Abstract
The family of orbits given in advance in the inverse problem of dynamics can be described in implicit or parametric form. It is proved that the similar curves expressed in parametric form can be rewritten implicitly, the corresponding Örst order partial di§erential equation satisÖed by the potential being integrable by quadratures. An example from astrophysics (Ögure-eight curves) is worked out to illustrate the theoretical results.
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Mira-Cristiana Anisiu
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Romania
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M.-C. Anisiu, Families of similar orbits in the inverse problem of Dynamics, Proceedings of the Itinerant Seminar T. Popoviciu of Functional Equations, Approximation and Convexity, Cluj-Napoca, 23-29 May 2000, ed. E. Popoviciu, Srima, 2000, 9-18 (pdf file here)
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[1] Anisiu, M.-C., Pal, A., Two-dimensional inverse problem of dynamics for families in parametric form, Inverse Problems 15 (1999), 135-140
[2] Anisiu, M.-C., Bozis, G., Programmed motion for a class of families of planar orbits, Inverse Problems 16 (2000), 19-32
[3] Bozis, G., Inverse problem with two parametric families of planar orbits, Celest. Mech. 31 (1983), 129-142
[4] Bozis, G., Caranicolas, N. D., The Earth and the Universe, Special Volume of the Aristotle University of Thessaloniki (ed. Asteriadis et al), 1997
[5] Bozis, G., Borghero, F., A new formulation of the two-dimensional inverse problem of dynamics, Inverse Problems 14 (1998), 41-51
[6] Caranicolas, N. D., Potentials for the central parts of a barred galaxy, Astronomy and Astrophysics 332 (1998), 88-92
[7] Grigoriadou, S., Bozis, G., Elmabsout, B., Solvable cases of Szebehelyís equation, Celest. Mech.74 (1999), 211-221
[8] Szebehely, V., On the determination of the potential by satellite observations, Proc. Int. Meeting on Earthís Rotations by Satellite Observations Ed. E. Proverbio (Bologna: Pitagora Editrice), 1974, 31-35
Families of Similar Orbits in the Inverse Problem of Dynamics
Abstract
The family of orbits given in advance in the inverse problem of dynamics can be described in implicit or parametric form. It is proved that the similar curves expressed in parametric form can be rewritten implicitly, the corresponding first order partial differential equation satisfied by the potential being integrable by quadratures. An example from astrophysics (figure-eight curves) is worked out to illustrate the theoretical results.
1 Introduction
The aim of the two-dimensional inverse problem of Dynamics is to find the potentials VV (or force fields) which can give rise to a prescribed monoparametric family of planar trajectories traced by a unit mass material point. The family of curves is given as
{:(1)f(x","y)=c:}\begin{equation*}
f(x, y)=c \tag{1}
\end{equation*}
the parameter cc varying along the family. An important tool for the inverse problem is Szebehely's linear first order partial differential equation
{:(2)f_(x)V_(x)+f_(y)V_(y)=2W(E-V)",":}\begin{equation*}
f_{x} V_{x}+f_{y} V_{y}=2 W(E-V), \tag{2}
\end{equation*}
where E=E(c)E=E(c) represents the energy dependence and W=W(x,y)W=W(x, y) is a function related to the curvature of the family, given by
W=(f_(xx)f_(y)^(2)-2f_(xy)f_(x)f_(y)+f_(yy)f_(x)^(2))/(f_(x)^(2)+f_(y)^(2)).W=\frac{f_{x x} f_{y}^{2}-2 f_{x y} f_{x} f_{y}+f_{y y} f_{x}^{2}}{f_{x}^{2}+f_{y}^{2}} .
