γ 2 3 / 2 K=|Gamma|//(1+gamma^(2))^(3//2)K=|\Gamma| /\left(1+\gamma^{2}\right)^{3 / 2}K=|Γ|/(1+γ2)3/2.
Under the action of a potential that satisfies equation (5), the curves (1) are traced by a material point only in the allowed region, defined by the inequality (Bozis and Ichtiaroglou 1994)
(6) V x + γ V y Γ ⩽ 0 . (6) V x + γ V y Γ ⩽ 0 . {:(6)(V_(x)+gammaV_(y))/(Gamma) <= 0.:}\begin{equation*} \frac{V_{x}+\gamma V_{y}}{\Gamma} \leqslant 0 . \tag{6} \end{equation*}(6)Vx+γVyΓ⩽0.
By eliminating the energy from (5) (using the fact that E y / E x = f y / f x E y / E x = f y / f x E_(y)//E_(x)=f_(y)//f_(x)E_{y} / E_{x}=f_{y} / f_{x}Ey/Ex=fy/fx ), Bozis (1984) obtained the equation of second order which is energy free
(7) − V x x + κ V x y + V y y = λ V x + μ V y (7) − V x x + κ V x y + V y y = λ V x + μ V y {:(7)-V_(xx)+kappaV_(xy)+V_(yy)=lambdaV_(x)+muV_(y):}\begin{equation*} -V_{x x}+\kappa V_{x y}+V_{y y}=\lambda V_{x}+\mu V_{y} \tag{7} \end{equation*}(7)−Vxx+κVxy+Vyy=λVx+μVy
where
(8) κ = 1 γ − γ , λ = Γ y − γ Γ x γ Γ , μ = λ γ + 3 Γ γ . (8) κ = 1 γ − γ , λ = Γ y − γ Γ x γ Γ , μ = λ γ + 3 Γ γ . {:(8)kappa=(1)/(gamma)-gamma","quad lambda=(Gamma_(y)-gammaGamma_(x))/(gamma Gamma)","quad mu=lambda gamma+(3Gamma)/(gamma).:}\begin{equation*} \kappa=\frac{1}{\gamma}-\gamma, \quad \lambda=\frac{\Gamma_{y}-\gamma \Gamma_{x}}{\gamma \Gamma}, \quad \mu=\lambda \gamma+\frac{3 \Gamma}{\gamma} . \tag{8} \end{equation*}(8)κ=1γ−γ,λ=Γy−γΓxγΓ,μ=λγ+3Γγ.
The basic equations (5) and (7) of the inverse problem of dynamics present the connection between geometry (described by γ γ gamma\gammaγ and Γ Γ Gamma\GammaΓ ) and dynamics (the planar potential V V VVV ). Their derivation and other related results are described by Bozis (1995) and by Anisiu (2003a, 2003b).
When we are facing an inverse problem related to the family of curves (1), we have to calculate the functions γ γ gamma\gammaγ and Γ Γ Gamma\GammaΓ from (4) and afterwards plug them into equation (5); from (8) we get κ , λ κ , λ kappa,lambda\kappa, \lambdaκ,λ and μ μ mu\muμ and insert them into (7). Therefore we have at our disposal
two partial differential equations in the unknown function V V VVV. If we can get some information on the energy (e.g. if we are interested in isoenergetic families, with E ( f ) = e = E ( f ) = e = E(f)=e=E(f)=e=E(f)=e= const, the case considered by Borghero and Bozis (2002)), we can use the first-order equation (5). Otherwise we are bound to work with the energy-free equation (7) in order to find the potentials (or at least some particular ones) which can give rise to the family of curves (1). The fact that equations (5) and (7) do not have a unique solution can be used to look for the potential in various classes of functions with physical significance, such as homogeneous or quasihomogeneous ones.
We remark that Szebehely (1974) obtained the first-order equation (3) intending to use it for the determination of the potential of the Earth by means of satellite observations, while Bozis (1984) used equation (7) to check if a given family of orbits may be generated in the plane of symmetry outside a material concentration.
In what follows we derive in a unified manner the two basic equations (5) and (7), as well as inequality (6). The special case of families of straight lines will also be treated.

