Families of orbits in conservative fields of Hénon-Heiles type

Mira Anisiu
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Abstract

The equations of the inverse problem of dynamics are used in order to obtain planar and spatial potentials of Hénon-Heiles type, which give rise to some special families of curves. The curves of such a family can be traced by a material point of unit mass, with suitable initial conditions, moving under the action of the specific potential. We determine the region where the motion is possible, as well as the total energy of the particle.

Authors

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

Keywords

celestial mechanics – inverse problem of dynamics – families of orbits.

Paper coordinates

M.-C. Anisiu, Families of orbits in conservative fields of Hénon-Heiles type, Rom. Astron. J. 16 suppl. (2006), Proceedings of the Scientific Session Scientific Programs and Astronomy Education in SEE and Ukraine, 75-80, https://www.astro.ro/~roaj/volume/abs16_3.html#11

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Families of orbits in planar anisotropic fields

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Journal

Astronomische Nacrichten

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Astronomical Notes

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1521-3994
Online ISSN

0004-6337

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asna.200410259 (1)

Families of orbits in planar anisotropic fields

M.-C. Anisiu and G. Bozis 1 1 ^(1){ }^{1}1 T. Popoviciu Institute of Numerical Analysis, Romanian Academy, PO Box 68, 400110 Cluj-Napoca, Romania 2 2 ^(2){ }^{2}2 Department of Physics, University of Thessaloniki, 54006 Thessaloniki, Greece

Received 13 May 2004; accepted 22 June 2004; published online 30 December 2004

Abstract

The aim of the planar inverse problem of dynamics is: given a monoparametric family of curves f ( x , y ) = c f ( x , y ) = c f(x,y)=cf(x, y)=cf(x,y)=c, find the potential V ( x , y ) V ( x , y ) V(x,y)V(x, y)V(x,y) under whose action a material point of unit mass can describe the curves of the family. In this study we look for V V VVV in the class of the anisotropic potentials V ( x , y ) = v ( a 2 x 2 + y 2 ) , ( a = V ( x , y ) = v a 2 x 2 + y 2 , ( a = V(x,y)=v(a^(2)x^(2)+y^(2)),(a=V(x, y)=v\left(a^{2} x^{2}+y^{2}\right),(a=V(x,y)=v(a2x2+y2),(a= constant ) ) ))). These potentials have been used lately in the search of connections between classical, quantum, and relativistic mechanics. We establish a general condition which must be satisfied by all the families produced by an anisotropic potential. We treat special cases regarding the families (e. g. families traced isoenergetically) and we present certain pertinent examples of compatible pairs of families of curves and anisotropic potentials.

Kev worde. celectial mechanics - ctellar dynamice

1. Introduction

The planar inverse problem of dynamics consists in finding potentials V ( x , y ) V ( x , y ) V(x,y)V(x, y)V(x,y) which can produce as orbits a preassigned monoparametric family of curves traced in the x y x y xyx yxy plane by a material point of unit mass. The partial differential equations satisfied by the potential, which will be described below, do not provide uniqueness for V V VVV. That is why it is desirable to look for solutions in specific classes of potentials.
In this paper we shall focus on anisotropic potentials, which appear in various mathematical models arising in Astronomy and Physics. We mention e.g. some of the first results associated to the anisotropic two-body problem for the Newtonian potential (Gutzwiller 1971; Will 1971; Vinti 1972), for the Manev potential (Craig et al. 1999) and for the Schwarzschild one (Mioc, Pérez-Chavela & Stavinschi 2003). The importance of these anisotropic models comes also from the fact that they were used in the search of connections between classical, quantum, and relativistic mechanics.
The potential V ( x , y ) V ( x , y ) V(x,y)V(x, y)V(x,y) which can generate the family
(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
traced with energy E = E ( f ) E = E ( f ) E=E(f)E=E(f)E=E(f) is given by a partial differential equation due to Szebehely (1974). In terms of the 'slope function' γ γ gamma\gammaγ, introduced by Bozis (1983) and given by
(2) γ = f y f x , (2) γ = f y f x , {:(2)gamma=(f_(y))/(f_(x))",":}\begin{equation*} \gamma=\frac{f_{y}}{f_{x}}, \tag{2} \end{equation*}(2)γ=fyfx,
Correspondence to: mira@math.ubbcluj.ro
this equation reads
(3) V x + γ V y + 2 Γ 1 + γ 2 ( E ( f ) − V ) = 0 . (3) V x + γ V y + 2 Γ 1 + γ 2 ( E ( f ) − V ) = 0 . {:(3)V_(x)+gammaV_(y)+(2Gamma)/(1+gamma^(2))(E(f)-V)=0.:}\begin{equation*} V_{x}+\gamma V_{y}+\frac{2 \Gamma}{1+\gamma^{2}}(E(f)-V)=0 . \tag{3} \end{equation*}(3)Vx+γVy+2Γ1+γ2(E(f)−V)=0.
The subscripts denote partial derivatives.
Remark 1. The linearity in V V VVV of Szebehely's equation implies that if V i V i V_(i)V_{i}Vi are solutions of (3) with the energy E i ( f ) E i ( f ) E_(i)(f)E_{i}(f)Ei(f), i = 1 , 2 i = 1 , 2 i=1,2i=1,2i=1,2, then V = V 1 + V 2 V = V 1 + V 2 V=V_(1)+V_(2)V=V_{1}+V_{2}V=V1+V2 is a solution of (3) with the energy E ( f ) = E 1 ( f ) + E 2 ( f ) E ( f ) = E 1 ( f ) + E 2 ( f ) E(f)=E_(1)(f)+E_(2)(f)E(f)=E_{1}(f)+E_{2}(f)E(f)=E1(f)+E2(f); if V 1 V 1 V_(1)V_{1}V1 is a solution of (3) with the energy E 1 ( f ) E 1 ( f ) E_(1)(f)E_{1}(f)E1(f), then c 1 V 1 c 1 V 1 c_(1)V_(1)c_{1} V_{1}c1V1 is also a solution with the energy c 1 E 1 ( f ) c 1 E 1 ( f ) c_(1)E_(1)(f)c_{1} E_{1}(f)c1E1(f), and c 1 V 1 + c 2 c 1 V 1 + c 2 c_(1)V_(1)+c_(2)c_{1} V_{1}+c_{2}c1V1+c2 is a solution with the energy c 1 E 1 ( f ) + c 2 c 1 E 1 ( f ) + c 2 c_(1)E_(1)(f)+c_(2)c_{1} E_{1}(f)+c_{2}c1E1(f)+c2.
We remark that relation (2) provides a one-to-one correspondence between γ γ gamma\gammaγ and the family (1).
The function Γ Γ Gamma\GammaΓ is proportional to the curvature of the family (1) and its value is
(4) Γ = γ γ x − γ y . (4) Γ = γ γ x − γ y . {:(4)Gamma=gammagamma_(x)-gamma_(y).:}\begin{equation*} \Gamma=\gamma \gamma_{x}-\gamma_{y} . \tag{4} \end{equation*}(4)Γ=γγx−γy.
The families of straight lines are characterized by Γ = 0 Γ = 0 Gamma=0\Gamma=0Γ=0. The potentials producing such a family must satisfy the equation (Bozis & Anisiu 2001)
(5) V x V y ( V x x − V y y ) = V x y ( V x 2 − V y 2 ) . (5) V x V y V x x − V y y = V x y V x 2 − V y 2 . {:(5)V_(x)V_(y)(V_(xx)-V_(yy))=V_(xy)(V_(x)^(2)-V_(y)^(2)).:}\begin{equation*} V_{x} V_{y}\left(V_{x x}-V_{y y}\right)=V_{x y}\left(V_{x}^{2}-V_{y}^{2}\right) . \tag{5} \end{equation*}(5)VxVy(Vxx−Vyy)=Vxy(Vx2−Vy2).
For the families (1) with Γ ≠ 0 Γ ≠ 0 Gamma!=0\Gamma \neq 0Γ≠0, Bozis (1984) obtained a free of energy second order partial differential equation
(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 = 1 γ Γ ( Γ y − γ Γ x ) , μ = λ γ + 3 Γ γ . (7) κ = 1 γ − γ , λ = 1 γ Γ Γ y − γ Γ x , μ = λ γ + 3 Γ γ . {:(7)kappa=(1)/(gamma)-gamma","lambda=(1)/(gamma Gamma)(Gamma_(y)-gammaGamma_(x))","mu=lambda gamma+(3Gamma)/(gamma).:}\begin{equation*} \kappa=\frac{1}{\gamma}-\gamma, \lambda=\frac{1}{\gamma \Gamma}\left(\Gamma_{y}-\gamma \Gamma_{x}\right), \mu=\lambda \gamma+\frac{3 \Gamma}{\gamma} . \tag{7} \end{equation*}(7)κ=1γ−γ,λ=1γΓ(Γy−γΓx),μ=λγ+3Γγ.
Real motion on the curves of the family (1) is possible only in the region (Bozis & Ichtiaroglou 1994) defined by the inequality
V x + γ V y Γ ≤ 0 V x + γ V y Γ ≤ 0 (V_(x)+gammaV_(y))/(Gamma) <= 0\frac{V_{x}+\gamma V_{y}}{\Gamma} \leq 0Vx+γVyΓ≤0.
Basic facts on the inverse problem of dynamics are to be found in Bozis (1995) and Anisiu (2003).