Szebehely (1974) showed that the potential VV, under whose action a unit mass material point describes the curves in family (1) with the energy dependence given in advance, verifies equation (2). This equation has been simplified by Bozis (1983) using a function gamma\gamma, representing the slope of the curves of the family orthogonal to (1). The simplified equation is
{:(3)V_(x)+gammaV_(y)+(2Gamma)/(1+gamma^(2))(E-V)=0:}\begin{equation*}
V_{x}+\gamma V_{y}+\frac{2 \Gamma}{1+\gamma^{2}}(E-V)=0 \tag{3}
\end{equation*}
where
{:(4)gamma=(f_(y))/(f_(x))","Gamma=gammagamma_(x)-gamma_(y).:}\begin{equation*}
\gamma=\frac{f_{y}}{f_{x}}, \Gamma=\gamma \gamma_{x}-\gamma_{y} . \tag{4}
\end{equation*}
Grigoriadou et al (1999) proved that equation (2), respectively (3), can be integrated by quadratures if the two equations
{:(5)(dy)/(dx)=gamma(x","y):}\begin{equation*}
\frac{d y}{d x}=\gamma(x, y) \tag{5}
\end{equation*}
and
{:(6)(dy)/(dx)=-(1)/(gamma(x,y)):}\begin{equation*}
\frac{d y}{d x}=-\frac{1}{\gamma(x, y)} \tag{6}
\end{equation*}
are solvable. They also listed some cases in which both equations (5) and (6) are solvable and offered some criteria to assure solvability in other cases.
For example, for
gamma(x,y)=sigma_(1)(x)sigma_(2)(y),\gamma(x, y)=\sigma_{1}(x) \sigma_{2}(y),
or gamma\gamma homogeneous of degree zero in xx and yy,
gamma(x,y)=sigma((y)/(x)),\gamma(x, y)=\sigma\left(\frac{y}{x}\right),
equations (5) and (6) are solvable and the solution of (3) can be obtained by quadratures.
2 Families given in parametric form
Recently the case of families of orbits given in parametric form was considered. The motivation is that some curves (as, for example, Lissajoux ones) can thus be described in a simpler way; some families (appearing in astrophysics) are more naturally modelled parametrically.
The family of curves is given by the parametric equations
{:(7)x=F(lambda","b)quad y=G(lambda","b):}\begin{equation*}
x=F(\lambda, b) \quad y=G(\lambda, b) \tag{7}
\end{equation*}
where the parameter bb varies from member to member of the family, as cc did for (1), while lambda\lambda varies along each specific curve for a fixed bb. The first order equation satisfied by the potential in this case was derived by Bozis and Borghero (1998), while the expression of a general force field (not necessarily conservative) was determined by Anisiu and Pal (1999).
Families of ellipses in parametric form were considered by Bozis and Caranicolas (1997) and other examples (Bozis and Borghero, 1998) arose in connection with isotach orbits (orbits along which the kinetic energy T=E-VT=E-V is constant), namely the family of homocentric circles
x=b cos lambdaquad y=b sin lambda,x=b \cos \lambda \quad y=b \sin \lambda,
or logarithmic spirals
x=be^(k lambda)cos lambdaquad y=be^(k lambda)sin lambda.x=b e^{k \lambda} \cos \lambda \quad y=b e^{k \lambda} \sin \lambda .
Other parametric families appeared in astrophysics, figure-eight curves
x=cos lambda,y=b sin 2lambdax=\cos \lambda, y=b \sin 2 \lambda
detected by NN - body simulations in barred galaxies being studied by Caranicolas (1998).
These special families of curves have in common the property that, under natural conditions, they can be transformed in the "classical" form (1) of families in the inverse problem. In the following it is shown that this is true for the class of similar curves.
These special families of curves have in common the property that, under natural conditions, they can be transformed in the "classical" form (1) of families in the inverse problem. In the following it is shown that this is true for the class of similar curves.
3 Families of similar curves
The curves in a family given in parametric form are called self-similar if the family is given by
{:(8)x=b phi(lambda)quad y=b psi(lambda)",":}\begin{equation*}
x=b \phi(\lambda) \quad y=b \psi(\lambda), \tag{8}
\end{equation*}
they are called xx-similar if the family is given by
{:(9)x=b phi(lambda)quad y=psi(lambda)",":}\begin{equation*}
x=b \phi(\lambda) \quad y=\psi(\lambda), \tag{9}
\end{equation*}
and yy-similar if
{:(10)x=phi(lambda)quad y=b psi(lambda)",":}\begin{equation*}
x=\phi(\lambda) \quad y=b \psi(\lambda), \tag{10}
\end{equation*}
with b inI_(0)subR^(+)b \in I_{0} \subset \mathbb{R}^{+}or R^(-),lambda in I subR\mathbb{R}^{-}, \lambda \in I \subset \mathbb{R} and phi,psi in C(I)\phi, \psi \in C(I).