2. Main results

Let us consider a particle whose motion is described by equations (2), where V V VVV is of C 2 C 2 C^(2)C^{2}C2 class on a domain of the x y x y xyx yxy plane. We shall use a procedure similar to that followed by Kasner (1906) while he obtained the differential equation of the trajectories corresponding to a general (not necessarily conservative) force field. By differentiating (1) with respect to t t ttt we get f x x ˙ + f y y ˙ = 0 f x x ˙ + f y y ˙ = 0 f_(x)x^(˙)+f_(y)y^(˙)=0f_{x} \dot{x}+f_{y} \dot{y}=0fxx˙+fyy˙=0, or, using the notation (4)
(9) γ = − x ˙ y ˙ . (9) γ = − x ˙ y ˙ . {:(9)gamma=-((x^(˙)))/((y^(˙))).:}\begin{equation*} \gamma=-\frac{\dot{x}}{\dot{y}} . \tag{9} \end{equation*}(9)γ=−x˙y˙.
By differentiating (9) we get γ x x ˙ + γ y y ˙ = ( x ˙ y ¨ − y ˙ x ¨ ) / y ˙ 2 γ x x ˙ + γ y y ˙ = ( x ˙ y ¨ − y ˙ x ¨ ) / y ˙ 2 gamma_(x)x^(˙)+gamma_(y)y^(˙)=(x^(˙)y^(¨)-y^(˙)x^(¨))//y^(˙)^(2)\gamma_{x} \dot{x}+\gamma_{y} \dot{y}=(\dot{x} \ddot{y}-\dot{y} \ddot{x}) / \dot{y}^{2}γxx˙+γyy˙=(x˙y¨−y˙x¨)/y˙2, or, using (4) again,
(10) − Γ = x ˙ y ¨ − y ˙ x ¨ y ˙ 3 . (10) − Γ = x ˙ y ¨ − y ˙ x ¨ y ˙ 3 . {:(10)-Gamma=((x^(˙))(y^(¨))-(y^(˙))(x^(¨)))/(y^(˙)^(3)).:}\begin{equation*} -\Gamma=\frac{\dot{x} \ddot{y}-\dot{y} \ddot{x}}{\dot{y}^{3}} . \tag{10} \end{equation*}(10)−Γ=x˙y¨−y˙x¨y˙3.
Inserting in (10) x ¨ x ¨ x^(¨)\ddot{x}x¨ and y ¨ y ¨ y^(¨)\ddot{y}y¨ from (2), and x ˙ x ˙ x^(˙)\dot{x}x˙ from (9) we obtain
Γ y ˙ 2 = − ( V x + γ V y ) . Γ y ˙ 2 = − V x + γ V y . Gammay^(˙)^(2)=-(V_(x)+gammaV_(y)).\Gamma \dot{y}^{2}=-\left(V_{x}+\gamma V_{y}\right) .Γy˙2=−(Vx+γVy).
If Γ = 0 Γ = 0 Gamma=0\Gamma=0Γ=0 (which corresponds to a family (1) of straight lines, studied by Bozis and Anisiu (2001)) we have by necessity
(11) V x + γ V y = 0 (11) V x + γ V y = 0 {:(11)V_(x)+gammaV_(y)=0:}\begin{equation*} V_{x}+\gamma V_{y}=0 \tag{11} \end{equation*}(11)Vx+γVy=0
which represents Szebehely's equation for this special case. The straight lines are traced with arbitrary energy.
Remark 2. The case of a family of straight lines appeared here as a special case in the mathematical reasoning. Another problem, namely that of Darboux integrability, revealed the importance of families of parallel or concurrent lines (Grigoriadou 1999). Isolated straight lines were found for the Hénon-Heiles model by Antonov and Timoshkova (1993) or van der Merwe (1991). Contopoulos and Zikides (1980), as well as Caranicolas and Innanen (1992), identified straight lines in galactic models.
Example 1. The central potential V ( x , y ) = v ( r ) V ( x , y ) = v ( r ) V(x,y)=v(r)V(x, y)=v(r)V(x,y)=v(r), where r = ( x 2 + y 2 ) 1 / 2 r = x 2 + y 2 1 / 2 r=(x^(2)+y^(2))^(1//2)r=\left(x^{2}+y^{2}\right)^{1 / 2}r=(x2+y2)1/2, is compatible with the family of straight lines γ = − x / y γ = − x / y gamma=-x//y\gamma=-x / yγ=−x/y which can be described equivalently by f ( x , y ) = y / x = c f ( x , y ) = y / x = c f(x,y)=y//x=cf(x, y)=y / x=cf(x,y)=y/x=c (Bozis and Anisiu 2001).
Let us consider now a general family (1) with Γ ≠ 0 Γ ≠ 0 Gamma!=0\Gamma \neq 0Γ≠0. In this case we have
(12) y ˙ 2 = − V x + γ V y Γ . (12) y ˙ 2 = − V x + γ V y Γ . {:(12)y^(˙)^(2)=-(V_(x)+gammaV_(y))/(Gamma).:}\begin{equation*} \dot{y}^{2}=-\frac{V_{x}+\gamma V_{y}}{\Gamma} . \tag{12} \end{equation*}(12)y˙2=−Vx+γVyΓ.
