Two and three-dimensional inverse problem of dynamics

Mira Anisiu
(original), paper

Abstract

For a given a monoparametric family of curves \(f(x,y)=c\), we present the partial differential equations satisfied by the potentials \(V=V(x,y)\) under whose action a particle of unit mass can describe the curves of the family. Szebehely’s equation depends on the total energy of the particle, while Bozis one relates merely the potential and the given family. Therefore the last one is also adequate for the direct problem of dynamics. A similar program is accomplished for a two-parametric spatial family of curves \(\varphi(x,y,z)=c_{1},\U{3c8} (x,y,z)=c_{2}\) and potentials \(V=V(x,y,z)\).

Authors

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

Keywords

Planar and spatial inverse problem of dynamics; energy-free equations.

Paper coordinates

M.-C. Anisiu, Two and three-dimensional inverse problem of dynamics, Studia Univ. Babeş-Bolyai, Math. XLIX (4) (2004), 13-26

PDF

https://www.cs.ubbcluj.ro/journal/studia-mathematica/archive/2004-4/anisiu.pdf

About this paper

Journal

Mathematica

Publisher Name

Studia Universitatis Babes-Bolyai Cluj-Napoca

DOI
Print ISSN

 0252-1938

Online ISSN

2065-961X

google scholar link

[1] M. C. Anisiu, PDEs in the inverse problem of dynamics, in V. Barbu et al (eds.), Analysis and Optimization of Differential Systems, Kluwer Academic Publishers 2003, 13-20
[2] M. C. Anisiu, The Equations of the Inverse Problem of Dynamics, House of the Book of Science, Cluj-Napoca 2003 (in Romanian).
[3] M. C. Anisiu, An alternative point of view on the equations of the inverse problem of dynamics, Inverse Problems 20 (2004), 1865-1872.
[4] M. C. Anisiu, , Energy-free equations of the 3D inverse problem of dynamics, 2004, submitted.
[5] M. C. Anisiu, A. Pal, Special families of orbits for the H´enon-Heiles type potential, Rom. Astron. J. 9 (1999), 179-185.
[6] M. C. Anisiu, C. Blaga, and G. Bozis, Special families of orbits in the direct problem of dynamics, Celest. Mech. Dyn. Astron. 88 (2004), 245-257.
[7] J. Bertrand, Sur la possibilit´e de d´eduire d’une seule des lois de Kepler le principe de l’attraction, Compt. Rend. 84 (1877), 671-673.
[8] G. Bozis, Inverse problem with two-parametric families of planar orbits, Celest. Mech. 31 (1983), 129-143.
[9] G. Bozis, Szebehely inverse problem for finite symmetrical material concentrations, Astron. Astrophys. 134(1984), 360-364.
[10] G. Bozis, The inverse problem of dynamics: basic facts, Inverse Problems, 11 (1995), 687-708.
[11] G. Bozis, and M. C. Anisiu, Families of straight lines in planar potentials, Rom. Astron. J. 11(2001), 2
[12] G. Bozis, and S. Ichtiaroglou, Boundary curves for families of planar orbits, Celest. Mech. Dyn. Astron., 58 (1994), 371-385.
[13] G. Bozis, and T. A. Kotoulas, Families of straight lines (FSL) in three-dimensional potentials, 2004, private communication.
[14] G. Bozis, and T. A. Kotoulas, Homogeneous two-parametric families of orbits in threedimensional homogeneous potentials, 2004, private communication.
[15] G. Bozis, and A. Nakhla, Solutions of the three-dimensional inverse problem, Celest. Mech. Dyn. Astron. 38(1986), 357-375.
[16] G. Bozis, M. C. Anisiu, and C. Blaga, Inhomogeneous potentials producing homogeneous orbits, Astron. Nachr. 318(1997), 313-318.
[17] G. Bozis, M. C. Anisiu, and C. Blaga, A solvable version of the direct problem of dynamics, Rom. Astron. J. 10(2000), 59-70.
[18] U. Dainelli, Sul movimento per una linea qualunque, G. Mat. 18(1880), 271-300.
[19] B. Erdi, A generalization of Szebehely’s equation for three dimensions,  Celest. Mech. 28(1982), 209-218.
[20] J. Favard, Cours d’Analyse de l’Ecole Polytechnique, Tome 32, p. 39, Gauthier-Villars,
Paris 1963.
[21] F. Gonzales-Gascon, A. Gonzales-Lopez, and P. J. Pascual-Broncano, On Szebehely’s equation and its connection with Dainelli’s-Whittaker’s equations, Celest. Mech. 33(1984), 85-97.
[22] E. Kasner, The trajectories of dynamics, Trans. Amer. Math. Soc. 7(1906), 401-424.
[23] E. Kasner, Dynamical trajectories: the motion of the particle in an arbitrary field of force, Trans. Amer. Math. Soc. 8(1907), 135-158.
[24] I. Newton, Philosophiae Naturalis Principia Mathematica, London 1687.
[25] F. Puel, Equations de Szebehely et principes variationnels,  Celest. Mech. 32(1984), 349-353.
[26] F. Puel, Formulation intrinseque de l’´equation de Szebehely, Celest. Mech. 32(1984), 209-212.
[27] F. Puel, Explicit solutions of the three-dimensional inverse problem of Dynamics, using the Frenet reference frame, Celest. Mech. Dyn. Astron. 53(1992), 207-218.
[28] S. G. Shorokhov, , Solution of an inverse problem of the Dynamics of a particle, Celest. Mech. Dyn. Astron. 44(1988), 193-206.
[29] V. Szebehely, On the determination of the potential by satellite observations, in G. Proverbio (ed.), Proc. of the Int. Meeting on Earth’s Rotation by Satellite Observation, The Univ. of Cagliari, Bologna, Italy 1974, 31-35.
[30] F. Varadi, and B. Erdi, Existence and solution of Szebehely’s equation in three dimension using a two-parametric family of orbits, Celest. Mech. 30(1983), 395-405.

2004-Anisiu-Studia

TWO- AND THREE-DIMENSIONAL INVERSE PROBLEM OF DYNAMICS

MIRA-CRISTIANA ANISIU

Abstract

For a given a monoparametric family of curves f ( x , y ) = c f ( x , y ) = c f(x,y)=cf(x, y)=cf(x,y)=c, we present the partial differential equations satisfied by the potentials V = V ( x , y ) V = V ( x , y ) V=V(x,y)V= V(x, y)V=V(x,y) under whose action a particle of unit mass can describe the curves of the family. Szebehely's equation depends on the total energy of the particle, while Bozis' one relates merely the potential and the given family. Therefore the last one is also adequate for the direct problem of dynamics. A similar program is accomplished for a two-parametric spatial family of curves φ ( x , y , z ) = c 1 , ψ ( x , y , z ) = c 2 φ ( x , y , z ) = c 1 , ψ ( x , y , z ) = c 2 varphi(x,y,z)=c_(1),psi(x,y,z)=c_(2)\varphi(x, y, z)=c_{1}, \psi(x, y, z)=c_{2}φ(x,y,z)=c1,ψ(x,y,z)=c2 and potentials V = V ( x , y , z ) V = V ( x , y , z ) V=V(x,y,z)\mathcal{V}=\mathcal{V}(x, y, z)V=V(x,y,z).