A family of curves can be described in the implicit way f(x,y)=c(1)f(x, y)=c(1) or in parametric form x=F(lambda,b)x=F(\lambda, b), y=G(lambda,b)y=G(\lambda, b) (7), the parameter bb varying along the family and lambda\lambda varying on each curve for a fixed bb. The parametric description of the family can be regarded as a transformation from a domain in the xyx y Cartesian plane to a domain in the lambda b\lambda b plane. As mentioned by Anisiu and Pal (1999), this transformation is one-to-one and with a C^(1)C^{1} inverse (at least locally) if it is of C^(1)C^{1} class and has the Jacobian
A family of curves can be described in the implicit way f(x,y)=c(1)f(x, y)=c(1) or in parametric form x=F(lambda,b)x=F(\lambda, b), y=G(lambda,b)y=G(\lambda, b) (7), the parameter bb varying along the family and lambda\lambda varying on each curve for a fixed bb. The parametric description of the family can be regarded as a transformation from a domain in the xyx y Cartesian plane to a domain in the lambda b\lambda b plane. As mentioned by Anisiu and Pal (1999), this transformation is one-to-one and with a C^(1)C^{1} inverse (at least locally) if it is of C^(1)C^{1} class and has the Jacobian
J=F_(lambda)G_(b)-F_(b)G_(lambda)J=F_{\lambda} G_{b}-F_{b} G_{\lambda}
different from zero.
Proposition 1 For phi,psi inC^(1)(I)\phi, \psi \in C^{1}(I) the Jacobians of the transformations (8), (9) and (10) are respectively
Proposition 1 For phi,psi inC^(1)(I)\phi, \psi \in C^{1}(I) the Jacobians of the transformations (8), (9) and (10) are respectively
{:[J_(1)=b(phi^(')(lambda)psi(lambda)-phi(lambda)psi^(')(lambda))],[J_(2)=-phi(lambda)psi^(')(lambda)],[J_(3)=phi^(')(lambda)psi(lambda)]:}\begin{gathered}
J_{1}=b\left(\phi^{\prime}(\lambda) \psi(\lambda)-\phi(\lambda) \psi^{\prime}(\lambda)\right) \\
J_{2}=-\phi(\lambda) \psi^{\prime}(\lambda) \\
J_{3}=\phi^{\prime}(\lambda) \psi(\lambda)
\end{gathered}
For these specific transformations we can give the expression of the inverse. This will allow us to find the corresponding families of curves in the xyx y plane which originate from families of similar curves.
4 The case of self-similar curves
Let the functions phi\phi and psi\psi in (8) be given so that the Jacobian J_(1)J_{1} is different from 0 and phi(lambda)!=0\phi(\lambda) \neq 0 for each lambda in I\lambda \in I; it follows that (psi )/(phi)\frac{\psi}{\phi} is one-to-one. A similar result holds for psi(lambda)!=0\psi(\lambda) \neq 0 for each lambda in I,(phi )/(psi)\lambda \in I, \frac{\phi}{\psi} being one-to-one. Then from (8) we obtain
(y)/(x)=(psi )/(phi)(lambda).\frac{y}{x}=\frac{\psi}{\phi}(\lambda) .