We differentiate (10), divide both members by y ˙ y ˙ y^(˙)\dot{y}y˙ and get
(13) γ Γ x − Γ y = y ˙ ( x ˙ y ⃛ − y ˙ x ¨ ) − 3 y ¨ ( x ˙ y ¨ − y ˙ x ¨ ) y ˙ 5 . (13) γ Γ x − Γ y = y ˙ ( x ˙ y ⃛ − y ˙ x ¨ ) − 3 y ¨ ( x ˙ y ¨ − y ˙ x ¨ ) y ˙ 5 . {:(13)gammaGamma_(x)-Gamma_(y)=((y^(˙))((x^(˙))(y^(⃛))-(y^(˙))(x^(¨)))-3(y^(¨))((x^(˙))(y^(¨))-(y^(˙))(x^(¨))))/(y^(˙)^(5)).:}\begin{equation*} \gamma \Gamma_{x}-\Gamma_{y}=\frac{\dot{y}(\dot{x} \dddot{y}-\dot{y} \ddot{x})-3 \ddot{y}(\dot{x} \ddot{y}-\dot{y} \ddot{x})}{\dot{y}^{5}} . \tag{13} \end{equation*}(13)γΓx−Γy=y˙(x˙y⃛−y˙x¨)−3y¨(x˙y¨−y˙x¨)y˙5.
As explained in remark 1, the functions γ γ gamma\gammaγ and Γ Γ Gamma\GammaΓ represent the geometry of the family of curves (1). The formulae (9), (10) and (13) relate these geometrical entities to the kinematics derivatives, namely to the velocity and acceleration of the particle describing the curves of the family.
Two additional equations are obtained by differentiating equations (2) with respect to t t ttt, namely
(14) x ⃛ = − ( V x x x ˙ + V x y y ˙ ) , y ⃛ = − ( V x y x ˙ + V y y y ˙ ) . (14) x ⃛ = − V x x x ˙ + V x y y ˙ , y ⃛ = − V x y x ˙ + V y y y ˙ . {:(14)x^(⃛)=-(V_(xx)(x^(˙))+V_(xy)(y^(˙)))","quady^(⃛)=-(V_(xy)(x^(˙))+V_(yy)(y^(˙))).:}\begin{equation*} \dddot{x}=-\left(V_{x x} \dot{x}+V_{x y} \dot{y}\right), \quad \dddot{y}=-\left(V_{x y} \dot{x}+V_{y y} \dot{y}\right) . \tag{14} \end{equation*}(14)x⃛=−(Vxxx˙+Vxyy˙),y⃛=−(Vxyx˙+Vyyy˙).
Now we eliminate the derivatives x ˙ , y ˙ , x ¨ , y ¨ , x ¨ , y ⃛ x ˙ , y ˙ , x ¨ , y ¨ , x ¨ , y ⃛ x^(˙),y^(˙),x^(¨),y^(¨),x^(¨),y^(⃛)\dot{x}, \dot{y}, \ddot{x}, \ddot{y}, \ddot{x}, \dddot{y}x˙,y˙,x¨,y¨,x¨,y⃛ between the seven relations in (2), (9), (12), (13) and (14), and get
Γ ( − γ V x x + V x y − γ 2 V x y + γ V y y ) = − ( V x + γ V y ) ( γ Γ x − Γ y ) + 3 V y Γ 2 Γ − γ V x x + V x y − γ 2 V x y + γ V y y = − V x + γ V y γ Γ x − Γ y + 3 V y Γ 2 Gamma(-gammaV_(xx)+V_(xy)-gamma^(2)V_(xy)+gammaV_(yy))=-(V_(x)+gammaV_(y))(gammaGamma_(x)-Gamma_(y))+3V_(y)Gamma^(2)\Gamma\left(-\gamma V_{x x}+V_{x y}-\gamma^{2} V_{x y}+\gamma V_{y y}\right)=-\left(V_{x}+\gamma V_{y}\right)\left(\gamma \Gamma_{x}-\Gamma_{y}\right)+3 V_{y} \Gamma^{2}Γ(−γVxx+Vxy−γ2Vxy+γVyy)=−(Vx+γVy)(γΓx−Γy)+3VyΓ2.
This is a differential equation which must be satisfied by all the potentials which admit as trajectories the curves of the family (1). After dividing both members by γ Γ γ Γ gamma Gamma\gamma \GammaγΓ we get Bozis' equation (7), with λ λ lambda\lambdaλ and μ μ mu\muμ given in (8).