1. Introduction

The first result concerning the inverse problem of dynamics is due to Newton [24], who presented the form of the gravitational potential on the basis of Kepler's laws. Kepler has had at his disposal the very accurate tables of observations made by Tycho Brache (whose assistant he was in Prague); these observations allowed him to discover that the orbit of Mars is an ellipse and to formulate the three laws of planetary motion.
Later on, Bertrand [7] showed that Kepler's first law suffices to derive the Newtonian universal force; Dainelli [18] obtained the expressions of general force fields producing given planar or spatial families of curves.
The two-dimensional problem, this time for conservative systems, has renewed the interest in the inverse problem of dynamics by means of Szebehely's [29] partial differential equation. This equation relates the potential to the given monoparametric family of curves and to the total energy. Puel [26] derived a Szebehely-type equation which is independent of the coordinate system. Another basic result for the twodimensional inverse problem is the energy-free partial differential equation obtained by Bozis [9] from Szebehely's equation, and later derived directly by Anisiu [3].
The conservative three-dimensional problem was considered by Érdi [19] for a monoparametric family of orbits, and then for two-parametric families by Váradi and Érdi [30]. Puel [25] used the least action principle of Maupertuis to obtain the equations satisfied by the potential in the two- and three-dimensional inverse problem of dynamics. The existence of such a potential and its relation with the energy in the three-dimensional case was subject to further papers, as those of Gonzales-Gascon et al [21], Bozis and Nakhla [15] and Shorokhov [28]. Puel [27] obtained the intrinsic equations of the three-dimensional inverse problem, using the Frenet reference frame. A review of the basic results in the inverse problem of dynamics, including the threedimensional ones, can be found in [10].

2. The planar inverse problem of dynamics

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 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¨=−Vxy¨=−Vy,
where the dots denote derivatives with respect to the time t t ttt and the subscripts partial derivatives. By making use of the energy integral, Szebehely [29] proved that the potential V V VVV is a solution of 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). Bozis [8] wrote Szebehely's equation in the simpler form
(4) V x + γ V y + 2 Γ ( E ( f ) − V ) 1 + γ 2 = 0 (4) V x + γ V y + 2 Γ ( E ( f ) − V ) 1 + γ 2 = 0 {:(4)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{4} \end{equation*}(4)Vx+γVy+2Γ(E(f)−V)1+γ2=0
making use of the functions
(5) γ = f y f x and Γ = γ γ x − γ y (5) γ = f y f x  and  Γ = γ γ x − γ y {:(5)gamma=(f_(y))/(f_(x))" and "Gamma=gammagamma_(x)-gamma_(y):}\begin{equation*} \gamma=\frac{f_{y}}{f_{x}} \text { and } \Gamma=\gamma \gamma_{x}-\gamma_{y} \tag{5} \end{equation*}(5)γ=fyfx and Γ=γγx−γy
related to the geometry of the family ( γ γ gamma\gammaγ representing the slope and Γ Γ Gamma\GammaΓ being proportional to the curvature). By eliminating the energy from (4) (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 [9] obtained the energy-free equation of second order
(6) − V x x + κ V x y + V y y = λ V x + μ V y (6) − V x x + κ V x y + V y y = λ V x + μ V y {:(6)-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{6} \end{equation*}(6)−Vxx+κVxy+Vyy=λVx+μVy
where
(7) κ = 1 γ − γ , λ = Γ y − γ Γ x γ Γ , μ = λ γ + 3 Γ γ . (7) κ = 1 γ − γ , λ = Γ y − γ Γ x γ Γ , μ = λ γ + 3 Γ γ . {:(7)kappa=(1)/(gamma)-gamma","lambda=(Gamma_(y)-gammaGamma_(x))/(gamma Gamma)","mu=lambda gamma+(3Gamma)/(gamma).:}\begin{equation*} \kappa=\frac{1}{\gamma}-\gamma, \lambda=\frac{\Gamma_{y}-\gamma \Gamma_{x}}{\gamma \Gamma}, \mu=\lambda \gamma+\frac{3 \Gamma}{\gamma} . \tag{7} \end{equation*}(7)κ=1γ−γ,λ=Γy−γΓxγΓ,μ=λγ+3Γγ.
The basic equations (4) and (6) of the planar inverse problem of dynamics present the connection between geometry and dynamics. Their derivation and other related results are exposed in [10], [2], [1], [3].
Szebehely obtained the first order equation intending to determine the potential of the earth by means of satellite observations, while Bozis used equation (6) to check if a given family of orbits may be generated in the plane of symmetry outside a material concentration.
2.1. Basic tools. 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 exposed by Anisiu [3], related to that followed by Kasner [22] while he has 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 notation (5),
(8) γ = − x ˙ y ˙ (8) γ = − x ˙ y ˙ {:(8)gamma=-((x^(˙)))/((y^(˙))):}\begin{equation*} \gamma=-\frac{\dot{x}}{\dot{y}} \tag{8} \end{equation*}(8)γ=−x˙y˙
By differentiating (8) we get
(9) − Γ = x ˙ y ¨ − y ˙ x ¨ y ˙ 3 (9) − Γ = x ˙ y ¨ − y ˙ x ¨ y ˙ 3 {:(9)-Gamma=((x^(˙))(y^(¨))-(y^(˙))(x^(¨)))/(y^(˙)^(3)):}\begin{equation*} -\Gamma=\frac{\dot{x} \ddot{y}-\dot{y} \ddot{x}}{\dot{y}^{3}} \tag{9} \end{equation*}(9)−Γ=x˙y¨−y˙x¨y˙3
Inserting in (9) x ¨ x ¨ x^(¨)\ddot{x}x¨ and y ¨ y ¨ y^(¨)\ddot{y}y¨ from (2), and x ˙ x ˙ x^(˙)\dot{x}x˙ from (8) 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).
The function Γ Γ Gamma\GammaΓ is related to the curvature K K KKK of the family (1) by K = | Γ | / ( γ 2 + 1 ) 3 / 2 K = | Γ | / γ 2 + 1 3 / 2 K=|Gamma|//(gamma^(2)+1)^(3//2)K= |\Gamma| /\left(\gamma^{2}+1\right)^{3 / 2}K=|Γ|/(γ2+1)3/2. It follows that Γ = 0 Γ = 0 Gamma=0\Gamma=0Γ=0 if and only if the family (1) contains only straight lines. In this case, which was studied in [11], we have by necessity
(10) V x + γ V y = 0 (10) V x + γ V y = 0 {:(10)V_(x)+gammaV_(y)=0:}\begin{equation*} V_{x}+\gamma V_{y}=0 \tag{10} \end{equation*}(10)Vx+γVy=0
which represents Szebehely's equation for this special case. The straight lines are traced with arbitrary energy.
Let us consider now a general family (1) with Γ ≠ 0 Γ ≠ 0 Gamma!=0\Gamma \neq 0Γ≠0. In this case we have
(11) y ˙ 2 = − V x + γ V y Γ (11) y ˙ 2 = − V x + γ V y Γ {:(11)y^(˙)^(2)=-(V_(x)+gammaV_(y))/(Gamma):}\begin{equation*} \dot{y}^{2}=-\frac{V_{x}+\gamma V_{y}}{\Gamma} \tag{11} \end{equation*}(11)y˙2=−Vx+γVyΓ
We differentiate (9), divide both members by y ˙ y ˙ y^(˙)\dot{y}y˙ and get
(12) γ Γ x − Γ y = y ˙ ( x ˙ y ⃛ − y ¨ x ⃛ ) − 3 y ¨ ( x ˙ y ¨ − y ˙ x ¨ ) y ˙ 5 (12) γ Γ x − Γ y = y ˙ ( x ˙ y ⃛ − y ¨ x ⃛ ) − 3 y ¨ ( x ˙ y ¨ − y ˙ x ¨ ) y ˙ 5 {:(12)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}-\ddot{y} \dddot{x})-3 \ddot{y}(\dot{x} \ddot{y}-\dot{y} \ddot{x})}{\dot{y}^{5}} \tag{12} \end{equation*}(12)γΓx−Γy=y˙(x˙y⃛−y¨x⃛)−3y¨(x˙y¨−y˙x¨)y˙5
We remark that (8), (9) and (12) express the relations between the geometry of the family of curves (1) and the kinematics derivatives.
Two additional equations are obtained by differentiating equations (2) with respect to t t ttt, namely
(13) x ⃛ = − ( V x x x ˙ + V x y y ˙ ) y ⃛ = − ( V x y x ˙ + V y y y ˙ ) . (13) x ⃛ = − V x x x ˙ + V x y y ˙ y ⃛ = − V x y x ˙ + V y y y ˙ . {:[(13)x^(⃛)=-(V_(xx)(x^(˙))+V_(xy)(y^(˙)))],[y^(⃛)=-(V_(xy)(x^(˙))+V_(yy)(y^(˙))).]:}\begin{align*} & \dddot{x}=-\left(V_{x x} \dot{x}+V_{x y} \dot{y}\right) \tag{13}\\ & \dddot{y}=-\left(V_{x y} \dot{x}+V_{y y} \dot{y}\right) . \end{align*}(13)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}, \dddot{x}, \dddot{y}x˙,y˙,x¨,y¨,x⃛,y⃛ between the seven relations in (2), (8), (11), (12) and (13), and get the partial differential equation
(14) 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 . (14) 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 . {:(14)Gamma(-gammaV_(xx)+V_(xy)-gamma^(2)V_(xy)+gammaV_(yy))=-(V_(x)+gammaV_(y))(gammaGamma_(x)-Gamma_(y))+3V_(y)Gamma^(2).:}\begin{equation*} \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} . \tag{14} \end{equation*}(14)Γ(−γVxx+Vxy−γ2Vxy+γVyy)=−(Vx+γVy)(γΓx−Γy)+3VyΓ2.
We divide both members of (14) by γ Γ γ Γ gamma Gamma\gamma \GammaγΓ and obtain Bozis' equation (6), with λ λ lambda\lambdaλ and μ μ mu\muμ given in (7).
A straightforward calculation shows that equation (6) can be written as
(15) γ W x − W y = 0 , (15) γ W x − W y = 0 , {:(15)gammaW_(x)-W_(y)=0",":}\begin{equation*} \gamma W_{x}-W_{y}=0, \tag{15} \end{equation*}(15)γWx−Wy=0,
where
(16) W = V − 1 + γ 2 2 Γ ( V x + γ V y ) . (16) W = V − 1 + γ 2 2 Γ V x + γ V y . {:(16)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{16} \end{equation*}(16)W=V−1+γ22Γ(Vx+γVy).
Equation (15) 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
(17) V − 1 + γ 2 2 Γ ( V x + γ V y ) = E ( f ) . (17) V − 1 + γ 2 2 Γ V x + γ V y = E ( f ) . {:(17)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{17} \end{equation*}(17)V−1+γ22Γ(Vx+γVy)=E(f).
In view of relations (2), (8) and (9) we obtain
(18) V + x ˙ 2 + y ˙ 2 2 = E ( f ) , (18) V + x ˙ 2 + y ˙ 2 2 = E ( f ) , {:(18)V+(x^(˙)^(2)+y^(˙)^(2))/(2)=E(f)",":}\begin{equation*} V+\frac{\dot{x}^{2}+\dot{y}^{2}}{2}=E(f), \tag{18} \end{equation*}(18)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 (17), obtained this time from Bozis' equation, is in fact Szebehely's equation. From (18) we obtain E ( f ) − V ≥ 0 E ( f ) − V ≥ 0 E(f)-V >= 0E(f)-V \geq 0E(f)−V≥0, and from (17) it follows that only the curves of the family (1) or parts of them which are situated in the plane region
(19) V x + γ V y Γ ≤ 0 (19) V x + γ V y Γ ≤ 0 {:(19)(V_(x)+gammaV_(y))/(Gamma) <= 0:}\begin{equation*} \frac{V_{x}+\gamma V_{y}}{\Gamma} \leq 0 \tag{19} \end{equation*}(19)Vx+γVyΓ≤0
can be described by the unit mass particle. Inequality (19) was obtained by Bozis and Ichtiaroglou [12].