So we get
{:(11)lambda=((psi )/(phi))^(-1)((y)/(x)):}\begin{equation*}
\lambda=\left(\frac{\psi}{\phi}\right)^{-1}\left(\frac{y}{x}\right) \tag{11}
\end{equation*}
and then
{:(12)b=sqrt((x^(2)+y^(2))/(phi^(2)(lambda)+psi^(2)(lambda)))",":}\begin{equation*}
b=\sqrt{\frac{x^{2}+y^{2}}{\phi^{2}(\lambda)+\psi^{2}(\lambda)}}, \tag{12}
\end{equation*}
if I subR^(+)I \subset \mathbb{R}^{+}(for I subR^(-),b=-sqrt((x^(2)+y^(2))/(phi^(2)(lambda)+psi^(2)(lambda)))I \subset \mathbb{R}^{-}, b=-\sqrt{\frac{x^{2}+y^{2}}{\phi^{2}(\lambda)+\psi^{2}(\lambda)}} ), lambda\lambda being given by (11). The case of self-similar curves gives rise to the family of curves (12) described in the classical way for the inverse problem of dynamics. The function ff which appears in this case can be written as
{:(13)f_(1)(x","y)=xF_(1)((y)/(x))","x > 0:}\begin{equation*}
f_{1}(x, y)=x F_{1}\left(\frac{y}{x}\right), x>0 \tag{13}
\end{equation*}
(respectively f_(1)(x,y)=-xF_(1)((y)/(x))f_{1}(x, y)=-x F_{1}\left(\frac{y}{x}\right) for x < 0x<0 ), where
{:(14)F_(1)(u)=sqrt((1+u^(2))/(phi^(2)(((psi )/(phi))^(-1)(u))+psi^(2)(((psi )/(phi))^(-1)(u)))):}\begin{equation*}
F_{1}(u)=\sqrt{\frac{1+u^{2}}{\phi^{2}\left(\left(\frac{\psi}{\phi}\right)^{-1}(u)\right)+\psi^{2}\left(\left(\frac{\psi}{\phi}\right)^{-1}(u)\right)}} \tag{14}
\end{equation*}
If the functions are of C^(1)C^{1} class and we calculate gamma=(f_(y))/(f_(x))\gamma=\frac{f_{y}}{f_{x}} corresponding to ff, we obtain
{:(15)gamma_(1)=(F_(1)^(')((y)/(x)))/(F_(1)((y)/(x))-F_(1)^(')((y)/(x))(y)/(x)):}\begin{equation*}
\gamma_{1}=\frac{F_{1}^{\prime}\left(\frac{y}{x}\right)}{F_{1}\left(\frac{y}{x}\right)-F_{1}^{\prime}\left(\frac{y}{x}\right) \frac{y}{x}} \tag{15}
\end{equation*}
The Jacobian can be written as
J_(1)=-bphi^(2)(lambda)((psi )/(phi))^(')(lambda)J_{1}=-b \phi^{2}(\lambda)\left(\frac{\psi}{\phi}\right)^{\prime}(\lambda)
We have proved
Theorem 2 Let phi,psi inC^(1)(I)\phi, \psi \in C^{1}(I) be given so that phi(lambda)!=0\phi(\lambda) \neq 0 and ((psi )/(phi))^(')(lambda)!=0\left(\frac{\psi}{\phi}\right)^{\prime}(\lambda) \neq 0 for each lambda in I\lambda \in I. The family (8) can be described in the form (1) with ff given by (13)-(14), and gamma\gamma given by the function in (15) which is homogeneous of degree zero in xx and yy.
5 The case of xx-similar curves
Let the functions phi\phi and psi\psi in (9) be given so that phi(lambda)!=0\phi(\lambda) \neq 0 and psi^(')(lambda)!=0\psi^{\prime}(\lambda) \neq 0 for each lambda in I\lambda \in I; it follows that psi\psi is one-to-one. From the second equality in (9) we obtain lambda=psi^(-1)(y)\lambda=\psi^{-1}(y), and from the first one
b=(x)/(phi@psi^(-1)(y))b=\frac{x}{\phi \circ \psi^{-1}(y)}
The function ff which can describe the family of orbits in this case is of the form
{:(16)f_(2)(x","y)=(x)/(F_(2)(y)):}\begin{equation*}
f_{2}(x, y)=\frac{x}{F_{2}(y)} \tag{16}
\end{equation*}
with
{:(17)F_(2)(y)=phi@psi^(-1)(y).:}\begin{equation*}
F_{2}(y)=\phi \circ \psi^{-1}(y) . \tag{17}
\end{equation*}
For gamma\gamma we obtain the value
{:(18)gamma_(2)=-(xF_(2)^(')(y))/(F_(2)(y)):}\begin{equation*}
\gamma_{2}=-\frac{x F_{2}^{\prime}(y)}{F_{2}(y)} \tag{18}
\end{equation*}
In this case we have
Theorem 3 Let phi,psi inC^(1)(I)\phi, \psi \in C^{1}(I) be given so that phi(lambda)!=0\phi(\lambda) \neq 0 and psi^(')(lambda)!=0\psi^{\prime}(\lambda) \neq 0 for each lambda in I\lambda \in I. The family (9) can be written in the form (1) with ff given by (16)-(17), gamma\gamma being given by the expression with separable variables in (18).