A straightforward calculation shows that equation (7) can be written as
(16) γ W x − W y = 0 , (16) γ W x − W y = 0 , {:(16)gammaW_(x)-W_(y)=0",":}\begin{equation*} \gamma W_{x}-W_{y}=0, \tag{16} \end{equation*}(16)γWx−Wy=0,
where
(17) W = V − 1 + γ 2 2 Γ ( V x + γ V y ) (17) W = V − 1 + γ 2 2 Γ V x + γ V y {:(17)W=V-(1+gamma^(2))/(2Gamma)(V_(x)+gammaV_(y)):}\begin{equation*} W=V-\frac{1+\gamma^{2}}{2 \Gamma}\left(V_{x}+\gamma V_{y}\right) \tag{17} \end{equation*}(17)W=V−1+γ22Γ(Vx+γVy)
But γ = f y / f x γ = f y / f x gamma=f_(y)//f_(x)\gamma=f_{y} / f_{x}γ=fy/fx implies f y W x − f x W y = 0 f y W x − f x W y = 0 f_(y)W_(x)-f_(x)W_(y)=0f_{y} W_{x}-f_{x} W_{y}=0fyWx−fxWy=0. This equation has the general solution W = E ( f ) W = E ( f ) W=E(f)W=E(f)W=E(f), where E E EEE denotes an arbitrary function. It follows that
(18) V − 1 + γ 2 2 Γ ( V x + γ V y ) = E ( f ) (18) V − 1 + γ 2 2 Γ V x + γ V y = E ( f ) {:(18)V-(1+gamma^(2))/(2Gamma)(V_(x)+gammaV_(y))=E(f):}\begin{equation*} V-\frac{1+\gamma^{2}}{2 \Gamma}\left(V_{x}+\gamma V_{y}\right)=E(f) \tag{18} \end{equation*}(18)V−1+γ22Γ(Vx+γVy)=E(f)
In view of relations (2), (9) and (10) we obtain
(19) V + x ˙ 2 + y ˙ 2 2 = E ( f ) (19) V + x ˙ 2 + y ˙ 2 2 = E ( f ) {:(19)V+(x^(˙)^(2)+y^(˙)^(2))/(2)=E(f):}\begin{equation*} V+\frac{\dot{x}^{2}+\dot{y}^{2}}{2}=E(f) \tag{19} \end{equation*}(19)V+x˙2+y˙22=E(f)
which means that E ( f ) E ( f ) E(f)E(f)E(f) represents the total energy, constant on each curve of the family (1). Therefore equation (18), obtained this time from Bozis' equation, is in fact Szebehely's equation. From (19) we obtain E ( f ) − V ⩾ 0 E ( f ) − V ⩾ 0 E(f)-V >= 0E(f)-V \geqslant 0E(f)−V⩾0, and from (18) it follows that only the curves of the family (1) or parts of them which are situated in the plane region (6) can be described by the unit mass particle.
Example 2. For the family of homocentric circles
(20) f ( x , y ) = x 2 + y 2 = c (20) f ( x , y ) = x 2 + y 2 = c {:(20)f(x","y)=x^(2)+y^(2)=c:}\begin{equation*} f(x, y)=x^{2}+y^{2}=c \tag{20} \end{equation*}(20)f(x,y)=x2+y2=c
and arbitrary energy E ( r ) ( r = x 2 + y 2 ) E ( r ) r = x 2 + y 2 E(r)(r=sqrt(x^(2)+y^(2)))E(r)\left(r=\sqrt{x^{2}+y^{2}}\right)E(r)(r=x2+y2), Broucke and Lass (1977) have found the general solution, in polar coordinates r , θ r , θ r,thetar, \thetar,θ, of Szebehely's equation (5) as
(21) V ( r , θ ) = g ( r ) + 1 r 2 h ( θ ) , (21) V ( r , θ ) = g ( r ) + 1 r 2 h ( θ ) , {:(21)V(r","theta)=g(r)+(1)/(r^(2))h(theta)",":}\begin{equation*} V(r, \theta)=g(r)+\frac{1}{r^{2}} h(\theta), \tag{21} \end{equation*}(21)V(r,θ)=g(r)+1r2h(θ),
with g g ggg and h h hhh arbitrary functions of their arguments. The energy in this case is E = g ( r ) + r g ′ ( r ) / 2 E = g ( r ) + r g ′ ( r ) / 2 E=g(r)+rg^(')(r)//2E=g(r)+ r g^{\prime}(r) / 2E=g(r)+rg′(r)/2, and inequality (6) becomes g ′ ( r ) ⩾ 2 h ( θ ) / r 3 g ′ ( r ) ⩾ 2 h ( θ ) / r 3 g^(')(r) >= 2h(theta)//r^(3)g^{\prime}(r) \geqslant 2 h(\theta) / r^{3}g′(r)⩾2h(θ)/r3. A special case of (21) is the Newtonian potential V = − 1 / r V = − 1 / r V=-1//rV=-1 / rV=−1/r (with g ( r ) = − 1 / r g ( r ) = − 1 / r g(r)=-1//rg(r)=-1 / rg(r)=−1/r and h ( θ ) = 0 h ( θ ) = 0 h(theta)=0h(\theta)=0h(θ)=0 ), under whose action the circles (20) are traced all over the plane. Another central potential compatible with the family (20) on the entire plane is Maneff's (1924) potential V = − 1 / r − α / r 2 V = − 1 / r − α / r 2 V=-1//r-alpha//r^(2)V=-1 / r-\alpha / r^{2}V=−1/r−α/r2 (with g ( r ) = − 1 / r g ( r ) = − 1 / r g(r)=-1//rg(r)=-1 / rg(r)=−1/r and h ( θ ) = − α , α > 0 h ( θ ) = − α , α > 0 h(theta)=-alpha,alpha > 0h(\theta)=-\alpha, \alpha>0h(θ)=−α,α>0 ).