MIRA-CRISTIANA ANISIU

Remark 1. Bozis [10] arranged equation (6) in a form adequate for the direct problem of dynamics, namely
(20) γ 2 γ x x − 2 γ γ x y + γ y y = h (20) γ 2 γ x x − 2 γ γ x y + γ y y = h {:(20)gamma^(2)gamma_(xx)-2gammagamma_(xy)+gamma_(yy)=h:}\begin{equation*} \gamma^{2} \gamma_{x x}-2 \gamma \gamma_{x y}+\gamma_{y y}=h \tag{20} \end{equation*}(20)γ2γxx−2γγxy+γyy=h
where
(21) h = γ γ x − γ y V y γ + V x ( − γ x V x + ( 2 γ γ x − 3 γ y ) V y + γ ( V x x − V y y ) + ( γ 2 − 1 ) V x y ) (21) h = γ γ x − γ y V y γ + V x − γ x V x + 2 γ γ x − 3 γ y V y + γ V x x − V y y + γ 2 − 1 V x y {:(21)h=(gammagamma_(x)-gamma_(y))/(V_(y)gamma+V_(x))(-gamma_(x)V_(x)+(2gammagamma_(x)-3gamma_(y))V_(y)+gamma(V_(xx)-V_(yy))+(gamma^(2)-1)V_(xy)):}\begin{equation*} h=\frac{\gamma \gamma_{x}-\gamma_{y}}{V_{y} \gamma+V_{x}}\left(-\gamma_{x} V_{x}+\left(2 \gamma \gamma_{x}-3 \gamma_{y}\right) V_{y}+\gamma\left(V_{x x}-V_{y y}\right)+\left(\gamma^{2}-1\right) V_{x y}\right) \tag{21} \end{equation*}(21)h=γγx−γyVyγ+Vx(−γxVx+(2γγx−3γy)Vy+γ(Vxx−Vyy)+(γ2−1)Vxy)
Relations (20)-(21) have been used to find families of curves satisfying auxiliary conditions, supposing that a potential is given, in [16], [17], [6].