Theorem 3 Let phi,psi inC^(1)(I)\phi, \psi \in C^{1}(I) be given so that phi(lambda)!=0\phi(\lambda) \neq 0 and psi^(')(lambda)!=0\psi^{\prime}(\lambda) \neq 0 for each lambda in I\lambda \in I. The family (9) can be written in the form (1) with ff given by (16)-(17), gamma\gamma being given by the expression with separable variables in (18).
6 The case of yy-similar curves
This case is analogous to the case of xx-similar curves. Considering phi\phi and psi\psi so that psi(lambda)!=0\psi(\lambda) \neq 0 and phi^(')(lambda)!=0\phi^{\prime}(\lambda) \neq 0 for each lambda in I\lambda \in I (hence phi\phi one-to-one), from the first equality (10) we obtain lambda=phi^(-1)(x)\lambda=\phi^{-1}(x), and from the second one
b=(y)/(psi@phi^(-1)(x))b=\frac{y}{\psi \circ \phi^{-1}(x)}
The function ff is then given by
{:(19)f_(3)(x","y)=(y)/(F_(3)(x)):}\begin{equation*}
f_{3}(x, y)=\frac{y}{F_{3}(x)} \tag{19}
\end{equation*}
where
{:(20)F_(3)(x)=psi@phi^(-1)(x):}\begin{equation*}
F_{3}(x)=\psi \circ \phi^{-1}(x) \tag{20}
\end{equation*}
gamma\gamma being
{:(21)gamma_(3)=-(F_(3)(x))/(yF_(3)^(')(x)):}\begin{equation*}
\gamma_{3}=-\frac{F_{3}(x)}{y F_{3}^{\prime}(x)} \tag{21}
\end{equation*}
We can state
Theorem 4 Let phi,psi inC^(1)(I)\phi, \psi \in C^{1}(I) be given so that psi(lambda)!=0\psi(\lambda) \neq 0 and phi^(')(lambda)!=0\phi^{\prime}(\lambda) \neq 0 for each lambda in I\lambda \in I. The family (10) can be written in the implicit form (1) with ff given by (19)-(20), gamma\gamma being a function with separable variables expressed in (21).
Theorem 4 Let phi,psi inC^(1)(I)\phi, \psi \in C^{1}(I) be given so that psi(lambda)!=0\psi(\lambda) \neq 0 and phi^(')(lambda)!=0\phi^{\prime}(\lambda) \neq 0 for each lambda in I\lambda \in I. The family (10) can be written in the implicit form (1) with ff given by (19)-(20), gamma\gamma being a function with separable variables expressed in (21).
7 Conclusions and examples
As it was mentioned in the introduction, Grigoriadou et al (1999) proved that the basic partial differential equation (3) of the inverse problem can be integrated by quadratures in the case that the function gamma\gamma is homogeneous of order 0 in xx and yy, or with separable variables. The results in the above theorems allow us to rewrite any family of similar curves (satisfying some natural conditions) in the classical form, and, more than that, to obtain the potential satisfying equation (3) by quadratures. So, for these types of families it is useful to deal with the classical description instead of the parametric one.