2. The inverse problem equations for anisotropic potentials

An anisotropic potential V V VVV is given by
V ( x , y ) = v ( a 2 x 2 + y 2 ) V ( x , y ) = v a 2 x 2 + y 2 V(x,y)=v(a^(2)x^(2)+y^(2))V(x, y)=v\left(a^{2} x^{2}+y^{2}\right)V(x,y)=v(a2x2+y2),
where a a aaa is a real number, a ∉ { − 1 , 0 , 1 } a ∉ { − 1 , 0 , 1 } a!in{-1,0,1}a \notin\{-1,0,1\}a∉{−1,0,1}. For a 2 = 1 a 2 = 1 a^(2)=1a^{2}=1a2=1, the potential becomes isotropic. We shall exclude the trivial case of constant potentials. As it was emphasized in Remark 1, if V V VVV is a solution of the inverse problem, so is c 1 V 1 + c 2 c 1 V 1 + c 2 c_(1)V_(1)+c_(2)c_{1} V_{1}+c_{2}c1V1+c2; the constant c 2 c 2 c_(2)c_{2}c2 will be omitted, but c 1 c 1 c_(1)c_{1}c1 can be chosen adequately to obtain simpler expressions, or, in view of (8), a suitable region of the plane for the orbits to lie.
Having no a priori information on the energy E ( f ) E ( f ) E(f)E(f)E(f), we intend to rely on eq. (6). So we shall settle at first the case Γ = Γ = Gamma=\Gamma=Γ= 0 , corresponding to families (1) of straight lines. Substituting V V VVV from (9) into (5) we obtain
( a 2 − 1 ) v ′ = 0 a 2 − 1 v ′ = 0 (a^(2)-1)v^(')=0\left(a^{2}-1\right) v^{\prime}=0(a2−1)v′=0,
where the prime denotes the derivative of v v vvv with respect to its unique argument w = a 2 x 2 + y 2 w = a 2 x 2 + y 2 w=a^(2)x^(2)+y^(2)w=a^{2} x^{2}+y^{2}w=a2x2+y2. It follows that no anisotropic potential allows for families of straight lines.
In what follows we shall consider Γ ≠ 0 Γ ≠ 0 Gamma!=0\Gamma \neq 0Γ≠0. We substitute V V VVV from (9) in eq. (6) and obtain the equation in v v vvv
(11) 2 ( − a 4 x 2 + a 2 x y κ + y 2 ) v ′ ′ = ( a 2 x λ + y μ + a 2 − 1 ) v ′ (11) 2 − a 4 x 2 + a 2 x y κ + y 2 v ′ ′ = a 2 x λ + y μ + a 2 − 1 v ′ {:[(11)2(-a^(4)x^(2)+a^(2)xy kappa+y^(2))v^('')=],[(a^(2)x lambda+y mu+a^(2)-1)v^(')]:}\begin{align*} & 2\left(-a^{4} x^{2}+a^{2} x y \kappa+y^{2}\right) v^{\prime \prime}= \tag{11}\\ & \left(a^{2} x \lambda+y \mu+a^{2}-1\right) v^{\prime} \end{align*}(11)2(−a4x2+a2xyκ+y2)v′′=(a2xλ+yμ+a2−1)v′
where κ , λ , μ κ , λ , μ kappa,lambda,mu\kappa, \lambda, \muκ,λ,μ are given by (7) in terms of γ γ gamma\gammaγ and its derivatives up to the second order.
Remark 2. For the two families of orbits γ 1 = y / ( a 2 x ) γ 1 = y / a 2 x gamma_(1)=y//(a^(2)x)\gamma_{1}=y /\left(a^{2} x\right)γ1=y/(a2x) and γ 2 = − a 2 x / y γ 2 = − a 2 x / y gamma_(2)=-a^(2)x//y\gamma_{2}=-a^{2} x / yγ2=−a2x/y the coefficient of v ′ ′ v ′ ′ v^('')v^{\prime \prime}v′′ in (11) becomes identically zero. For each of them, eq. (11) reduces to ( a 2 − 1 ) v ′ = 0 a 2 − 1 v ′ = 0 (a^(2)-1)v^(')=0\left(a^{2}-1\right) v^{\prime}=0(a2−1)v′=0, hence no nontrivial anisotropic potential (9) gives rise to such families.
Remark 3. If γ γ gamma\gammaγ is a solution of
a 2 x λ + y μ + a 2 − 1 = 0 a 2 x λ + y μ + a 2 − 1 = 0 a^(2)x lambda+y mu+a^(2)-1=0a^{2} x \lambda+y \mu+a^{2}-1=0a2xλ+yμ+a2−1=0,
eq. (11) becomes v ′ ′ = 0 v ′ ′ = 0 v^('')=0v^{\prime \prime}=0v′′=0 and from (9) we obtain the potential V ( x , y ) = c 1 ( a 2 x 2 + y 2 ) V ( x , y ) = c 1 a 2 x 2 + y 2 V(x,y)=c_(1)(a^(2)x^(2)+y^(2))V(x, y)=c_{1}\left(a^{2} x^{2}+y^{2}\right)V(x,y)=c1(a2x2+y2), which produces the family of curves corresponding to γ γ gamma\gammaγ.
From now on we shall exclude from our study the family of conics f 1 ( x , y ) = a 2 x 2 + y 2 f 1 ( x , y ) = a 2 x 2 + y 2 f_(1)(x,y)=a^(2)x^(2)+y^(2)f_{1}(x, y)=a^{2} x^{2}+y^{2}f1(x,y)=a2x2+y2 and the family f 2 ( x , y ) = x − 1 / a 2 y f 2 ( x , y ) = x − 1 / a 2 y f_(2)(x,y)=x^(-1//a^(2))yf_{2}(x, y)= x^{-1 / a^{2}} yf2(x,y)=x−1/a2y, corresponding
to γ 1 , γ 2 γ 1 , γ 2 gamma_(1),gamma_(2)\gamma_{1}, \gamma_{2}γ1,γ2 in Remark 2.
Equation (11) can be written as
(13) v ′ ′ v ′ = U ( x , y ; a ) (13) v ′ ′ v ′ = U ( x , y ; a ) {:(13)(v^(''))/(v^('))=U(x","y;a):}\begin{equation*} \frac{v^{\prime \prime}}{v^{\prime}}=U(x, y ; a) \tag{13} \end{equation*}(13)v′′v′=U(x,y;a)
where
U ( x , y ; a ) = a 2 x λ + y μ + a 2 − 1 2 ( − a 4 x 2 + a 2 x y κ + y 2 ) U ( x , y ; a ) = a 2 x λ + y μ + a 2 − 1 2 − a 4 x 2 + a 2 x y κ + y 2 U(x,y;a)=(a^(2)x lambda+y mu+a^(2)-1)/(2(-a^(4)x^(2)+a^(2)xy kappa+y^(2)))U(x, y ; a)=\frac{a^{2} x \lambda+y \mu+a^{2}-1}{2\left(-a^{4} x^{2}+a^{2} x y \kappa+y^{2}\right)}U(x,y;a)=a2xλ+yμ+a2−12(−a4x2+a2xyκ+y2).
The condition for (13) to admit of a solution of the form (9) is
y U x − a 2 x U y = 0 y U x − a 2 x U y = 0 yU_(x)-a^(2)xU_(y)=0y U_{x}-a^{2} x U_{y}=0yUx−a2xUy=0.
