Pencils of straight lines in logarithmic potentials

Mira Anisiu
(original), paper

Abstract

The aim of the planar inverse problem of dynamics is to find the potentials under whose action a material point of unit mass, with appropriate initial conditions, describes the curves in a given family. We solve the following special problem: determine the finite Borel measures, with support in the unit circle, whose logarithmic potentials give rise to a family of lines passing through a given point.

Authors

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

Valeriu Anisiu
Universitatea ,,Babe¸s-Bolyai”

Keywords

Inverse problem of dynamics; logarithmic potential

Paper coordinates

M.-C. Anisiu, V. Anisiu, Pencils of straight lines in logarithmic potentials, Mathematica (Cluj) 48 (71), No. 2 (2006), 121-125

PDF

https://math.ubbcluj.ro/~mathjour/fulltext/2006-2/anisiu-anisiu.pdf

About this paper

Journal

Mathematica

Publisher Name

Publishing House of the Romanian Academy

DOI
Print ISSN

12229016

Online ISSN

2601744X

google scholar link

[1] Anisiu, M.-C., The Equations of the Inverse Problem of Dynamics, House of the Book of Science, Cluj-Napoca, 2003 (Romanian).
[2] Anisiu, M.-C., An alternative point of view on the equations of the inverse problem of dynamics, Inverse Problems, 20 (2004), 1865–1872.
[3] Antonov, V. A. and Timoshkova, E. I., Simple trajectories in a rotationally symmetric gravitational field, Astronom. Rep., 37 (1993), 138–144.
[4] Betsakos, D. and Grigoriadou, S., On the determination of a measure by the orbits generated by its logarithmic potential, Proc. Amer. Math. Soc., 134 (2006), 541–548.
[5] Bozis, G., Inverse problem with two-parametric families of planar orbits, Celest. Mech., 31 (1983), 129–143.
[6] Bozis, G., The inverse problem of dynamics. Basic facts, Inverse Problems, 11 (1995), 687–708.
[7] Bozis, G. and Anisiu, M.-C., Families of straight lines in planar potentials, Rom. Astron. J., 11 (2001), 27–43.
[8] Caranicolas, N. D. and Innanen, K. A., Periodic motion in perturbed elliptic oscillators, Astronom. J., 103 (1992), 1308–1312.
[9] Contopoulos, G. and Zikides, M., Periodic orbits and ergodic components of a resonant dynamical system, Astronom. Astrophys., 90 (1980), 198–203.
[10] Galiullin, A. S., Inverse Problem of Dynamics, Mir Publishers, Moscow, 1984.
[11] Grigoriadou, S., The inverse problem of dynamics and Darboux’s integrability criterion, Inverse Problems, 15 (1999), 1621–1637.
[12] HeNon, M. and Heiles, C., The applicability of the third integral of motion: some numerical experiments, Astronom. J., 69 (1964), 73–79.
[13] van der Merwe, P. du T., Solvable forms of a generalized H´enon-Heiles system, Physics Letters A, 156 (1991), 216–220.
[14] Ransford, T., Potential Theory in the Complex Plane, Cambridge University Press, 1995.
[15] Rudin, W., Real and Complex Analysis, Third Edition, McGraw-Hill, 1987.
[16] Szebehely, V., On the determination of the potential by satellite observation, Proceedings of the International Meeting on Earth’s Rotations by Satellite Observations, University of Cagliari, Italy, ed. E. Proverbio, 1974, 31–35.

2006-anisiu-anisiu_pencils

PENCILS OF STRAIGHT LINES IN LOGARITHMIC POTENTIALS

MIRA-CRISTIANA ANISIU and VALERIU ANISIU

Abstract

The aim of the planar inverse problem of dynamics is to find the potentials under whose action a material point of unit mass, with appropriate initial conditions, describes the curves in a given family. We solve the following special problem: determine the finite Borel measures, with support in the unit circle, whose logarithmic potentials give rise to a family of lines passing through a given point.

MSC 2000. 31A05, 70D05.
Key words. Inverse problem of dynamics, logarithmic potential.