Let us consider the family of figure-eight curves
x=cos lambda,y=b sin 2lambdax=\cos \lambda, y=b \sin 2 \lambda
studied by Caranicolas (1997), for which an approximate potential was found (containing the first terms of a power series in the small parameter bb ). Applying theorem 4 for this yy-similar family of curves with b in(0,oo),lambda in(0,(pi)/(2))b \in(0, \infty), \lambda \in\left(0, \frac{\pi}{2}\right) one obtains f_(3)(x,y)=(y)/(xsqrt(1-x^(2))),x in(0,1),y in(0,oo)f_{3}(x, y)=\frac{y}{x \sqrt{1-x^{2}}}, x \in(0,1), y \in(0, \infty), which gives rise to the family
{:(22)(y^(2))/(x^(2)(1-x^(2)))=c:}\begin{equation*}
\frac{y^{2}}{x^{2}\left(1-x^{2}\right)}=c \tag{22}
\end{equation*}
the functions gamma\gamma and Gamma\Gamma in (4) are in this case
gamma=-(x(x^(2)-1))/(y(2x^(2)-1)),quad Gamma=-(x^(3)(x^(2)-1)(2x^(2)-3))/(y^(2)(2x^(2)-1)^(3))\gamma=-\frac{x\left(x^{2}-1\right)}{y\left(2 x^{2}-1\right)}, \quad \Gamma=-\frac{x^{3}\left(x^{2}-1\right)\left(2 x^{2}-3\right)}{y^{2}\left(2 x^{2}-1\right)^{3}}
The first equation of the subsidiary system of ordinary differential equations for Szebehely's partial differential equation (3) is
(dy)/(dx)=-(x(x^(2)-1))/(y(2x^(2)-1))\frac{d y}{d x}=-\frac{x\left(x^{2}-1\right)}{y\left(2 x^{2}-1\right)}
which gives the integral
{:(23)y^(2)+(1)/(2)x^(2)-(1)/(4)ln|(2x^(2)-1)/(2)|=c_(1):}\begin{equation*}
y^{2}+\frac{1}{2} x^{2}-\frac{1}{4} \ln \left|\frac{2 x^{2}-1}{2}\right|=c_{1} \tag{23}
\end{equation*}
The second equation is
(dV)/(dx)=u(x,c_(1))V-u(x,c_(1)) bar(E)(x,c_(1))\frac{d V}{d x}=u\left(x, c_{1}\right) V-u\left(x, c_{1}\right) \bar{E}\left(x, c_{1}\right)
where u(x,c_(1))u\left(x, c_{1}\right) and bar(E)(x,c_(1))\bar{E}\left(x, c_{1}\right) are obtained from (2Gamma(x,y))/(1+gamma^(2)(x,y))\frac{2 \Gamma(x, y)}{1+\gamma^{2}(x, y)}, respectively E(f(x,y))E(f(x, y)), by substituting y^(2)=c_(1)-(1)/(2)x^(2)+(1)/(4)ln|(2x^(2)-1)/(2)|y^{2}=c_{1}- \frac{1}{2} x^{2}+\frac{1}{4} \ln \left|\frac{2 x^{2}-1}{2}\right|. We obtain
V=exp(int^(x)u(s,c_(1))ds)(c_(2)-int^(x)u(s,c_(1))( bar(E))(s,c_(1))exp(-int^(s)u(t,c_(1))dt)ds)V=\exp \left(\int^{x} u\left(s, c_{1}\right) d s\right)\left(c_{2}-\int^{x} u\left(s, c_{1}\right) \bar{E}\left(s, c_{1}\right) \exp \left(-\int^{s} u\left(t, c_{1}\right) d t\right) d s\right)
The calculations show that the function uu has the form
u(x,c_(1))=(v^(')(x,c_(1)))/(v(x,c_(1)))u\left(x, c_{1}\right)=\frac{v^{\prime}\left(x, c_{1}\right)}{v\left(x, c_{1}\right)}
where v^(')v^{\prime} denotes the derivative with respect to xx of the function
v(x,c_(1))=c_(1)-(1)/(2)x^(2)+(1)/(4)ln|(2x^(2)-1)/(2)|+(x^(2)(x^(2)-1)^(2))/((2x^(2)-1)^(2))v\left(x, c_{1}\right)=c_{1}-\frac{1}{2} x^{2}+\frac{1}{4} \ln \left|\frac{2 x^{2}-1}{2}\right|+\frac{x^{2}\left(x^{2}-1\right)^{2}}{\left(2 x^{2}-1\right)^{2}}
The expression of VV can be written
V=v(x,c_(1))(c_(2)-int^(x)(v^(')(s,c_(1)))/(v^(2)(s,c_(1)))( bar(E))(s,c_(1))ds)V=v\left(x, c_{1}\right)\left(c_{2}-\int^{x} \frac{v^{\prime}\left(s, c_{1}\right)}{v^{2}\left(s, c_{1}\right)} \bar{E}\left(s, c_{1}\right) d s\right)
After an integration by parts we get
V= bar(E)(x,c_(1))+v(x,c_(1))(c_(2)+I(x,c_(1))),V=\bar{E}\left(x, c_{1}\right)+v\left(x, c_{1}\right)\left(c_{2}+I\left(x, c_{1}\right)\right),
where
{:(24)I(x,c_(1))=-int^(x)E^(')((c_(1)-(1)/(2)s^(2)+(1)/(4)ln|(2s^(2)-1)/(2)|)/(s^(2)(1-s^(2))))(2(2s^(2)-1))/(s^(3)(s^(2)-1)^(2))ds:}\begin{equation*}
I\left(x, c_{1}\right)=-\int^{x} E^{\prime}\left(\frac{c_{1}-\frac{1}{2} s^{2}+\frac{1}{4} \ln \left|\frac{2 s^{2}-1}{2}\right|}{s^{2}\left(1-s^{2}\right)}\right) \frac{2\left(2 s^{2}-1\right)}{s^{3}\left(s^{2}-1\right)^{2}} d s \tag{24}
\end{equation*}
and E^(')E^{\prime} denotes the derivative of the function EE with respect to its argument.