The special form of the family in example 2 allowed the integration of Szebehely's equation; when we do not have at our disposal information on the energy, Bozis' equation is more suitable. It was used, e.g., by Anisiu and Pal (1999) to find out of the Hénon-Heiles type potentials
(22) V ( x , y ) = x 2 + a 1 y 2 + a 2 x 2 y + a 3 y 3 , a 1 , a 2 , a 3 ∈ R , a 1 > 0 (22) V ( x , y ) = x 2 + a 1 y 2 + a 2 x 2 y + a 3 y 3 , a 1 , a 2 , a 3 ∈ R , a 1 > 0 {:(22)V(x","y)=x^(2)+a_(1)y^(2)+a_(2)x^(2)y+a_(3)y^(3)","quada_(1)","a_(2)","a_(3)inR","quada_(1) > 0:}\begin{equation*} V(x, y)=x^{2}+a_{1} y^{2}+a_{2} x^{2} y+a_{3} y^{3}, \quad a_{1}, a_{2}, a_{3} \in \mathbb{R}, \quad a_{1}>0 \tag{22} \end{equation*}(22)V(x,y)=x2+a1y2+a2x2y+a3y3,a1,a2,a3∈R,a1>0
those which are compatible with a family of polytropic curves. This kind of potential was introduced by Hénon and Heiles (1964) as a model for the motion of a star in a galaxy; it can be used to represent the gravitational field of the Earth, other planets and their satellites (Agekian 2003).
Example 3. The curves of the family
(23) f ( x , y ) = x − 4 y = c (23) f ( x , y ) = x − 4 y = c {:(23)f(x","y)=x^(-4)y=c:}\begin{equation*} f(x, y)=x^{-4} y=c \tag{23} \end{equation*}(23)f(x,y)=x−4y=c
can be traced by a unit mass particle under the action of the potential
V ( x , y ) = x 2 + 16 y 2 + a 2 x 2 y + ( 16 / 3 ) a 2 y 3 V ( x , y ) = x 2 + 16 y 2 + a 2 x 2 y + ( 16 / 3 ) a 2 y 3 V(x,y)=x^(2)+16y^(2)+a_(2)x^(2)y+(16//3)a_(2)y^(3)V(x, y)=x^{2}+16 y^{2}+a_{2} x^{2} y+(16 / 3) a_{2} y^{3}V(x,y)=x2+16y2+a2x2y+(16/3)a2y3
with the energy E ( f ) = − a 2 / ( 24 f ) E ( f ) = − a 2 / ( 24 f ) E(f)=-a_(2)//(24 f)E(f)=-a_{2} /(24 f)E(f)=−a2/(24f), in the region described by the inequality ( a 2 ( x 2 + 8 y 2 ) + 24 y ) y ⩽ 0 a 2 x 2 + 8 y 2 + 24 y ) y ⩽ 0 (a_(2)(x^(2)+8y^(2))+:}24 y)y <= 0\left(a_{2}\left(x^{2}+8 y^{2}\right)+\right. 24 y) y \leqslant 0(a2(x2+8y2)+24y)y⩽0. This result can be obtained by inserting γ = − x / ( 4 y ) , Γ = − 3 x / ( 16 y 2 ) γ = − x / ( 4 y ) , Γ = − 3 x / 16 y 2 gamma=-x//(4y),Gamma=-3x//(16y^(2))\gamma=-x /(4 y), \Gamma=-3 x /\left(16 y^{2}\right)γ=−x/(4y),Γ=−3x/(16y2) and V V VVV from (22) in equation (7), and selecting adequately the coefficients in V V VVV. Afterwards the energy is determined from Szebehely's equation (5) and the allowed region from (6).