2.2. Examples.

Example 2. From the class of Hénon-Heiles potentials
(22) V ( x , y ) = a x 2 + b y 2 + c x 2 y + d y 3 (22) V ( x , y ) = a x 2 + b y 2 + c x 2 y + d y 3 {:(22)V(x","y)=ax^(2)+by^(2)+cx^(2)y+dy^(3):}\begin{equation*} V(x, y)=a x^{2}+b y^{2}+c x^{2} y+d y^{3} \tag{22} \end{equation*}(22)V(x,y)=ax2+by2+cx2y+dy3
with a , b , c , d ∈ R , a , b > 0 a , b , c , d ∈ R , a , b > 0 a,b,c,d inR,a,b > 0a, b, c, d \in \mathbb{R}, a, b>0a,b,c,d∈R,a,b>0, Anisiu and Pal [5] looked for those compatible with the family of polytropic curves f ( x , y ) = x − p y f ( x , y ) = x − p y f(x,y)=x^(-p)yf(x, y)=x^{-p} yf(x,y)=x−py, where p ∈ Z ∖ { 0 , 1 } p ∈ Z ∖ { 0 , 1 } p inZ\\{0,1}p \in \mathbb{Z} \backslash\{0,1\}p∈Z∖{0,1}. The potential
V 1 ( x , y ) = a ( x 2 + 16 y 2 ) + c ( x 2 + ( 16 / 3 ) y 2 ) y V 1 ( x , y ) = a x 2 + 16 y 2 + c x 2 + ( 16 / 3 ) y 2 y V_(1)(x,y)=a(x^(2)+16y^(2))+c(x^(2)+(16//3)y^(2))yV_{1}(x, y)=a\left(x^{2}+16 y^{2}\right)+c\left(x^{2}+(16 / 3) y^{2}\right) yV1(x,y)=a(x2+16y2)+c(x2+(16/3)y2)y
was found to generate the family f 1 ( x , y ) = x − 4 y f 1 ( x , y ) = x − 4 y f_(1)(x,y)=x^(-4)yf_{1}(x, y)=x^{-4} yf1(x,y)=x−4y in the region described by y ( c x 2 + 8 c y 2 + 24 a y ) ≤ 0 y c x 2 + 8 c y 2 + 24 a y ≤ 0 y(cx^(2)+:}{:8cy^(2)+24 ay) <= 0y\left(c x^{2}+\right. \left.8 c y^{2}+24 a y\right) \leq 0y(cx2+8cy2+24ay)≤0, with the energy E 1 ( f 1 ) = − c / ( 24 f 1 ) E 1 f 1 = − c / 24 f 1 E_(1)(f_(1))=-c//(24f_(1))E_{1}\left(f_{1}\right)=-c /\left(24 f_{1}\right)E1(f1)=−c/(24f1). Another potential is
V 2 ( x , y ) = a ( x 2 + 4 y 2 ) + d y 3 V 2 ( x , y ) = a x 2 + 4 y 2 + d y 3 V_(2)(x,y)=a(x^(2)+4y^(2))+dy^(3)V_{2}(x, y)=a\left(x^{2}+4 y^{2}\right)+d y^{3}V2(x,y)=a(x2+4y2)+dy3
which produces the family f 2 ( x , y ) = x 2 y f 2 ( x , y ) = x 2 y f_(2)(x,y)=x^(2)yf_{2}(x, y)=x^{2} yf2(x,y)=x2y in the region d y + 4 a ≤ 0 d y + 4 a ≤ 0 dy+4a <= 0d y+4 a \leq 0dy+4a≤0, with the energy E 2 ( f 2 ) = − d f 2 / 4 E 2 f 2 = − d f 2 / 4 E_(2)(f_(2))=-df_(2)//4E_{2}\left(f_{2}\right)=-d f_{2} / 4E2(f2)=−df2/4.
It was shown in [11] that no potential of the form (22) allows for families of straight lines.
Example 3. For the family f = y − 1 / x 2 f = y − 1 / x 2 f=y-1//x^(2)f=y-1 / x^{2}f=y−1/x2, the potential
V ( x , y ) = 8 y 2 + 4 x 2 y − x 8 − 6 x 2 V ( x , y ) = 8 y 2 + 4 x 2 y − x 8 − 6 x 2 V(x,y)=8y^(2)+4x^(2)y-x^(8)-6x^(2)V(x, y)=8 y^{2}+4 x^{2} y-x^{8}-6 x^{2}V(x,y)=8y2+4x2y−x8−6x2
was found in [17]. The particle describes the curves of the given family in the region y ≤ x 4 + 1 / ( 2 x 2 ) y ≤ x 4 + 1 / 2 x 2 y <= x^(4)+1//(2x^(2))y \leq x^{4}+1 /\left(2 x^{2}\right)y≤x4+1/(2x2) with the energy E ( f ) = 8 f 2 E ( f ) = 8 f 2 E(f)=8f^(2)E(f)=8 f^{2}E(f)=8f2.

3. The three-dimensional inverse problem

We consider the three-dimensional family of curves
(23) φ ( x , y , z ) = c 1 , ψ ( x , y , z ) = c 2 (23) φ ( x , y , z ) = c 1 , ψ ( x , y , z ) = c 2 {:(23)varphi(x","y","z)=c_(1)","quad psi(x","y","z)=c_(2):}\begin{equation*} \varphi(x, y, z)=c_{1}, \quad \psi(x, y, z)=c_{2} \tag{23} \end{equation*}(23)φ(x,y,z)=c1,ψ(x,y,z)=c2
with φ , ψ φ , ψ varphi,psi\varphi, \psiφ,ψ of C 3 C 3 C^(3)C^{3}C3-class and with
(24) | φ y φ z ψ y ψ z | ≠ 0 (24) φ y φ z ψ y ψ z ≠ 0 {:(24)|[varphi_(y),varphi_(z)],[psi_(y),psi_(z)]|!=0:}\left|\begin{array}{ll} \varphi_{y} & \varphi_{z} \tag{24}\\ \psi_{y} & \psi_{z} \end{array}\right| \neq 0(24)|φyφzψyψz|≠0
We can suppose that any other determinant (containing derivatives with respect to x x xxx and y y yyy, or to x x xxx and z z zzz ) is different from zero, and proceed accordingly.
We deal with the following version of the inverse problem: find the potentials V ( x , y , z ) V ( x , y , z ) V(x,y,z)\mathcal{V}(x, y, z)V(x,y,z) under whose action, for appropriate initial conditions, a material point of unit mass, whose motion is described by
(25) x ¨ = − V x y ¨ = − V y z ¨ = − V z , (25) x ¨ = − V x y ¨ = − V y z ¨ = − V z , {:(25)x^(¨)=-V_(x)quady^(¨)=-V_(y)quadz^(¨)=-V_(z)",":}\begin{equation*} \ddot{x}=-\mathcal{V}_{x} \quad \ddot{y}=-\mathcal{V}_{y} \quad \ddot{z}=-\mathcal{V}_{z}, \tag{25} \end{equation*}(25)x¨=−Vxy¨=−Vyz¨=−Vz,
will trace the curves of the family (23). The partial differential equations satisfied by V V V\mathcal{V}V will be derived as in [4], where the geometrical methods used by Kasner [23] were adapted to this problem.
3.1. Basic tools. In order to obtain the equations satisfied by V V V\mathcal{V}V, we differentiate both sides of equations (23) with respect to t t ttt, and get
(26) y ˙ x ˙ = α , z ˙ x ˙ = β , (26) y ˙ x ˙ = α , z ˙ x ˙ = β , {:(26)((y^(˙)))/((x^(˙)))=alpha","quad((z^(˙)))/((x^(˙)))=beta",":}\begin{equation*} \frac{\dot{y}}{\dot{x}}=\alpha, \quad \frac{\dot{z}}{\dot{x}}=\beta, \tag{26} \end{equation*}(26)y˙x˙=α,z˙x˙=β,
where
(27) α = φ z ψ x − φ x ψ z φ y ψ z − φ z ψ y , β = φ x ψ y − φ y ψ x φ y ψ z − φ z ψ y (27) α = φ z ψ x − φ x ψ z φ y ψ z − φ z ψ y , β = φ x ψ y − φ y ψ x φ y ψ z − φ z ψ y {:(27)alpha=(varphi_(z)psi_(x)-varphi_(x)psi_(z))/(varphi_(y)psi_(z)-varphi_(z)psi_(y))","quad beta=(varphi_(x)psi_(y)-varphi_(y)psi_(x))/(varphi_(y)psi_(z)-varphi_(z)psi_(y)):}\begin{equation*} \alpha=\frac{\varphi_{z} \psi_{x}-\varphi_{x} \psi_{z}}{\varphi_{y} \psi_{z}-\varphi_{z} \psi_{y}}, \quad \beta=\frac{\varphi_{x} \psi_{y}-\varphi_{y} \psi_{x}}{\varphi_{y} \psi_{z}-\varphi_{z} \psi_{y}} \tag{27} \end{equation*}(27)α=φzψx−φxψzφyψz−φzψy,β=φxψy−φyψxφyψz−φzψy
We remark that at least one of the functions α α alpha\alphaα and β β beta\betaβ, say α α alpha\alphaα, is not identically null (otherwise condition (24) fails to be fulfilled).
The notation (27) was introduced by Bozis and Kotoulas [13], where it was emphasized that the family (23) leads to a unique pair α , β α , β alpha,beta\alpha, \betaα,β and, conversely, the pair α , β α , β alpha,beta\alpha, \betaα,β determines uniquely the family (23).