The general solution of the partial differential equation (3) will be given by c_(2)=A(c_(1))c_{2}=A\left(c_{1}\right) with AA an arbitrary function of c_(1)c_{1} from (23). For the family of curves (22) traced with a preassigned energy E(f)E(f), the potentials creating it are given by
The general solution of the partial differential equation (3) will be given by c_(2)=A(c_(1))c_{2}=A\left(c_{1}\right) with AA an arbitrary function of c_(1)c_{1} from (23). For the family of curves (22) traced with a preassigned energy E(f)E(f), the potentials creating it are given by
{:(25)V(x","y)=E(f(x","y))+(y^(2)+(x^(2)(x^(2)-1)^(2))/((2x^(2)-1)^(2)))(A(c_(1))+I(x,c_(1)))",":}\begin{equation*}
V(x, y)=E(f(x, y))+\left(y^{2}+\frac{x^{2}\left(x^{2}-1\right)^{2}}{\left(2 x^{2}-1\right)^{2}}\right)\left(A\left(c_{1}\right)+I\left(x, c_{1}\right)\right), \tag{25}
\end{equation*}
where II is given by (24) and c_(1)=y^(2)+(1)/(2)x^(2)-(1)/(4)ln|(2x^(2)-1)/(2)|c_{1}=y^{2}+\frac{1}{2} x^{2}-\frac{1}{4} \ln \left|\frac{2 x^{2}-1}{2}\right|.
It is known that during the motion of a material point of unit mass along an orbit of the family the inequality
It is known that during the motion of a material point of unit mass along an orbit of the family the inequality
{:(26)E(f(x","y))-V(x","y) >= 0",":}\begin{equation*}
E(f(x, y))-V(x, y) \geq 0, \tag{26}
\end{equation*}
must be observed, so real motion will take place only in the region where
{:(27)A(c_(1))+I(x,c_(1)) <= 0",":}\begin{equation*}
A\left(c_{1}\right)+I\left(x, c_{1}\right) \leq 0, \tag{27}
\end{equation*}
c_(1)c_{1} being given by (23).
If we are interested in finding the potentials producing families of orbits with constant energy (i.e. the energy has the same value ee for all the curves in the family), we obtain
If we are interested in finding the potentials producing families of orbits with constant energy (i.e. the energy has the same value ee for all the curves in the family), we obtain
V(x,y)=e+(y^(2)+(x^(2)(x^(2)-1)^(2))/((2x^(2)-1)^(2)))A(y^(2)+(1)/(2)x^(2)-(1)/(4)ln|(2x^(2)-1)/(2)|),V(x, y)=e+\left(y^{2}+\frac{x^{2}\left(x^{2}-1\right)^{2}}{\left(2 x^{2}-1\right)^{2}}\right) A\left(y^{2}+\frac{1}{2} x^{2}-\frac{1}{4} \ln \left|\frac{2 x^{2}-1}{2}\right|\right),
in this case II being identically null.
If we choose E(f)=f+1E(f)=f+1, it follows E^(')(f)=1E^{\prime}(f)=1 and I(x,c_(1))I\left(x, c_{1}\right) will depend only on xx,
If we choose E(f)=f+1E(f)=f+1, it follows E^(')(f)=1E^{\prime}(f)=1 and I(x,c_(1))I\left(x, c_{1}\right) will depend only on xx,
I(x,c_(1))=(1)/(x^(2)(x^(2)-1)).I\left(x, c_{1}\right)=\frac{1}{x^{2}\left(x^{2}-1\right)} .