Remark 3. As expected, the general solution of the second-order equation (7) will depend on two arbitrary functions; the same situation occurs for equation (5), one arbitrary function being the energy. So, even if the general solution cannot be found, sometimes it is useful to look for the potential in certain classes of functions (e.g. homogeneous (Borghero and Bozis 2002), or quasihomogeneous, as in example 3). Several pairs ( f , V f , V f,Vf, Vf,V ) can be found in the papers of Bozis (1995), Anisiu (2003a) and in the references therein.

3. The case of a general force field

Bertrand (1877) raised the problem of finding the force, not necessarily conservative, depending merely on the position ( x , y ) ( x , y ) (x,y)(x, y)(x,y) of the planets moving on conic sections under the action of that force. Dainelli (1880) solved the problem of Bernard for arbitrary families of curves (1) and obtained, using different notation, formulae similar to (32) and (33). In what follows we derive a partial differential equation satisfied by the force components, and find the region where real motion is possible; finally we provide the formulae for the components of the most general force which is compatible with the family of curves (1). These formulae can be useful whenever the force field is not supposed a priori to be conservative. The advantage of working with general force fields is that we do not have to integrate partial differential equations, because we dispose of formulae (32) and (33).
We apply the procedure in section 2 for the system
(24) x ¨ = X , y ¨ = Y , (24) x ¨ = X , y ¨ = Y , {:(24)x^(¨)=X","quady^(¨)=Y",":}\begin{equation*} \ddot{x}=X, \quad \ddot{y}=Y, \tag{24} \end{equation*}(24)x¨=X,y¨=Y,
the force components X X XXX and Y Y YYY being of C 1 C 1 C^(1)C^{1}C1-class on a domain of the plane x y x y xyx yxy. If the family (1) consists of straight lines ( Γ = 0 Γ = 0 Gamma=0\Gamma=0Γ=0 ), instead of (11) we obtain
(25) X + γ Y = 0 , (25) X + γ Y = 0 , {:(25)X+gamma Y=0",":}\begin{equation*} X+\gamma Y=0, \tag{25} \end{equation*}(25)X+γY=0,
this being the relation satisfied by the components of the force field in this special case.
Let us consider now a general family (1) with Γ ≠ 0 Γ ≠ 0 Gamma!=0\Gamma \neq 0Γ≠0. Instead of (12) we have this time
(26) y ˙ 2 = X + γ Y Γ . (26) y ˙ 2 = X + γ Y Γ . {:(26)y^(˙)^(2)=(X+gamma Y)/(Gamma).:}\begin{equation*} \dot{y}^{2}=\frac{X+\gamma Y}{\Gamma} . \tag{26} \end{equation*}(26)y˙2=X+γYΓ.
The differentiation of equations (24) with respect to t t ttt gives
(27) x ⃛ = X x x ˙ + X y y ˙ , y ¨ = Y x x ˙ + Y y y ˙ . (27) x ⃛ = X x x ˙ + X y y ˙ , y ¨ = Y x x ˙ + Y y y ˙ . {:(27)x^(⃛)=X_(x)x^(˙)+X_(y)y^(˙)","quady^(¨)=Y_(x)x^(˙)+Y_(y)y^(˙).:}\begin{equation*} \dddot{x}=X_{x} \dot{x}+X_{y} \dot{y}, \quad \ddot{y}=Y_{x} \dot{x}+Y_{y} \dot{y} . \tag{27} \end{equation*}(27)x⃛=Xxx˙+Xyy˙,y¨=Yxx˙+Yyy˙.