It follows that the families of curves which can be generated under the action of an anisotropic potential are those determined by the solutions γ γ gamma\gammaγ of the differential relation (15). For such a γ γ gamma\gammaγ we have U ( x , y ; a ) = u ( a 2 x 2 + y 2 ) U ( x , y ; a ) = u a 2 x 2 + y 2 U(x,y;a)=u(a^(2)x^(2)+y^(2))U(x, y ; a)=u\left(a^{2} x^{2}+y^{2}\right)U(x,y;a)=u(a2x2+y2), and from (13) we get
v = c 1 ∫ exp ⁡ ( ∫ u ) + c 2 v = c 1 ∫ exp ⁡ ∫ u + c 2 v=c_(1)int exp(int u)+c_(2)v=c_{1} \int \exp \left(\int u\right)+c_{2}v=c1∫exp⁡(∫u)+c2.
Working on condition (15), we obtain
(17) x 4 λ y a 8 + s 6 a 6 + s 4 a 4 + s 2 a 2 + y 4 μ x = 0 (17) x 4 λ y a 8 + s 6 a 6 + s 4 a 4 + s 2 a 2 + y 4 μ x = 0 {:(17)x^(4)lambda_(y)a^(8)+s_(6)a^(6)+s_(4)a^(4)+s_(2)a^(2)+y^(4)mu_(x)=0:}\begin{equation*} x^{4} \lambda_{y} a^{8}+s_{6} a^{6}+s_{4} a^{4}+s_{2} a^{2}+y^{4} \mu_{x}=0 \tag{17} \end{equation*}(17)x4λya8+s6a6+s4a4+s2a2+y4μx=0
where
s 6 = x ( x 2 y ( λ κ y − κ λ y + μ y − λ x ) + x 2 ( κ λ + μ ) + x y ( λ + κ y ) + x κ + 2 y ) s 4 = x 2 y 2 ( κ λ x − λ κ x + μ κ y − κ μ y − λ y − μ x ) (18) + x 2 y ( 2 λ − κ y ) + x y 2 ( 2 μ − κ x ) − ( x 2 + y 2 ) κ s 2 = y ( x y 2 ( κ μ x − μ κ x + λ x − μ y ) + y 2 ( λ − κ μ ) + x y ( μ + κ x ) − 2 x + y κ . ) s 6 = x x 2 y λ κ y − κ λ y + μ y − λ x + x 2 ( κ λ + μ ) + x y λ + κ y + x κ + 2 y s 4 = x 2 y 2 κ λ x − λ κ x + μ κ y − κ μ y − λ y − μ x (18) + x 2 y 2 λ − κ y + x y 2 2 μ − κ x − x 2 + y 2 κ s 2 = y x y 2 κ μ x − μ κ x + λ x − μ y + y 2 ( λ − κ μ ) + x y μ + κ x − 2 x + y κ . {:[s_(6)=x(x^(2)y(lambdakappa_(y)-kappalambda_(y)+mu_(y)-lambda_(x))+x^(2)(kappa lambda+mu):}],[{:+xy(lambda+kappa_(y))+x kappa+2y)],[s_(4)=x^(2)y^(2)(kappalambda_(x)-lambdakappa_(x)+mukappa_(y)-kappamu_(y)-lambda_(y)-mu_(x))],[(18)+x^(2)y(2lambda-kappa_(y))+xy^(2)(2mu-kappa_(x))-(x^(2)+y^(2))kappa],[s_(2)=y(xy^(2)(kappamu_(x)-mukappa_(x)+lambda_(x)-mu_(y))+y^(2)(lambda-kappa mu):}],[{:+xy(mu+kappa_(x))-2x+y kappa.)]:}\begin{align*} & s_{6}=x\left(x^{2} y\left(\lambda \kappa_{y}-\kappa \lambda_{y}+\mu_{y}-\lambda_{x}\right)+x^{2}(\kappa \lambda+\mu)\right. \\ & \left.+x y\left(\lambda+\kappa_{y}\right)+x \kappa+2 y\right) \\ & s_{4}=x^{2} y^{2}\left(\kappa \lambda_{x}-\lambda \kappa_{x}+\mu \kappa_{y}-\kappa \mu_{y}-\lambda_{y}-\mu_{x}\right) \\ & +x^{2} y\left(2 \lambda-\kappa_{y}\right)+x y^{2}\left(2 \mu-\kappa_{x}\right)-\left(x^{2}+y^{2}\right) \kappa \tag{18}\\ & s_{2}=y\left(x y^{2}\left(\kappa \mu_{x}-\mu \kappa_{x}+\lambda_{x}-\mu_{y}\right)+y^{2}(\lambda-\kappa \mu)\right. \\ & \left.+x y\left(\mu+\kappa_{x}\right)-2 x+y \kappa .\right) \end{align*}s6=x(x2y(λκy−κλy+μy−λx)+x2(κλ+μ)+xy(λ+κy)+xκ+2y)s4=x2y2(κλx−λκx+μκy−κμy−λy−μx)(18)+x2y(2λ−κy)+xy2(2μ−κx)−(x2+y2)κs2=y(xy2(κμx−μκx+λx−μy)+y2(λ−κμ)+xy(μ+κx)−2x+yκ.)
The coefficients in (18) are expressed in terms of the functions κ , λ , μ κ , λ , μ kappa,lambda,mu\kappa, \lambda, \muκ,λ,μ from (7) and of their first order partial derivatives. Equation (17) represents the necessary condition to be fulfilled by a family (1) in order to be produced by an anisotropic potential.
Remark 4. For a = 1 a = 1 a=1a=1a=1 the condition (17) reduces to
x y ( x λ + y μ ) ( x κ y − y κ x ) (19) + ( x 2 − x y κ − y 2 ) ( x ( x λ y − y λ x ) + y ( x μ y − y μ x ) ) + ( κ x 3 + 3 x 2 y + y 3 ) λ + ( x 3 + 3 x y 2 − κ y 3 ) μ = 0 x y ( x λ + y μ ) x κ y − y κ x (19) + x 2 − x y κ − y 2 x x λ y − y λ x + y x μ y − y μ x + κ x 3 + 3 x 2 y + y 3 λ + x 3 + 3 x y 2 − κ y 3 μ = 0 {:[xy(x lambda+y mu)(xkappa_(y)-ykappa_(x))],[(19)+(x^(2)-xy kappa-y^(2))(x(xlambda_(y)-ylambda_(x))+y(xmu_(y)-ymu_(x)))],[+(kappax^(3)+3x^(2)y+y^(3))lambda+(x^(3)+3xy^(2)-kappay^(3))mu=0]:}\begin{align*} & x y(x \lambda+y \mu)\left(x \kappa_{y}-y \kappa_{x}\right) \\ & +\left(x^{2}-x y \kappa-y^{2}\right)\left(x\left(x \lambda_{y}-y \lambda_{x}\right)+y\left(x \mu_{y}-y \mu_{x}\right)\right) \tag{19}\\ & +\left(\kappa x^{3}+3 x^{2} y+y^{3}\right) \lambda+\left(x^{3}+3 x y^{2}-\kappa y^{3}\right) \mu=0 \end{align*}xy(xλ+yμ)(xκy−yκx)(19)+(x2−xyκ−y2)(x(xλy−yλx)+y(xμy−yμx))+(κx3+3x2y+y3)λ+(x3+3xy2−κy3)μ=0
Equation (19) gives the totality of families (1) produced by central potentials V = v ( r ) , r = ( x 2 + y 2 ) 1 / 2 V = v ( r ) , r = x 2 + y 2 1 / 2 V=v(r),r=(x^(2)+y^(2))^(1//2)V=v(r), r=\left(x^{2}+y^{2}\right)^{1 / 2}V=v(r),r=(x2+y2)1/2 and is in agreement with pertinent findings by Borghero, Bozis & Melis (1999).