1. INTRODUCTION

The goal of the classical inverse problem of dynamics is to find the planar potentials V = V ( x , y ) V = V ( x , y ) V=V(x,y)V=V(x, y)V=V(x,y) creating preassigned families of orbits, traced by a material point of unit mass. Reviews of this and of other versions of the inverse problem can be found in [10], [6] and [1].
The equations governing the motion of the particle are
(1) x ¨ = − V x y ¨ = − V y . (1) x ¨ = − V x y ¨ = − V y . {:(1)x^(¨)=-V_(x)quady^(¨)=-V_(y).:}\begin{equation*} \ddot{x}=-V_{x} \quad \ddot{y}=-V_{y} . \tag{1} \end{equation*}(1)x¨=−Vxy¨=−Vy.
The very simple families of straight lines have been considered only recently. The interest in such families was raised by the fact that isolated straight line solutions have been found in galactic models by Contopoulos and Zikides [9] and by Caranicolas and Innanen [8]. Straight lines appear also in the HénonHeiles model [12] (van der Merwe [13], Antonov and Timoshkova [3]). Some families of straight lines were studied by Grigoriadou [11] in connection with the problem of Darboux integrability.
A monoparametric family of curves
(2) f ( x , y ) = c (2) f ( x , y ) = c {:(2)f(x","y)=c:}\begin{equation*} f(x, y)=c \tag{2} \end{equation*}(2)f(x,y)=c
is determined by the slope function
(3) γ = f y f x , (3) γ = f y f x , {:(3)gamma=(f_(y))/(f_(x))",":}\begin{equation*} \gamma=\frac{f_{y}}{f_{x}}, \tag{3} \end{equation*}(3)γ=fyfx,
where the subscripts denote partial differentiation. To each f f fff there corresponds obviously one γ γ gamma\gammaγ and to each γ γ gamma\gammaγ there corresponds just one monoparametric family (2). We define also the function
(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
which can be expressed in terms of the derivatives of f f fff as
Γ = 2 f x y f x f y − f x x f y 2 − f y y f x 2 f x 3 Γ = 2 f x y f x f y − f x x f y 2 − f y y f x 2 f x 3 Gamma=(2f_(xy)f_(x)f_(y)-f_(xx)f_(y)^(2)-f_(yy)f_(x)^(2))/(f_(x)^(3))\Gamma=\frac{2 f_{x y} f_{x} f_{y}-f_{x x} f_{y}^{2}-f_{y y} f_{x}^{2}}{f_{x}^{3}}Γ=2fxyfxfy−fxxfy2−fyyfx2fx3
It follows that the curvature of the orbits in (2) is given by
K = | Γ | / ( 1 + γ 2 ) 3 / 2 K = | Γ | / 1 + γ 2 3 / 2 K=|Gamma|//(1+gamma^(2))^(3//2)K=|\Gamma| /\left(1+\gamma^{2}\right)^{3 / 2}K=|Γ|/(1+γ2)3/2
therefore the family (2) consists of straight lines if and only if Γ = 0 Γ = 0 Gamma=0\Gamma=0Γ=0. In view of (4) this condition may be written as
(5) γ γ x − γ y = 0 (5) γ γ x − γ y = 0 {:(5)gammagamma_(x)-gamma_(y)=0:}\begin{equation*} \gamma \gamma_{x}-\gamma_{y}=0 \tag{5} \end{equation*}(5)γγx−γy=0
The potentials which produce the family of straight lines (2) (for which (5) is fulfilled) satisfy the linear first order equation
(6) V x + γ V y = 0 (6) V x + γ V y = 0 {:(6)V_(x)+gammaV_(y)=0:}\begin{equation*} V_{x}+\gamma V_{y}=0 \tag{6} \end{equation*}(6)Vx+γVy=0
Equation (6) was derived in [7] as a consequence of the equation of Szebehely [16], written in terms of γ γ gamma\gammaγ and Γ Γ Gamma\GammaΓ in [5]; later it was obtained directly in [2]. Expressing γ γ gamma\gammaγ from (6) and introducing its value into (5), a nonlinear partial differential equation
(7) V x V y ( V x x − V y y ) = V x y ( V x 2 − V y 2 ) (7) V x V y V x x − V y y = V x y V x 2 − V y 2 {:(7)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{7} \end{equation*}(7)VxVy(Vxx−Vyy)=Vxy(Vx2−Vy2)
was given in [7], which must be satisfied by all potentials creating (among other orbits) a family of straight lines. It is obvious that the potential will not be uniquely determined.
It can be easily checked that for a family of straight lines through a fixed point ( x 0 , y 0 ) x 0 , y 0 (x_(0),y_(0))\left(x_{0}, y_{0}\right)(x0,y0) we have γ = − ( x − x 0 ) / ( y − y 0 ) γ = − x − x 0 / y − y 0 gamma=-(x-x_(0))//(y-y_(0))\gamma=-\left(x-x_{0}\right) /\left(y-y_{0}\right)γ=−(x−x0)/(y−y0), and from (6) we obtain V ( x , y ) = v ( ( x − x 0 ) 2 + ( y − y 0 ) 2 V ( x , y ) = v x − x 0 2 + y − y 0 2 V(x,y)=v((x-x_(0))^(2)+(y-y_(0))^(2):}V(x, y)= v\left(\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}\right.V(x,y)=v((x−x0)2+(y−y0)2 ), hence the potential is an arbitrary function of the distance to the point ( x 0 , y 0 ) x 0 , y 0 (x_(0),y_(0))\left(x_{0}, y_{0}\right)(x0,y0).