MIRA-CRISTIANA ANISIU

We differentiate both relations in (26) and get
(28) x ˙ y ¨ − x ¨ y ˙ x ˙ 3 = A , x ˙ z ¨ − x ¨ z ˙ x ˙ 3 = B , (28) x ˙ y ¨ − x ¨ y ˙ x ˙ 3 = A , x ˙ z ¨ − x ¨ z ˙ x ˙ 3 = B , {:(28)((x^(˙))(y^(¨))-(x^(¨))(y^(˙)))/(x^(˙)^(3))=A","quad((x^(˙))(z^(¨))-(x^(¨))(z^(˙)))/(x^(˙)^(3))=B",":}\begin{equation*} \frac{\dot{x} \ddot{y}-\ddot{x} \dot{y}}{\dot{x}^{3}}=A, \quad \frac{\dot{x} \ddot{z}-\ddot{x} \dot{z}}{\dot{x}^{3}}=B, \tag{28} \end{equation*}(28)x˙y¨−x¨y˙x˙3=A,x˙z¨−x¨z˙x˙3=B,
where
(29) A = α x + α α y + β α z , B = β x + α β y + β β z (29) A = α x + α α y + β α z , B = β x + α β y + β β z {:(29)A=alpha_(x)+alphaalpha_(y)+betaalpha_(z)","quad B=beta_(x)+alphabeta_(y)+betabeta_(z):}\begin{equation*} A=\alpha_{x}+\alpha \alpha_{y}+\beta \alpha_{z}, \quad B=\beta_{x}+\alpha \beta_{y}+\beta \beta_{z} \tag{29} \end{equation*}(29)A=αx+ααy+βαz,B=βx+αβy+ββz
Using (26) and equations (25), we obtain from (28)
(30) α V x − V y x ˙ 2 = A , β V x − V z x ˙ 2 = B (30) α V x − V y x ˙ 2 = A , β V x − V z x ˙ 2 = B {:(30)(alphaV_(x)-V_(y))/(x^(˙)^(2))=A","quad(betaV_(x)-V_(z))/(x^(˙)^(2))=B:}\begin{equation*} \frac{\alpha \mathcal{V}_{x}-\mathcal{V}_{y}}{\dot{x}^{2}}=A, \quad \frac{\beta \mathcal{V}_{x}-\mathcal{V}_{z}}{\dot{x}^{2}}=B \tag{30} \end{equation*}(30)αVx−Vyx˙2=A,βVx−Vzx˙2=B
We have to analyze the special case when A = B = 0 A = B = 0 A=B=0A=B=0A=B=0. It is obvious that, in view of relation (28), it follows that also y ˙ z ¨ − y ¨ z ˙ = 0 y ˙ z ¨ − y ¨ z ˙ = 0 y^(˙)z^(¨)-y^(¨)z^(˙)=0\dot{y} \ddot{z}-\ddot{y} \dot{z}=0y˙z¨−y¨z˙=0, hence the curvature K = | r ¯ ˙ × r ¯ ¨ | / | r ¯ ˙ | 3 K = | r ¯ ˙ × r ¯ ¨ | / | r ¯ ˙ | 3 K=| bar(r)^(˙)xx bar(r)^(¨)|//| bar(r)^(˙)|^(3)K= |\dot{\bar{r}} \times \ddot{\bar{r}}| /|\dot{\bar{r}}|^{3}K=|r¯˙×r¯¨|/|r¯˙|3 of each member of the family (23) vanishes. We have denoted by r ¯ = x ( t ) i ¯ + y ( t ) j ¯ + z ( t ) k ¯ r ¯ = x ( t ) i ¯ + y ( t ) j ¯ + z ( t ) k ¯ bar(r)=x(t) bar(i)+y(t) bar(j)+z(t) bar(k)\bar{r}= x(t) \bar{i}+y(t) \bar{j}+z(t) \bar{k}r¯=x(t)i¯+y(t)j¯+z(t)k¯, where i ¯ , j ¯ , k ¯ i ¯ , j ¯ , k ¯ bar(i), bar(j), bar(k)\bar{i}, \bar{j}, \bar{k}i¯,j¯,k¯ are unit vectors along the axes O x , O y , O z O x , O y , O z Ox,Oy,OzO x, O y, O zOx,Oy,Oz.
It follows that we have A = B = 0 A = B = 0 A=B=0A=B=0A=B=0 if and only if the family (23) consists of straight lines. This case was analyzed in detail in [13]. Relations (30) give rise to two linear partial differential equations to be necessarily satisfied by V V V\mathcal{V}V, namely
(31) α V x − V y = 0 , β V x − V z = 0 . (31) α V x − V y = 0 , β V x − V z = 0 . {:(31)alphaV_(x)-V_(y)=0","quad betaV_(x)-V_(z)=0.:}\begin{equation*} \alpha \mathcal{V}_{x}-\mathcal{V}_{y}=0, \quad \beta \mathcal{V}_{x}-\mathcal{V}_{z}=0 . \tag{31} \end{equation*}(31)αVx−Vy=0,βVx−Vz=0.
These equations will admit of a solution only if α α alpha\alphaα and β β beta\betaβ satisfy, besides the two equations obtained from (29) for A = B = 0 A = B = 0 A=B=0A=B=0A=B=0, a supplementary equation (see [20])
(32) α β x − β α x = β y − α z . (32) α β x − β α x = β y − α z . {:(32)alphabeta_(x)-betaalpha_(x)=beta_(y)-alpha_(z).:}\begin{equation*} \alpha \beta_{x}-\beta \alpha_{x}=\beta_{y}-\alpha_{z} . \tag{32} \end{equation*}(32)αβx−βαx=βy−αz.
So, generally, the inverse problem is not expected to have a solution for arbitrary families of straight lines.
Let us consider now A ≠ 0 A ≠ 0 A!=0A \neq 0A≠0 and B ≠ 0 B ≠ 0 B!=0B \neq 0B≠0. By eliminating x ˙ 2 x ˙ 2 x^(˙)^(2)\dot{x}^{2}x˙2 between the two relations in (30) we obtain a first necessary condition to be satisfied by V V V\mathcal{V}V,
(33) α V x − V y A = β V x − V z B , (33) α V x − V y A = β V x − V z B , {:(33)(alphaV_(x)-V_(y))/(A)=(betaV_(x)-V_(z))/(B)",":}\begin{equation*} \frac{\alpha \mathcal{V}_{x}-\mathcal{V}_{y}}{A}=\frac{\beta \mathcal{V}_{x}-\mathcal{V}_{z}}{B}, \tag{33} \end{equation*}(33)αVx−VyA=βVx−VzB,
where α , β α , β alpha,beta\alpha, \betaα,β from (27) and A , B A , B A,BA, BA,B from (29) depend on the derivatives of φ φ varphi\varphiφ and ψ ψ psi\psiψ up to the second order. Because of x ˙ 2 ≥ 0 x ˙ 2 ≥ 0 x^(˙)^(2) >= 0\dot{x}^{2} \geq 0x˙2≥0, it follows that the motion is possible only in the region determined by
(34) α V x − V y A ≥ 0 . (34) α V x − V y A ≥ 0 . {:(34)(alphaV_(x)-V_(y))/(A) >= 0.:}\begin{equation*} \frac{\alpha \mathcal{V}_{x}-\mathcal{V}_{y}}{A} \geq 0 . \tag{34} \end{equation*}(34)αVx−VyA≥0.
Differentiating both members of the equality x ˙ 2 = ( α V x − V y ) / A x ˙ 2 = α V x − V y / A x^(˙)^(2)=(alphaV_(x)-V_(y))//A\dot{x}^{2}=\left(\alpha \mathcal{V}_{x}-\mathcal{V}_{y}\right) / Ax˙2=(αVx−Vy)/A with respect to t t ttt and replacing x ¨ x ¨ x^(¨)\ddot{x}x¨ from the first equation in (25), respectively y ˙ / x ˙ y ˙ / x ˙ y^(˙)//x^(˙)\dot{y} / \dot{x}y˙/x˙ and z ˙ / x ˙ z ˙ / x ˙ z^(˙)//x^(˙)\dot{z} / \dot{x}z˙/x˙ from (26), we obtain a second differential relation to be satisfied by V V V\mathcal{V}V