3. The two-dimensional anisotropic harmonic oscillator

The potential V ( x , y ) = ( a 2 x 2 + y 2 ) / 2 V ( x , y ) = a 2 x 2 + y 2 / 2 V(x,y)=(a^(2)x^(2)+y^(2))//2V(x, y)=\left(a^{2} x^{2}+y^{2}\right) / 2V(x,y)=(a2x2+y2)/2, analyzed in detail by Iro (2002), is one of the simplest anisotropic potentials. In this case the equations of motion
x ¨ + a 2 x = 0 x ¨ + a 2 x = 0 x^(¨)+a^(2)x=0\ddot{x}+a^{2} x=0x¨+a2x=0
(20) y ¨ + y = 0 (20) y ¨ + y = 0 {:(20)y^(¨)+y=0:}\begin{equation*} \ddot{y}+y=0 \tag{20} \end{equation*}(20)y¨+y=0
are not coupled, and the solutions for the initial values x 0 = b 1 , y 0 = b 3 , x ˙ 0 = a b 2 , y ˙ 0 = b 4 x 0 = b 1 , y 0 = b 3 , x ˙ 0 = a b 2 , y ˙ 0 = b 4 x_(0)=b_(1),y_(0)=b_(3),x^(˙)_(0)=ab_(2),y^(˙)_(0)=b_(4)x_{0}= b_{1}, y_{0}=b_{3}, \dot{x}_{0}=a b_{2}, \dot{y}_{0}=b_{4}x0=b1,y0=b3,x˙0=ab2,y˙0=b4 are
(21) x ( t ) = b 1 cos ⁡ a t + b 2 sin ⁡ a t (21) x ( t ) = b 1 cos ⁡ a t + b 2 sin ⁡ a t {:(21)x(t)=b_(1)cos at+b_(2)sin at:}\begin{equation*} x(t)=b_{1} \cos a t+b_{2} \sin a t \tag{21} \end{equation*}(21)x(t)=b1cos⁡at+b2sin⁡at
y ( t ) = b 3 cos ⁡ t + b 4 sin ⁡ t y ( t ) = b 3 cos ⁡ t + b 4 sin ⁡ t y(t)=b_(3)cos t+b_(4)sin ty(t)=b_{3} \cos t+b_{4} \sin ty(t)=b3cos⁡t+b4sin⁡t.
It is known that the motion in the configuration plane consists of Lissajous' figures. If a a aaa is rational, the orbit is closed; for a a aaa irrational, the orbit fills entirely a region of the plane.
We try to obtain families of orbits by eliminating t t ttt between the two equations (21), hence we shall consider a = q / s , q , s a = q / s , q , s a=q//s,q,sa= q / s, q, sa=q/s,q,s natural numbers.
From the first equation in (21) and its derivative with respect to t t ttt we get cos ⁡ a t = ( a b 1 x + b 2 x ˙ ) / ( a ( b 1 2 + b 2 2 ) ) cos ⁡ a t = a b 1 x + b 2 x ˙ / a b 1 2 + b 2 2 cos at=(ab_(1)x+b_(2)(x^(˙)))//(a(b_(1)^(2)+b_(2)^(2)))\cos a t=\left(a b_{1} x+b_{2} \dot{x}\right) /\left(a\left(b_{1}^{2}+b_{2}^{2}\right)\right)cos⁡at=(ab1x+b2x˙)/(a(b12+b22)) and sin ⁡ a t = ( a b 2 x − b 1 x ˙ ) / ( a ( b 1 2 + b 2 2 ) ) sin ⁡ a t = a b 2 x − b 1 x ˙ / a b 1 2 + b 2 2 sin at=(ab_(2)x-b_(1)(x^(˙)))//(a(b_(1)^(2)+b_(2)^(2)))\sin a t=\left(a b_{2} x-b_{1} \dot{x}\right) /\left(a\left(b_{1}^{2}+b_{2}^{2}\right)\right)sin⁡at=(ab2x−b1x˙)/(a(b12+b22)), hence
t = 1 a arctan ⁡ a b 2 x − b 1 x ˙ a b 1 x + b 2 x ˙ t = 1 a arctan ⁡ a b 2 x − b 1 x ˙ a b 1 x + b 2 x ˙ t=(1)/(a)arctan((ab_(2)x-b_(1)(x^(˙)))/(ab_(1)x+b_(2)(x^(˙))))t=\frac{1}{a} \arctan \frac{a b_{2} x-b_{1} \dot{x}}{a b_{1} x+b_{2} \dot{x}}t=1aarctan⁡ab2x−b1x˙ab1x+b2x˙.
Similarly, from the second equation in (21) and its derivative we get
t = arctan ⁡ b 4 y − b 3 y ˙ b 3 y + b 4 y ˙ t = arctan ⁡ b 4 y − b 3 y ˙ b 3 y + b 4 y ˙ t=arctan((b_(4)y-b_(3)(y^(˙)))/(b_(3)y+b_(4)(y^(˙))))t=\arctan \frac{b_{4} y-b_{3} \dot{y}}{b_{3} y+b_{4} \dot{y}}t=arctan⁡b4y−b3y˙b3y+b4y˙,
and, eliminating t t ttt between the two equations (22) and (23) we obtain
a arctan ⁡ b 4 y − b 3 y ˙ b 3 y + b 4 y ˙ − arctan ⁡ a b 2 x − b 1 x ˙ a b 1 x + b 2 x ˙ = 0 a arctan ⁡ b 4 y − b 3 y ˙ b 3 y + b 4 y ˙ − arctan ⁡ a b 2 x − b 1 x ˙ a b 1 x + b 2 x ˙ = 0 a arctan((b_(4)y-b_(3)(y^(˙)))/(b_(3)y+b_(4)(y^(˙))))-arctan((ab_(2)x-b_(1)(x^(˙)))/(ab_(1)x+b_(2)(x^(˙))))=0a \arctan \frac{b_{4} y-b_{3} \dot{y}}{b_{3} y+b_{4} \dot{y}}-\arctan \frac{a b_{2} x-b_{1} \dot{x}}{a b_{1} x+b_{2} \dot{x}}=0aarctan⁡b4y−b3y˙b3y+b4y˙−arctan⁡ab2x−b1x˙ab1x+b2x˙=0.
The energy is conserved in each direction, i. e.
x ˙ 2 + a 2 x 2 = a 2 ( b 1 2 + b 2 2 ) , y ˙ 2 + y 2 = b 3 2 + b 4 2 x ˙ 2 + a 2 x 2 = a 2 b 1 2 + b 2 2 , y ˙ 2 + y 2 = b 3 2 + b 4 2 x^(˙)^(2)+a^(2)x^(2)=a^(2)(b_(1)^(2)+b_(2)^(2)),y^(˙)^(2)+y^(2)=b_(3)^(2)+b_(4)^(2)\dot{x}^{2}+a^{2} x^{2}=a^{2}\left(b_{1}^{2}+b_{2}^{2}\right), \dot{y}^{2}+y^{2}=b_{3}^{2}+b_{4}^{2}x˙2+a2x2=a2(b12+b22),y˙2+y2=b32+b42.