The elimination of the derivatives of x x xxx and y y yyy between the seven relations in (24), (9), (26), (13) and (27) leads to
(28) Γ ( γ X x − X y + γ 2 Y x − γ Y y ) = ( X + γ Y ) ( γ Γ x − Γ y ) − 3 Y Γ 2 , (28) Γ γ X x − X y + γ 2 Y x − γ Y y = ( X + γ Y ) γ Γ x − Γ y − 3 Y Γ 2 , {:(28)Gamma(gammaX_(x)-X_(y)+gamma^(2)Y_(x)-gammaY_(y))=(X+gamma Y)(gammaGamma_(x)-Gamma_(y))-3YGamma^(2)",":}\begin{equation*} \Gamma\left(\gamma X_{x}-X_{y}+\gamma^{2} Y_{x}-\gamma Y_{y}\right)=(X+\gamma Y)\left(\gamma \Gamma_{x}-\Gamma_{y}\right)-3 Y \Gamma^{2}, \tag{28} \end{equation*}(28)Γ(γXx−Xy+γ2Yx−γYy)=(X+γY)(γΓx−Γy)−3YΓ2,
a differential relation satisfied by the force field in order to admit as trajectories the curves of the family (1). After dividing both members by γ Γ γ Γ gamma Gamma\gamma \GammaγΓ we get
(29) − X x + 1 γ X y − γ Y x + Y y = λ X + μ Y , (29) − X x + 1 γ X y − γ Y x + Y y = λ X + μ Y , {:(29)-X_(x)+(1)/(gamma)X_(y)-gammaY_(x)+Y_(y)=lambda X+mu Y",":}\begin{equation*} -X_{x}+\frac{1}{\gamma} X_{y}-\gamma Y_{x}+Y_{y}=\lambda X+\mu Y, \tag{29} \end{equation*}(29)−Xx+1γXy−γYx+Yy=λX+μY,
where λ λ lambda\lambdaλ and μ μ mu\muμ are given in (8). This equation was obtained by Bozis (1983), using a different method. From (26) it follows that the motion of the particle is possible only in the plane region (Bozis 1994) described by the inequality
(30) X + γ Y Γ ⩾ 0 . (30) X + γ Y Γ ⩾ 0 . {:(30)(X+gamma Y)/(Gamma) >= 0.:}\begin{equation*} \frac{X+\gamma Y}{\Gamma} \geqslant 0 . \tag{30} \end{equation*}(30)X+γYΓ⩾0.
It is obvious that Bozis' equation (7) and the inequality (6) found by Bozis and Ichtiaroglou (1994) follow from (29), respectively from (30), after replacing X = − V x X = − V x X=-V_(x)X=-V_{x}X=−Vx and Y = − V y Y = − V y Y=-V_(y)Y=-V_{y}Y=−Vy.
We remark that, if we denote by
(31) ξ = X + γ Y Γ , (31) ξ = X + γ Y Γ , {:(31)xi=(X+gamma Y)/(Gamma)",":}\begin{equation*} \xi=\frac{X+\gamma Y}{\Gamma}, \tag{31} \end{equation*}(31)ξ=X+γYΓ,
equation (29) can be written as γ ξ x − ξ y = − 2 Y γ ξ x − ξ y = − 2 Y gammaxi_(x)-xi_(y)=-2Y\gamma \xi_{x}-\xi_{y}=-2 Yγξx−ξy=−2Y, or
(32) Y = − 1 2 γ ξ x + 1 2 ξ y . (32) Y = − 1 2 γ ξ x + 1 2 ξ y . {:(32)Y=-(1)/(2)gammaxi_(x)+(1)/(2)xi_(y).:}\begin{equation*} Y=-\frac{1}{2} \gamma \xi_{x}+\frac{1}{2} \xi_{y} . \tag{32} \end{equation*}(32)Y=−12γξx+12ξy.
From relation (31) we get then
(33) X = 1 2 γ 2 ξ x − 1 2 γ ξ y + Γ ξ . (33) X = 1 2 γ 2 ξ x − 1 2 γ ξ y + Γ ξ . {:(33)X=(1)/(2)gamma^(2)xi_(x)-(1)/(2)gammaxi_(y)+Gamma xi.:}\begin{equation*} X=\frac{1}{2} \gamma^{2} \xi_{x}-\frac{1}{2} \gamma \xi_{y}+\Gamma \xi . \tag{33} \end{equation*}(33)X=12γ2ξx−12γξy+Γξ.
Therefore for an arbitrary positive function ξ ξ xi\xiξ we obtain the components of the force given by (33) and (32), which were found by a different method by Bozis (1983).