(35) − V x x + k V x y + V y y + p V y z + q V x z = l V x + m V y (35) − V x x + k V x y + V y y + p V y z + q V x z = l V x + m V y {:(35)-V_(xx)+kV_(xy)+V_(yy)+pV_(yz)+qV_(xz)=lV_(x)+mV_(y):}\begin{equation*} -\mathcal{V}_{x x}+k \mathcal{V}_{x y}+\mathcal{V}_{y y}+p \mathcal{V}_{y z}+q \mathcal{V}_{x z}=l \mathcal{V}_{x}+m \mathcal{V}_{y} \tag{35} \end{equation*}(35)−Vxx+kVxy+Vyy+pVyz+qVxz=lVx+mVy
where
k = 1 α − α , p = β α , q = − β (36) l = 3 A α − α m , m = A x + α A y + β A z α A k = 1 α − α , p = β α , q = − β (36) l = 3 A α − α m , m = A x + α A y + β A z α A {:[k=(1)/(alpha)-alpha","p=(beta )/(alpha)","q=-beta],[(36)l=(3A)/(alpha)-alpha m","m=(A_(x)+alphaA_(y)+betaA_(z))/(alpha A)]:}\begin{align*} & k=\frac{1}{\alpha}-\alpha, p=\frac{\beta}{\alpha}, q=-\beta \\ & l=\frac{3 A}{\alpha}-\alpha m, m=\frac{A_{x}+\alpha A_{y}+\beta A_{z}}{\alpha A} \tag{36} \end{align*}k=1α−α,p=βα,q=−β(36)l=3Aα−αm,m=Ax+αAy+βAzαA
Summarizing the above reasoning, we assert that a potential which produces as orbits the curves of the family (23) satisfies by necessity the two differential relations (33) and (35), the motion of the particle being possible in the region determined by inequality (34). We remark that equation (35) is of second order in V V V\mathcal{V}V and does not involve the energy (constant on each curve of the family), hence it is the corresponding for the three-dimensional case of Bozis' equation (6) satisfied by planar potentials.
In the following we shall derive the equation from which the total energy can be expressed. Denoting by
(37) W = ( 1 + α 2 + β 2 ) α V x − V y 2 A + V (37) W = 1 + α 2 + β 2 α V x − V y 2 A + V {:(37)W=(1+alpha^(2)+beta^(2))(alphaV_(x)-V_(y))/(2A)+V:}\begin{equation*} \mathcal{W}=\left(1+\alpha^{2}+\beta^{2}\right) \frac{\alpha \mathcal{V}_{x}-\mathcal{V}_{y}}{2 A}+\mathcal{V} \tag{37} \end{equation*}(37)W=(1+α2+β2)αVx−Vy2A+V
one can check by direct calculation that (35) is equivalent to
(38) W x + α W y + β W z = 0 (38) W x + α W y + β W z = 0 {:(38)W_(x)+alphaW_(y)+betaW_(z)=0:}\begin{equation*} \mathcal{W}_{x}+\alpha \mathcal{W}_{y}+\beta \mathcal{W}_{z}=0 \tag{38} \end{equation*}(38)Wx+αWy+βWz=0
The characteristic system for (38) is
d x φ y ψ z − φ z ψ y = d y ψ x φ z − φ x ψ z = d z φ x ψ y − φ y ψ x d x φ y ψ z − φ z ψ y = d y ψ x φ z − φ x ψ z = d z φ x ψ y − φ y ψ x (dx)/(varphi_(y)psi_(z)-varphi_(z)psi_(y))=(dy)/(psi_(x)varphi_(z)-varphi_(x)psi_(z))=(dz)/(varphi_(x)psi_(y)-varphi_(y)psi_(x))\frac{d x}{\varphi_{y} \psi_{z}-\varphi_{z} \psi_{y}}=\frac{d y}{\psi_{x} \varphi_{z}-\varphi_{x} \psi_{z}}=\frac{d z}{\varphi_{x} \psi_{y}-\varphi_{y} \psi_{x}}dxφyψz−φzψy=dyψxφz−φxψz=dzφxψy−φyψx
and one obtains easily that φ x d x + φ y d y + φ z d z = 0 φ x d x + φ y d y + φ z d z = 0 varphi_(x)dx+varphi_(y)dy+varphi_(z)dz=0\varphi_{x} d x+\varphi_{y} d y+\varphi_{z} d z=0φxdx+φydy+φzdz=0 and ψ x d x + ψ y d y + ψ z d z = 0 ψ x d x + ψ y d y + ψ z d z = 0 psi_(x)dx+psi_(y)dy+psi_(z)dz=0\psi_{x} d x+\psi_{y} d y+\psi_{z} d z=0ψxdx+ψydy+ψzdz=0. It follows that φ ( x , y , z ) = c 1 φ ( x , y , z ) = c 1 varphi(x,y,z)=c_(1)\varphi(x, y, z)=c_{1}φ(x,y,z)=c1 and ψ ( x , y , z ) = c 2 ψ ( x , y , z ) = c 2 psi(x,y,z)=c_(2)\psi(x, y, z)=c_{2}ψ(x,y,z)=c2 are integrals, hence the general solution of (38) is W = E ( φ , ψ ) W = E ( φ , ψ ) W=E(varphi,psi)\mathcal{W}=\mathcal{E}(\varphi, \psi)W=E(φ,ψ) with E E E\mathcal{E}E an arbitrary function.
In view of relations (26) and (30), we get from (37) that
(39) E ( φ , ψ ) = ( x ˙ 2 + y ˙ 2 + z ˙ 2 ) / 2 + V (39) E ( φ , ψ ) = x ˙ 2 + y ˙ 2 + z ˙ 2 / 2 + V {:(39)E(varphi","psi)=(x^(˙)^(2)+y^(˙)^(2)+z^(˙)^(2))//2+V:}\begin{equation*} \mathcal{E}(\varphi, \psi)=\left(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2}\right) / 2+\mathcal{V} \tag{39} \end{equation*}(39)E(φ,ψ)=(x˙2+y˙2+z˙2)/2+V
i.e. W = E ( φ , ψ ) W = E ( φ , ψ ) W=E(varphi,psi)\mathcal{W}=\mathcal{E}(\varphi, \psi)W=E(φ,ψ) is the total energy, constant on each curve of the family (23). It follows that the equation
(40) E ( φ , ψ ) = ( 1 + α 2 + β 2 ) α V x − V y 2 A + V (40) E ( φ , ψ ) = 1 + α 2 + β 2 α V x − V y 2 A + V {:(40)E(varphi","psi)=(1+alpha^(2)+beta^(2))(alphaV_(x)-V_(y))/(2A)+V:}\begin{equation*} \mathcal{E}(\varphi, \psi)=\left(1+\alpha^{2}+\beta^{2}\right) \frac{\alpha \mathcal{V}_{x}-\mathcal{V}_{y}}{2 A}+\mathcal{V} \tag{40} \end{equation*}(40)E(φ,ψ)=(1+α2+β2)αVx−Vy2A+V
which was derived by Váradi and Érdi [30] using the energy integral (and which corresponds to Szebehely's planar equation), can be obtained as a consequence of the second order partial differential equation (35).