The relations (25) allow us to eliminate x ˙ x ˙ x^(˙)\dot{x}x˙ and y ˙ y ˙ y^(˙)\dot{y}y˙ from (24).
In so doing, we obtain a family of orbits which, except for a a aaa, includes (not independently, of course) the four constants b 1 , b 2 , b 3 , b 4 b 1 , b 2 , b 3 , b 4 b_(1),b_(2),b_(3),b_(4)b_{1}, b_{2}, b_{3}, b_{4}b1,b2,b3,b4. Indeed, from (2) we find
γ = − a c 1 2 − x 2 c 2 2 − y 2 γ = − a c 1 2 − x 2 c 2 2 − y 2 gamma=-(asqrt(c_(1)^(2)-x^(2)))/(sqrt(c_(2)^(2)-y^(2)))\gamma=-\frac{a \sqrt{c_{1}^{2}-x^{2}}}{\sqrt{c_{2}^{2}-y^{2}}}γ=−ac12−x2c22−y2,
where
c 1 2 = b 1 2 + b 2 2 , c 2 2 = b 3 2 + b 4 2 c 1 2 = b 1 2 + b 2 2 , c 2 2 = b 3 2 + b 4 2 c_(1)^(2)=b_(1)^(2)+b_(2)^(2),quadc_(2)^(2)=b_(3)^(2)+b_(4)^(2)c_{1}^{2}=b_{1}^{2}+b_{2}^{2}, \quad c_{2}^{2}=b_{3}^{2}+b_{4}^{2}c12=b12+b22,c22=b32+b42,
and from (3)
E = 1 2 ( a 2 c 1 2 + c 2 2 ) E = 1 2 a 2 c 1 2 + c 2 2 E=(1)/(2)(a^(2)c_(1)^(2)+c_(2)^(2))E=\frac{1}{2}\left(a^{2} c_{1}^{2}+c_{2}^{2}\right)E=12(a2c12+c22).
For each a a aaa, (26) offers a two-parametric set of slope functions γ γ gamma\gammaγ (i.e. of monoparametric families) compatible with V = ( a 2 x 2 + y 2 ) / 2 V = a 2 x 2 + y 2 / 2 V=(a^(2)x^(2)+y^(2))//2V=\left(a^{2} x^{2}+y^{2}\right) / 2V=(a2x2+y2)/2 and, as expected in view of the Remark 3, with eq. (12) as well. For specific values of a , c 1 , c 2 a , c 1 , c 2 a,c_(1),c_(2)a, c_{1}, c_{2}a,c1,c2 the monoparametric (in c c ccc ) family corresponding to (26) is
f ( x , y ) = q arcsin ⁡ y c 2 − s arcsin ⁡ x c 1 = c f ( x , y ) = q arcsin ⁡ y c 2 − s arcsin ⁡ x c 1 = c f(x,y)=q arcsin((y)/(c_(2)))-s arcsin((x)/(c_(1)))=cf(x, y)=q \arcsin \frac{y}{c_{2}}-s \arcsin \frac{x}{c_{1}}=cf(x,y)=qarcsin⁡yc2−sarcsin⁡xc1=c.
In fact (29) is three-parametric and includes all orbits produced by the anisotropic harmonic oscillator. Of course, out of the set (29), one may extract e.g. infinitely many twoparametric subsets by imposing any relation φ ( c , c 1 , c 2 ) = 0 φ c , c 1 , c 2 = 0 varphi(c,c_(1),c_(2))=0\varphi\left(c, c_{1}, c_{2}\right)=0φ(c,c1,c2)=0 between the three parameters c , c 1 , c 2 c , c 1 , c 2 c,c_(1),c_(2)c, c_{1}, c_{2}c,c1,c2. However, solving for either of the remaining two parameters is neither an easy nor always an accomplishable task. For this reason we may search directly for particular solutions of (12). In this manner we found (with b b bbb constant) e.g. for a = ± 2 a = ± 2 a=+-2a= \pm 2a=±2,
γ = b − 4 x 2 y , f = y 2 4 x − b , E = − b f + b 2 8 , 4 x + b 4 x − b ≤ 0 γ = b − 4 x 2 y , f = y 2 4 x − b , E = − b f + b 2 8 , 4 x + b 4 x − b ≤ 0 gamma=(b-4x)/(2y),f=(y^(2))/(4x-b),E=-bf+(b^(2))/(8),(4x+b)/(4x-b) <= 0\gamma=\frac{b-4 x}{2 y}, f=\frac{y^{2}}{4 x-b}, E=-b f+\frac{b^{2}}{8}, \frac{4 x+b}{4 x-b} \leq 0γ=b−4x2y,f=y24x−b,E=−bf+b28,4x+b4x−b≤0, and
(30) γ = b y , f = b y 2 + 2 x , E = f 2 2 + f b , 4 x + b y 2 b ≥ 0 ; (30) γ = b y , f = b y 2 + 2 x , E = f 2 2 + f b , 4 x + b y 2 b ≥ 0 ; {:(30)gamma=by","f=by^(2)+2x","E=(f^(2))/(2)+(f)/(b)","(4x+by^(2))/(b) >= 0;:}\begin{equation*} \gamma=b y, f=b y^{2}+2 x, E=\frac{f^{2}}{2}+\frac{f}{b}, \frac{4 x+b y^{2}}{b} \geq 0 ; \tag{30} \end{equation*}(30)γ=by,f=by2+2x,E=f22+fb,4x+by2b≥0;
for: a = ± 1 2 a = ± 1 2 a=+-(1)/(2)a= \pm \frac{1}{2}a=±12,
γ = b − x 2 2 x y , f = y x 2 − b , E = b 2 2 f 2 + b 4 , x 2 − 2 b ≤ 0 γ = b − x 2 2 x y , f = y x 2 − b , E = b 2 2 f 2 + b 4 , x 2 − 2 b ≤ 0 gamma=(b-x^(2))/(2xy),f=(y)/(x^(2)-b),E=(b^(2))/(2)f^(2)+(b)/(4),x^(2)-2b <= 0\gamma=\frac{b-x^{2}}{2 x y}, f=\frac{y}{x^{2}-b}, E=\frac{b^{2}}{2} f^{2}+\frac{b}{4}, x^{2}-2 b \leq 0γ=b−x22xy,f=yx2−b,E=b22f2+b4,x2−2b≤0.