Example 4. For the monoparametric family
(34) f ( x , y ) = x − x 2 − y 2 x 2 + y 2 = c (34) f ( x , y ) = x − x 2 − y 2 x 2 + y 2 = c {:(34)f(x","y)=x-(x^(2)-y^(2))/(sqrt(x^(2)+y^(2)))=c:}\begin{equation*} f(x, y)=x-\frac{x^{2}-y^{2}}{\sqrt{x^{2}+y^{2}}}=c \tag{34} \end{equation*}(34)f(x,y)=x−x2−y2x2+y2=c
we obtain from (4)
γ = y ( 3 x 2 + y 2 ) ( x 2 + y 2 ) 3 − x ( x 2 + 3 y 2 ) and Γ = γ γ x − γ y . γ = y 3 x 2 + y 2 x 2 + y 2 3 − x x 2 + 3 y 2  and  Γ = γ γ x − γ y . gamma=(y(3x^(2)+y^(2)))/(sqrt((x^(2)+y^(2))^(3))-x(x^(2)+3y^(2)))quad" and "quad Gamma=gammagamma_(x)-gamma_(y).\gamma=\frac{y\left(3 x^{2}+y^{2}\right)}{\sqrt{\left(x^{2}+y^{2}\right)^{3}}-x\left(x^{2}+3 y^{2}\right)} \quad \text { and } \quad \Gamma=\gamma \gamma_{x}-\gamma_{y} .γ=y(3x2+y2)(x2+y2)3−x(x2+3y2) and Γ=γγx−γy.
For an arbitrary function ξ ξ xi\xiξ we get from formulae (33) and (32) the components X , Y X , Y X,YX, YX,Y of the force compatible with the family (34).
Specifying the following value of the arbitrary function
ξ = ( x ( x 2 + y 2 ) 3 − x 4 + y 4 ) ( x 3 + 3 x y 2 − ( x 2 + y 2 ) 3 ) 2 ( x 2 + y 2 ) 7 { 2 x 4 + ( y 2 − x 2 ) ( 2 x x 2 + y 2 + y 2 ) } , ξ = x x 2 + y 2 3 − x 4 + y 4 x 3 + 3 x y 2 − x 2 + y 2 3 2 x 2 + y 2 7 2 x 4 + y 2 − x 2 2 x x 2 + y 2 + y 2 , xi=((xsqrt((x^(2)+y^(2))^(3))-x^(4)+y^(4))(x^(3)+3xy^(2)-sqrt((x^(2)+y^(2))^(3)))^(2))/(sqrt((x^(2)+y^(2))^(7)){2x^(4)+(y^(2)-x^(2))(2xsqrt(x^(2)+y^(2))+y^(2))}),\xi=\frac{\left(x \sqrt{\left(x^{2}+y^{2}\right)^{3}}-x^{4}+y^{4}\right)\left(x^{3}+3 x y^{2}-\sqrt{\left(x^{2}+y^{2}\right)^{3}}\right)^{2}}{\sqrt{\left(x^{2}+y^{2}\right)^{7}}\left\{2 x^{4}+\left(y^{2}-x^{2}\right)\left(2 x \sqrt{x^{2}+y^{2}}+y^{2}\right)\right\}},ξ=(x(x2+y2)3−x4+y4)(x3+3xy2−(x2+y2)3)2(x2+y2)7{2x4+(y2−x2)(2xx2+y2+y2)},
we obtain the obviously nonconservative force with components
(35) X = x ( y 2 − x 2 ) ( x 2 + y 2 ) 5 and Y = y ( y 2 − x 2 ) ( x 2 + y 2 ) 5 . (35) X = x y 2 − x 2 x 2 + y 2 5  and  Y = y y 2 − x 2 x 2 + y 2 5 . {:(35)X=(x(y^(2)-x^(2)))/(sqrt((x^(2)+y^(2))^(5)))quad" and "quad Y=(y(y^(2)-x^(2)))/(sqrt((x^(2)+y^(2))^(5))).:}\begin{equation*} X=\frac{x\left(y^{2}-x^{2}\right)}{\sqrt{\left(x^{2}+y^{2}\right)^{5}}} \quad \text { and } \quad Y=\frac{y\left(y^{2}-x^{2}\right)}{\sqrt{\left(x^{2}+y^{2}\right)^{5}}} . \tag{35} \end{equation*}(35)X=x(y2−x2)(x2+y2)5 and Y=y(y2−x2)(x2+y2)5.
The force (35) was considered by Borghero et al (1999) in view of the direct problem; they proved its compatibility with the family (34).

4. Conclusions

We assert that Szebehely's and Bozis' equations are of equal importance for the inverse problem attached to a family (1) and a system (2); when we have no a priori information on the energy, it is useful (and fully justified) to start working with equation (7) and then to obtain the energy from equation (18).
We have derived the basic equations of the inverse problem in a simple and natural way, by a process of elimination of the time derivatives of x x xxx and y y yyy. Doing so, the case of families of straight lines presented its particularities and the allowed region emerged.
This unifying consideration of conservative and general force field systems explains also the connection (already mentioned by Bozis (1995)) between Bozis' equation and the differential relation (29).

Acknowledgments

I express my thanks to an unknown referee for carefully reading the paper and for making several useful suggestions.

References

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2004

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