The two equations (33) and (35) for a single unknown function V V V\mathcal{V}V will not have always a solution; compatibility conditions are to be checked. The advantage of this formulation consists in the fact that it is free of energy.
Remark 4. Equations (33) and (35) are suitable for the direct problem of dynamics: given a three-dimensional potential, find families of curves of the form (23) generated by it. We can rearrange the mentioned equations and obtain a linear partial differential equation of first order in α α alpha\alphaα and β β beta\betaβ
(41) ( V x β − V z ) ( α x + α α y + β α z ) − ( V x α − V y ) ( β x + α β y + β β z ) = 0 (41) V x β − V z α x + α α y + β α z − V x α − V y β x + α β y + β β z = 0 {:(41)(V_(x)beta-V_(z))(alpha_(x)+alphaalpha_(y)+betaalpha_(z))-(V_(x)alpha-V_(y))(beta_(x)+alphabeta_(y)+betabeta_(z))=0:}\begin{equation*} \left(\mathcal{V}_{x} \beta-\mathcal{V}_{z}\right)\left(\alpha_{x}+\alpha \alpha_{y}+\beta \alpha_{z}\right)-\left(\mathcal{V}_{x} \alpha-\mathcal{V}_{y}\right)\left(\beta_{x}+\alpha \beta_{y}+\beta \beta_{z}\right)=0 \tag{41} \end{equation*}(41)(Vxβ−Vz)(αx+ααy+βαz)−(Vxα−Vy)(βx+αβy+ββz)=0
and a nonlinear one of second order
α x x + α 2 α y y + β 2 α z z + 2 α α x y + 2 β α x z + 2 α β α y z = (42) A V x α − V y ⋅ ( 3 V x α x + ( 2 V x α + V y ) α y + ( 2 V x β + V z ) α z + V x x α − V x y ( 1 − α 2 ) − V y y α − V y z β + V x z α β ) α x x + α 2 α y y + β 2 α z z + 2 α α x y + 2 β α x z + 2 α β α y z = (42) A V x α − V y ⋅ 3 V x α x + 2 V x α + V y α y + 2 V x β + V z α z + V x x α − V x y 1 − α 2 − V y y α − V y z β + V x z α β {:[alpha_(xx)+alpha^(2)alpha_(yy)+beta^(2)alpha_(zz)+2alphaalpha_(xy)+2betaalpha_(xz)+2alpha betaalpha_(yz)=],[(42)(A)/(V_(x)alpha-V_(y))*(3V_(x)alpha_(x)+(2V_(x)alpha+V_(y))alpha_(y)+(2V_(x)beta+V_(z))alpha_(z):}],[{:+V_(xx)alpha-V_(xy)(1-alpha^(2))-V_(yy)alpha-V_(yz)beta+V_(xz)alpha beta)]:}\begin{align*} & \alpha_{x x}+\alpha^{2} \alpha_{y y}+\beta^{2} \alpha_{z z}+2 \alpha \alpha_{x y}+2 \beta \alpha_{x z}+2 \alpha \beta \alpha_{y z}= \\ & \frac{A}{\mathcal{V}_{x} \alpha-\mathcal{V}_{y}} \cdot\left(3 \mathcal{V}_{x} \alpha_{x}+\left(2 \mathcal{V}_{x} \alpha+\mathcal{V}_{y}\right) \alpha_{y}+\left(2 \mathcal{V}_{x} \beta+\mathcal{V}_{z}\right) \alpha_{z}\right. \tag{42}\\ & \left.+\mathcal{V}_{x x} \alpha-\mathcal{V}_{x y}\left(1-\alpha^{2}\right)-\mathcal{V}_{y y} \alpha-\mathcal{V}_{y z} \beta+\mathcal{V}_{x z} \alpha \beta\right) \end{align*}αxx+α2αyy+β2αzz+2ααxy+2βαxz+2αβαyz=(42)AVxα−Vy⋅(3Vxαx+(2Vxα+Vy)αy+(2Vxβ+Vz)αz+Vxxα−Vxy(1−α2)−Vyyα−Vyzβ+Vxzαβ)
If B = 0 B = 0 B=0B=0B=0 and A ≠ 0 A ≠ 0 A!=0A \neq 0A≠0, we still have inequality (34); instead of (33), the relation β V x − V z = 0 β V x − V z = 0 betaV_(x)-V_(z)=0\beta \mathcal{V}_{x}-\mathcal{V}_{z}=0βVx−Vz=0 holds, beside the second order partial differential equation (35).
If A = 0 A = 0 A=0A=0A=0 and B ≠ 0 B ≠ 0 B!=0B \neq 0B≠0, the inequality to be satisfied is ( β V x − V z ) / B ≥ 0 β V x − V z / B ≥ 0 (betaV_(x)-V_(z))//B >= 0\left(\beta \mathcal{V}_{x}-\mathcal{V}_{z}\right) / B \geq 0(βVx−Vz)/B≥0, and (33) is replaced by α V x − V y = 0 α V x − V y = 0 alphaV_(x)-V_(y)=0\alpha \mathcal{V}_{x}-\mathcal{V}_{y}=0αVx−Vy=0. Starting with x ˙ 2 = ( β V x − V z ) / B x ˙ 2 = β V x − V z / B x^(˙)^(2)=(betaV_(x)-V_(z))//B\dot{x}^{2}=\left(\beta \mathcal{V}_{x}-\mathcal{V}_{z}\right) / Bx˙2=(βVx−Vz)/B, we follow the steps from the case when both A A AAA and B B BBB were different from zero and obtain instead of (35)
(43) − V x x + k ~ V x z + V z z + p ~ V y z + q ~ V x y = l ~ V x + m ~ V z (43) − V x x + k ~ V x z + V z z + p ~ V y z + q ~ V x y = l ~ V x + m ~ V z {:(43)-V_(xx)+ tilde(k)V_(xz)+V_(zz)+ tilde(p)V_(yz)+ tilde(q)V_(xy)= tilde(l)V_(x)+ tilde(m)V_(z):}\begin{equation*} -\mathcal{V}_{x x}+\tilde{k} \mathcal{V}_{x z}+\mathcal{V}_{z z}+\tilde{p} \mathcal{V}_{y z}+\tilde{q} \mathcal{V}_{x y}=\tilde{l} \mathcal{V}_{x}+\tilde{m} \mathcal{V}_{z} \tag{43} \end{equation*}(43)−Vxx+k~Vxz+Vzz+p~Vyz+q~Vxy=l~Vx+m~Vz
where
k ~ = 1 β − β , p ~ = α β , q ~ = − α (44) l ~ = 3 B β − β m ~ , m ~ = B x + α B y + β B z β B k ~ = 1 β − β , p ~ = α β , q ~ = − α (44) l ~ = 3 B β − β m ~ , m ~ = B x + α B y + β B z β B {:[ tilde(k)=(1)/(beta)-beta"," tilde(p)=(alpha )/(beta)"," tilde(q)=-alpha],[(44) tilde(l)=(3B)/(beta)-beta tilde(m)"," tilde(m)=(B_(x)+alphaB_(y)+betaB_(z))/(beta B)]:}\begin{align*} & \tilde{k}=\frac{1}{\beta}-\beta, \tilde{p}=\frac{\alpha}{\beta}, \tilde{q}=-\alpha \\ & \tilde{l}=\frac{3 B}{\beta}-\beta \tilde{m}, \tilde{m}=\frac{B_{x}+\alpha B_{y}+\beta B_{z}}{\beta B} \tag{44} \end{align*}k~=1β−β,p~=αβ,q~=−α(44)l~=3Bβ−βm~,m~=Bx+αBy+βBzβB