Beside the family, we gave the energy dependence E E EEE and the region (8) where real motion is allowed.
The potential of the harmonic oscillator being homogeneous of order two, it is natural to look for compatible homogeneous families (for which γ γ gamma\gammaγ is homogeneous of order zero),
γ = g ( z ) γ = g ( z ) gamma=g(z)\gamma=g(z)γ=g(z) with z = y / x z = y / x z=y//xz=y / xz=y/x.
In this case, eq. (12) reads
( z g + a 2 ) ( z g + 1 ) g ¨ + ( 2 z g + 3 a 2 − 1 ) g g ˙ (33) − z ( 2 z g − a 2 + 3 ) g ˙ 2 = 0 , z g + a 2 ( z g + 1 ) g ¨ + 2 z g + 3 a 2 − 1 g g ˙ (33) − z 2 z g − a 2 + 3 g ˙ 2 = 0 , {:[(zg+a^(2))(zg+1)g^(¨)+(2zg+3a^(2)-1)gg^(˙)],[(33)-z(2zg-a^(2)+3)g^(˙)^(2)=0","]:}\begin{align*} & \left(z g+a^{2}\right)(z g+1) \ddot{g}+\left(2 z g+3 a^{2}-1\right) g \dot{g} \\ & -z\left(2 z g-a^{2}+3\right) \dot{g}^{2}=0, \tag{33} \end{align*}(zg+a2)(zg+1)g¨+(2zg+3a2−1)gg˙(33)−z(2zg−a2+3)g˙2=0,
where the dot denotes the derivative with respect to z z zzz.
From the solutions of the form g ( z ) = b z m g ( z ) = b z m g(z)=bz^(m)g(z)=b z^{m}g(z)=bzm the only ones which verify (33) (with a 2 ≠ 1 a 2 ≠ 1 a^(2)!=1a^{2} \neq 1a2≠1 ) are obtained for m = − 1 m = − 1 m=-1m=-1m=−1 and are given by g = ± a / z g = ± a / z g=+-a//zg= \pm a / zg=±a/z; they correspond to the families
f ( x , y ) = x ∓ 1 / a y f ( x , y ) = x ∓ 1 / a y f(x,y)=x^(∓1//a)yf(x, y)=x^{\mp 1 / a} yf(x,y)=x∓1/ay,
traced with zero energy. The inequality (8) becomes y 2 ≤ 0 y 2 ≤ 0 y^(2) <= 0y^{2} \leq 0y2≤0, which means that real motion is not allowed on the curves of the family (34) under the action of the potential of an anisotropic harmonic oscillator; these curves can be described under a potential with opposite sign, namely V = − ( a 2 x 2 + y 2 ) / 2 V = − a 2 x 2 + y 2 / 2 V=-(a^(2)x^(2)+y^(2))//2V= -\left(a^{2} x^{2}+y^{2}\right) / 2V=−(a2x2+y2)/2.
Specifying a a aaa, we can obtain further solutions of (33), which give rise to families f f fff traced with the specified energy, under the action of the potential of the harmonic oscillator, in certain regions of the plane (again b b bbb denotes a constant):
a = ± 2 a = ± 2 a=+-2a= \pm 2a=±2 :
γ = − 2 z + b z , f = b y 2 − x 2 y 4 , E = b + b 2 2 f , 1 x 2 − b y 2 ≤ 0 ; γ = − 2 z + b z , f = b y 2 − x 2 y 4 , E = b + b 2 2 f , 1 x 2 − b y 2 ≤ 0 ; gamma=-(2)/(z)+bz,f=(by^(2)-x^(2))/(y^(4)),E=(b+b^(2))/(2f),(1)/(x^(2)-by^(2)) <= 0;\gamma=-\frac{2}{z}+b z, f=\frac{b y^{2}-x^{2}}{y^{4}}, E=\frac{b+b^{2}}{2 f}, \frac{1}{x^{2}-b y^{2}} \leq 0 ;γ=−2z+bz,f=by2−x2y4,E=b+b22f,1x2−by2≤0;
a = ± 3 a = ± 3 a=+-3a= \pm 3a=±3 :
(35) γ = b − 3 z , f = 2 x − b y y 3 , E = − 4 b + b 3 6 f , 6 x + b y 2 x − b y ≤ 0 ; (35) γ = b − 3 z , f = 2 x − b y y 3 , E = − 4 b + b 3 6 f , 6 x + b y 2 x − b y ≤ 0 ; {:(35)gamma=b-(3)/(z)","f=(2x-by)/(y^(3))","E=-(4b+b^(3))/(6f)","(6x+by)/(2x-by) <= 0;:}\begin{equation*} \gamma=b-\frac{3}{z}, f=\frac{2 x-b y}{y^{3}}, E=-\frac{4 b+b^{3}}{6 f}, \frac{6 x+b y}{2 x-b y} \leq 0 ; \tag{35} \end{equation*}(35)γ=b−3z,f=2x−byy3,E=−4b+b36f,6x+by2x−by≤0;
a = ± 1 3 a = ± 1 3 a=+-(1)/(3)a= \pm \frac{1}{3}a=±13 :
γ = − b 3 ( b z − 1 ) , f = 2 b y − 3 x x 3 , E = − 9 + 4 b 2 18 b 2 f , x + 2 b y 3 x − 2 b y ≥ 0 γ = − b 3 ( b z − 1 ) , f = 2 b y − 3 x x 3 , E = − 9 + 4 b 2 18 b 2 f , x + 2 b y 3 x − 2 b y ≥ 0 gamma=-(b)/(3(bz-1)),f=(2by-3x)/(x^(3)),E=-(9+4b^(2))/(18b^(2)f),(x+2by)/(3x-2by) >= 0\gamma=-\frac{b}{3(b z-1)}, f=\frac{2 b y-3 x}{x^{3}}, E=-\frac{9+4 b^{2}}{18 b^{2} f}, \frac{x+2 b y}{3 x-2 b y} \geq 0γ=−b3(bz−1),f=2by−3xx3,E=−9+4b218b2f,x+2by3x−2by≥0.