The value of the potential will be obtained from (25)
V(x,y)=(x^(2)(4x^(2)-3))/((2x^(2)-1)^(2))+(y^(2)+(x^(2)(x^(2)-1)^(2))/((2x^(2)-1)^(2)))A(y^(2)+(1)/(2)x^(2)-(1)/(4)ln|(2x^(2)-1)/(2)|).V(x, y)=\frac{x^{2}\left(4 x^{2}-3\right)}{\left(2 x^{2}-1\right)^{2}}+\left(y^{2}+\frac{x^{2}\left(x^{2}-1\right)^{2}}{\left(2 x^{2}-1\right)^{2}}\right) A\left(y^{2}+\frac{1}{2} x^{2}-\frac{1}{4} \ln \left|\frac{2 x^{2}-1}{2}\right|\right) .
In this case, considering the special value of the arbitrary function A,A(z)=4A, A(z)=4 we obtain the potential
V_(p)(x,y)=4y^(2)+x^(2)V_{p}(x, y)=4 y^{2}+x^{2}
which is a two-dimensional harmonic oscillator potential with the ratio of frequencies 1:21: 2. This represents a very simple potential producing figure-eight orbits.
We have proved that the case of orbits (22) given in parametric form can be successfully treated by using the classical implicit description of the family. The most general form of potentials producing this family is given by (25) and, choosing the dependence of the energy on the family, we can obtain specific potentials. The inequality (27) allows us to programme motion in some regions of the space chosen in advance, having at our disposal the arbitrary function AA. The possibility of programming motion was studied in detail by Anisiu and Bozis (2000) for the case of families of curves f(x,y)=y-h(x)f(x, y)=y-h(x); for that type of families the integration by quadratures of Szebehely's equation can also be accomplished.
References
[1] Anisiu, M.-C., Pal, A., Two-dimensional inverse problem of dynamics for families in parametric form, Inverse Problems 15 (1999), 135-140
[2] Anisiu, M.-C., Bozis, G., Programmed motion for a class of families of planar orbits, Inverse Problems 16 (2000), 19-32
[3] Bozis, G., Inverse problem with two parametric families of planar orbits, Celest. Mech. 31 (1983), 129-142
[4] Bozis, G., Caranicolas, N. D., The Earth and the Universe, Special Volume of the Aristotle University of Thessaloniki (ed. Asteriadis et al), 1997
[5] Bozis, G., Borghero, F., A new formulation of the two-dimensional inverse problem of dynamics, Inverse Problems 14 (1998), 41-51
[6] Caranicolas, N. D., Potentials for the central parts of a barred galaxy, Astronomy and Astrophysics 332 (1998), 88-92
[7] Grigoriadou, S., Bozis, G., Elmabsout, B., Solvable cases of Szebehely's equation, Celest. Mech. 74 (1999), 211-221
[8] Szebehely, V., On the determination of the potential by satellite observations, Proc. Int. Meeting on Earth's Rotations by Satellite Observations Ed. E. Proverbio (Bologna: Pitagora Editrice), 1974, 31-35
[2] Anisiu, M.-C., Bozis, G., Programmed motion for a class of families of planar orbits, Inverse Problems 16 (2000), 19-32
[3] Bozis, G., Inverse problem with two parametric families of planar orbits, Celest. Mech. 31 (1983), 129-142
[4] Bozis, G., Caranicolas, N. D., The Earth and the Universe, Special Volume of the Aristotle University of Thessaloniki (ed. Asteriadis et al), 1997
[5] Bozis, G., Borghero, F., A new formulation of the two-dimensional inverse problem of dynamics, Inverse Problems 14 (1998), 41-51
[6] Caranicolas, N. D., Potentials for the central parts of a barred galaxy, Astronomy and Astrophysics 332 (1998), 88-92
[7] Grigoriadou, S., Bozis, G., Elmabsout, B., Solvable cases of Szebehely's equation, Celest. Mech. 74 (1999), 211-221
[8] Szebehely, V., On the determination of the potential by satellite observations, Proc. Int. Meeting on Earth's Rotations by Satellite Observations Ed. E. Proverbio (Bologna: Pitagora Editrice), 1974, 31-35