3.2. Examples.

Example 5. The two-parametric family of straight lines
y x = c 1 , z x = c 2 y x = c 1 , z x = c 2 (y)/(x)=c_(1),quad(z)/(x)=c_(2)\frac{y}{x}=c_{1}, \quad \frac{z}{x}=c_{2}yx=c1,zx=c2
was found in [13] to be compatible with the (central) potential
V ( x , y , z ) = F ( x 2 + y 2 + z 2 ) V ( x , y , z ) = F x 2 + y 2 + z 2 V(x,y,z)=F(x^(2)+y^(2)+z^(2))\mathcal{V}(x, y, z)=F\left(x^{2}+y^{2}+z^{2}\right)V(x,y,z)=F(x2+y2+z2)
where F F FFF is an arbitrary function of its argument.
Shorokhov [28] presented a family of straight lines
x y = c 1 , y + z = c 2 x y = c 1 , y + z = c 2 (x)/(y)=c_(1),quad y+z=c_(2)\frac{x}{y}=c_{1}, \quad y+z=c_{2}xy=c1,y+z=c2
which cannot be described by a particle under the action of any potential. This family has α = y / x α = y / x alpha=y//x\alpha=y / xα=y/x and β = − y / x β = − y / x beta=-y//x\beta=-y / xβ=−y/x, hence condition (31) does not hold.
Example 6. The family of curves
z x = c 1 , x 2 + y 2 = c 2 z x = c 1 , x 2 + y 2 = c 2 (z)/(x)=c_(1),quadx^(2)+y^(2)=c_(2)\frac{z}{x}=c_{1}, \quad x^{2}+y^{2}=c_{2}zx=c1,x2+y2=c2
was considered in [30] and [15]. It can be traced all over the space under the action of the potential
V ( x , y , z ) = ( x 2 + y 2 + z 2 ) / 2 V ( x , y , z ) = x 2 + y 2 + z 2 / 2 V(x,y,z)=(x^(2)+y^(2)+z^(2))//2\mathcal{V}(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right) / 2V(x,y,z)=(x2+y2+z2)/2
with the energy E ( φ , ψ ) = ψ ( φ 2 + 2 ) / 2 E ( φ , ψ ) = ψ φ 2 + 2 / 2 E(varphi,psi)=psi(varphi^(2)+2)//2\mathcal{E}(\varphi, \psi)=\psi\left(\varphi^{2}+2\right) / 2E(φ,ψ)=ψ(φ2+2)/2. This example illustrates the case A ≠ 0 A ≠ 0 A!=0A \neq 0A≠0, B = 0 B = 0 B=0B=0B=0 。
Example 7. For the family of curves
x 2 + y 2 = c 1 , x 2 − y 2 z = c 2 x 2 + y 2 = c 1 , x 2 − y 2 z = c 2 x^(2)+y^(2)=c_(1),quad(x^(2)-y^(2))/(z)=c_(2)x^{2}+y^{2}=c_{1}, \quad \frac{x^{2}-y^{2}}{z}=c_{2}x2+y2=c1,x2−y2z=c2
one has A ≠ 0 A ≠ 0 A!=0A \neq 0A≠0 and B ≠ 0 B ≠ 0 B!=0B \neq 0B≠0. The potential
V ( x , y , z ) = x 2 + y 2 + 4 z 2 V ( x , y , z ) = x 2 + y 2 + 4 z 2 V(x,y,z)=x^(2)+y^(2)+4z^(2)\mathcal{V}(x, y, z)=x^{2}+y^{2}+4 z^{2}V(x,y,z)=x2+y2+4z2

MIRA-CRISTIANA ANISIU

given in [14] produces the given family with the energy E ( φ , ψ ) = 2 φ ( 2 φ + ψ 2 ) / ψ 2 E ( φ , ψ ) = 2 φ 2 φ + ψ 2 / ψ 2 E(varphi,psi)=2varphi(2varphi+psi^(2))//psi^(2)\mathcal{E}(\varphi, \psi)=2 \varphi\left(2 \varphi+\psi^{2}\right) / \psi^{2}E(φ,ψ)=2φ(2φ+ψ2)/ψ2.

4. Conclusions

The energy-free equations have a basic role in the inverse problem of dynamics. When we have no a priori information on the energy of the given family, it is natural to work with equations (6), respectively (33) and (35) in order to obtain potentials compatible with the given family. These equations can be used also when the search of the potentials is restricted to a class of theoretical or practical interest.

References

[1] M. C. Anisiu, PDEs in the inverse problem of dynamics, in V. Barbu et al (eds.), Analysis and Optimization of Differential Systems, Kluwer Academic Publishers 2003, 13-20.
[2] M. C. Anisiu, The Equations of the Inverse Problem of Dynamics, House of the Book of Science, Cluj-Napoca 2003 (in Romanian).
[3] M. C. Anisiu, An alternative point of view on the equations of the inverse problem of dynamics, Inverse Problems 20 (2004), 1865-1872.
[4] M. C. Anisiu, , Energy-free equations of the 3D inverse problem of dynamics, 2004, submitted.
[5] M. C. Anisiu, A. Pal, Special families of orbits for the Hénon-Heiles type potential, Rom. Astron. J. 9 (1999), 179-185.
[6] M. C. Anisiu, C. Blaga, and G. Bozis, Special families of orbits in the direct problem of dynamics, Celest. Mech. Dyn. Astron. 88 (2004), 245-257.
[7] J. Bertrand, Sur la possibilité de déduire d'une seule des lois de Kepler le principe de l'attraction, Compt. Rend. 84 (1877), 671-673.
[8] G. Bozis, Inverse problem with two-parametric families of planar orbits, Celest. Mech. 31 (1983), 129-143.
[9] G. Bozis, Szebehely inverse problem for finite symmetrical material concentrations, Astron. Astrophys. 134(1984), 360-364.
[10] G. Bozis, The inverse problem of dynamics: basic facts, Inverse Problems, 11 (1995), 687-708.
[11] G. Bozis, and M. C. Anisiu, Families of straight lines in planar potentials, Rom. Astron. J. 11(2001), 27-43.
TWO- AND THREE-DIMENSIONAL INVERSE PROBLEM OF DYNAMICS
[12] G. Bozis, and S. Ichtiaroglou, Boundary curves for families of planar orbits, Celest. Mech. Dyn. Astron., 58 (1994), 371-385.
[13] G. Bozis, and T. A. Kotoulas, Families of straight lines (FSL) in three-dimensional potentials, 2004, private communication.
[14] G. Bozis, and T. A. Kotoulas, Homogeneous two-parametric families of orbits in threedimensional homogeneous potentials, 2004, private communication.
[15] G. Bozis, and A. Nakhla, Solutions of the three-dimensional inverse problem, Celest. Mech. Dyn. Astron. 38(1986), 357-375.
[16] G. Bozis, M. C. Anisiu, and C. Blaga, Inhomogeneous potentials producing homogeneous orbits, Astron. Nachr. 318(1997), 313-318.
[17] G. Bozis, M. C. Anisiu, and C. Blaga, A solvable version of the direct problem of dynamics, Rom. Astron. J. 10(2000), 59-70.
[18] U. Dainelli, Sul movimento per una linea qualunque, G. Mat. 18(1880), 271-300.
[19] B. Érdi, A generalization of Szebehely's equation for three dimensions, Celest. Mech. 28(1982), 209-218.
[20] J. Favard, Cours d'Analyse de l'École Polytechnique, Tome 3², p. 39, Gauthier-Villars, Paris 1963.
[21] F. Gonzales-Gascon, A. Gonzales-Lopez, and P. J. Pascual-Broncano, On Szebehely's equation and its connection with Dainelli's-Whittaker's equations, Celest. Mech. 33(1984), 85-97.
[22] E. Kasner, The trajectories of dynamics, Trans. Amer. Math. Soc. 7(1906), 401-424.
[23] E. Kasner, Dynamical trajectories: the motion of the particle in an arbitrary field of force, Trans. Amer. Math. Soc. 8(1907), 135-158.
[24] I. Newton, Philosophiae Naturalis Principia Mathematica, London 1687.
[25] F. Puel, Équations de Szebehely et principes variationnels, Celest. Mech. 32(1984), 349-353.
[26] F. Puel, Formulation intrinseque de l'équation de Szebehely, Celest. Mech. 32(1984), 209-212.
[27] F. Puel, Explicit solutions of the three-dimensional inverse problem of Dynamics, using the Frenet reference frame, Celest. Mech. Dyn. Astron. 53(1992), 207-218.
[28] S. G. Shorokhov, , Solution of an inverse problem of the Dynamics of a particle, Celest. Mech. Dyn. Astron. 44(1988), 193-206.
[29] V. Szebehely, On the determination of the potential by satellite observations, in G. Proverbio (ed.), Proc. of the Int. Meeting on Earth's Rotation by Satellite Observation, The Univ. of Cagliari, Bologna, Italy 1974, 31-35.

MIRA-CRISTIANA ANISIU

[30] F. Váradi, and B. Érdi, Existence and solution of Szebehely's equation in three dimension using a two-parametric family of orbits, Celest. Mech. 30(1983), 395-405.
T. Popoviciu Institute of Numerical Analysis, Romanian Academy P.O. Box 68, 400110 Cluj-Napoca, Romania
E-mail address: mira@math.ubbcluj.ro
  1. Received by the editors: 06.12.2004.
    2000 Mathematics Subject Classification. 34L40.
    Key words and phrases. Planar and spatial inverse problem of dynamics, energy-free equations.
    This paper was presented at International Conference on Nonlinear Operators, Differential Equations and Applications held in Cluj-Napoca (Romania) from August 24 to August 27, 2004.
2004

Related Posts