4. Isoenergetic families

The isoenergetic families have the total energy E ( f ) = e E ( f ) = e E(f)=eE(f)=eE(f)=e, e e eee being a constant which can be considered zero. For the anisotropic potential V V VVV given by (9), Szebehely's equation (3) can be written as
v ′ v = Γ ( y γ + a 2 x ) ( γ 2 + 1 ) v ′ v = Γ y γ + a 2 x γ 2 + 1 (v^('))/(v)=(Gamma)/((y gamma+a^(2)x)(gamma^(2)+1))\frac{v^{\prime}}{v}=\frac{\Gamma}{\left(y \gamma+a^{2} x\right)\left(\gamma^{2}+1\right)}v′v=Γ(yγ+a2x)(γ2+1).
The condition that the right hand side is a function of a 2 x 2 + y 2 a 2 x 2 + y 2 a^(2)x^(2)+y^(2)a^{2} x^{2}+ y^{2}a2x2+y2 reads
y ( ( y γ + a 2 x ) ( γ 2 + 1 ) Γ ) x y y γ + a 2 x γ 2 + 1 Γ x y(((y gamma+a^(2)x)(gamma^(2)+1))/(Gamma))_(x)y\left(\frac{\left(y \gamma+a^{2} x\right)\left(\gamma^{2}+1\right)}{\Gamma}\right)_{x}y((yγ+a2x)(γ2+1)Γ)x
− a 2 x ( ( y γ + a 2 x ) ( γ 2 + 1 ) Γ ) y x = 0 − a 2 x y γ + a 2 x γ 2 + 1 Γ y x = 0 -a^(2)x(((y gamma+a^(2)x)(gamma^(2)+1))/(Gamma))_(y)^(x)=0-a^{2} x\left(\frac{\left(y \gamma+a^{2} x\right)\left(\gamma^{2}+1\right)}{\Gamma}\right)_{y}^{x}=0−a2x((yγ+a2x)(γ2+1)Γ)yx=0,
or
( y γ + a 2 x ) ( γ 2 + 1 ) ( a 2 x Γ y − y Γ x ) + Γ ( − 2 a 4 x 2 γ γ y (37) + a 2 ( ( γ 2 + 1 ) ( y − x γ ) + x y ( 2 γ γ x − ( 3 γ 2 + 1 ) γ y ) ) + y 2 ( 3 γ 2 + 1 ) γ x ) = 0 y γ + a 2 x γ 2 + 1 a 2 x Γ y − y Γ x + Γ − 2 a 4 x 2 γ γ y (37) + a 2 γ 2 + 1 ( y − x γ ) + x y 2 γ γ x − 3 γ 2 + 1 γ y + y 2 3 γ 2 + 1 γ x = 0 {:[(y gamma+a^(2)x)(gamma^(2)+1)(a^(2)xGamma_(y)-yGamma_(x))+Gamma(-2a^(4)x^(2)gammagamma_(y):}],[(37)+a^(2)((gamma^(2)+1)(y-x gamma)+xy(2gammagamma_(x)-(3gamma^(2)+1)gamma_(y)))],[{:+y^(2)(3gamma^(2)+1)gamma_(x))=0]:}\begin{align*} & \left(y \gamma+a^{2} x\right)\left(\gamma^{2}+1\right)\left(a^{2} x \Gamma_{y}-y \Gamma_{x}\right)+\Gamma\left(-2 a^{4} x^{2} \gamma \gamma_{y}\right. \\ & +a^{2}\left(\left(\gamma^{2}+1\right)(y-x \gamma)+x y\left(2 \gamma \gamma_{x}-\left(3 \gamma^{2}+1\right) \gamma_{y}\right)\right) \tag{37}\\ & \left.+y^{2}\left(3 \gamma^{2}+1\right) \gamma_{x}\right)=0 \end{align*}(yγ+a2x)(γ2+1)(a2xΓy−yΓx)+Γ(−2a4x2γγy(37)+a2((γ2+1)(y−xγ)+xy(2γγx−(3γ2+1)γy))+y2(3γ2+1)γx)=0
If γ γ gamma\gammaγ satisfies the differential condition (37), the anisotropic potential will be given by v = c 1 exp ⁡ ( ∫ F ) v = c 1 exp ⁡ ∫ F v=c_(1)exp(int F)v=c_{1} \exp \left(\int F\right)v=c1exp⁡(∫F), where F ( a 2 x 2 + y 2 ) = Γ / ( ( y γ + a 2 x ) ( γ 2 + 1 ) ) F a 2 x 2 + y 2 = Γ / y γ + a 2 x γ 2 + 1 F(a^(2)x^(2)+y^(2))=Gamma//((y gamma+a^(2)x)(gamma^(2)+1))F\left(a^{2} x^{2}+y^{2}\right)=\Gamma /\left(\left(y \gamma+a^{2} x\right)\left(\gamma^{2}+1\right)\right)F(a2x2+y2)=Γ/((yγ+a2x)(γ2+1)).
It can be checked that γ = y / x γ = y / x gamma=y//x\gamma=y / xγ=y/x, representing a family of homocentric circles
x 2 + y 2 = c x 2 + y 2 = c x^(2)+y^(2)=cx^{2}+y^{2}=cx2+y2=c,
is a solution of eq. (37). Equation (36) becomes v ′ / v = − 1 / w v ′ / v = − 1 / w v^(')//v=-1//wv^{\prime} / v= -1 / wv′/v=−1/w, with w = a 2 x 2 + y 2 w = a 2 x 2 + y 2 w=a^(2)x^(2)+y^(2)w=a^{2} x^{2}+y^{2}w=a2x2+y2 and has the solution v = c 1 / w v = c 1 / w v=c_(1)//wv=c_{1} / wv=c1/w, corresponding to
V ( x , y ) = c 1 a 2 x 2 + y 2 V ( x , y ) = c 1 a 2 x 2 + y 2 V(x,y)=(c_(1))/(a^(2)x^(2)+y^(2))V(x, y)=\frac{c_{1}}{a^{2} x^{2}+y^{2}}V(x,y)=c1a2x2+y2.
The circles are traced isoenergetically all over the plane for c 1 < 0 c 1 < 0 c_(1) < 0c_{1}<0c1<0. The anisotropic potential V V VVV given by (39) is a member of the totality of homogeneous potentials, found by Borghero & Bozis (2002), which produce isoenergetically the family (38).
Remark 5. It is known that the Newtonian potential is compatible with the family of circles (38); it follows by Remark 1 that the perturbed Newtonian potential
V = − 1 x 2 + y 2 + c 1 a 2 x 2 + y 2 , c 1 < 0 V = − 1 x 2 + y 2 + c 1 a 2 x 2 + y 2 , c 1 < 0 V=-(1)/(sqrt(x^(2)+y^(2)))+(c_(1))/(a^(2)x^(2)+y^(2)),c_(1) < 0V=-\frac{1}{\sqrt{x^{2}+y^{2}}}+\frac{c_{1}}{a^{2} x^{2}+y^{2}}, c_{1}<0V=−1x2+y2+c1a2x2+y2,c1<0
can give rise to family (38) all over the plane with
E = − 1 / ( 2 x 2 + y 2 ) E = − 1 / 2 x 2 + y 2 E=-1//(2sqrt(x^(2)+y^(2)))E=-1 /\left(2 \sqrt{x^{2}+y^{2}}\right)E=−1/(2x2+y2).

5. Concluding remarks

We studied real anisotropic potentials V ( x , y ) = v ( a 2 x 2 + y 2 ) V ( x , y ) = v a 2 x 2 + y 2 V(x,y)=v(a^(2)x^(2)+y^(2))V(x, y)= v\left(a^{2} x^{2}+y^{2}\right)V(x,y)=v(a2x2+y2) in the light of the planar inverse problem of dynamics and mainly from the viewpoint of the monoparametric families (1) which they can produce. We established the general differential condition (17) which must be satisfied by all the families γ = γ ( x , y ) γ = γ ( x , y ) gamma=gamma(x,y)\gamma=\gamma(x, y)γ=γ(x,y) compatible with such potentials. Written explicitly the condition (17) would include the slope function γ ( x , y ) γ ( x , y ) gamma(x,y)\gamma(x, y)γ(x,y) and partial derivatives of it up to the third order. As such, it is a highly nonlinear PDE in the unknown function γ ( x , y ) γ ( x , y ) gamma(x,y)\gamma(x, y)γ(x,y).
The (superintegrable case of the) two-dimensional anisotropic harmonic oscillator V = ( a 2 x 2 + y 2 ) / 2 V = a 2 x 2 + y 2 / 2 V=(a^(2)x^(2)+y^(2))//2V=\left(a^{2} x^{2}+y^{2}\right) / 2V=(a2x2+y2)/2 was reviewed from the same viewpoint. The pertinent threeparametric family of orbits was given by the eq. (29). There exist, of course, infinitely many ways of extracting out of (29) families with one or two parameters but this task is not always possible, depending on the value of the constant a a aaa at hand. For this reason, and for specific values of a a aaa, we found certain examples by direct reference to the PDE (6).
To aid the algebra, we assumed either homogeneity of the orbits (i. e. families of the form γ = g ( y / x ) γ = g ( y / x ) gamma=g(y//x)\gamma=g(y / x)γ=g(y/x) ) or isoenergeticity of the families (i. e. all the members of each family are traced with the same total energy, say E = 0 E = 0 E=0E=0E=0 ). Sporadic findings are given by (30), (31) and (35), respectively.

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2006

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