2. LOGARITHMIC POTENTIALS ASSOCIATED TO A BOREL MEASURE

Betsakos and Grigoriadou [4] considered the following type of inverse problem of dynamics: Given a monoparametric family of planar curves, find the finite Borel measures supported in the unit circle, whose logarithmic potentials generate the curves of the family. The problem was solved for families of straight lines through the origin or through the point ( 1 , 0 ) ( 1 , 0 ) (1,0)(1,0)(1,0), as well as for the family of circles centered at the origin. In what follows we shall consider the problem for pencils of lines through an arbitrary point of the plane.
Let σ σ sigma\sigmaσ be a finite Borel measure with support in a compact set K ⊂ C K ⊂ C K subCK \subset \mathbb{C}K⊂C. The logarithmic potential V σ : C → ( − ∞ , ∞ ] V σ : C → ( − ∞ , ∞ ] V_(sigma):Crarr(-oo,oo]V_{\sigma}: \mathbb{C} \rightarrow(-\infty, \infty]Vσ:C→(−∞,∞] is given by
(8) V σ ( z ) = ∫ K log ⁡ 1 | z − ζ | d σ ( ζ ) (8) V σ ( z ) = ∫ K   log ⁡ 1 | z − ζ | d σ ( ζ ) {:(8)V_(sigma)(z)=int_(K)log((1)/(|z-zeta|))dsigma(zeta):}\begin{equation*} V_{\sigma}(z)=\int_{K} \log \frac{1}{|z-\zeta|} \mathrm{d} \sigma(\zeta) \tag{8} \end{equation*}(8)Vσ(z)=∫Klog⁡1|z−ζ|dσ(ζ)
and is harmonic in the complement of its support ([14], Ch. 3).
Using a reflection principle for harmonic functions, the following theorem was proved in [4].
Theorem 1. [4] Let σ σ sigma\sigmaσ be a finite Borel measure with compact support K ⊂ K ⊂ K subK \subsetK⊂ C. Suppose that the logarithmic potential (8) generates an orbit α ∈ C ( I ) α ∈ C ( I ) alpha in C(I)\alpha \in C(I)α∈C(I) given by
α ( t ) = x ( t ) + i y ( t ) , t ∈ I α ( t ) = x ( t ) + i y ( t ) , t ∈ I alpha(t)=x(t)+iy(t),quad t in I\alpha(t)=x(t)+i y(t), \quad t \in Iα(t)=x(t)+iy(t),t∈I
( I I III being a real interval) that lies on a straight line ℓ ℓ ℓ\ellℓ. Then V σ V σ V_(sigma)V_{\sigma}Vσ is locally symmetric with respect to ℓ ℓ ℓ\ellℓ, i. e. V σ ( z ) = V σ ( z ^ ) V σ ( z ) = V σ ( z ^ ) V_(sigma)(z)=V_(sigma)( hat(z))V_{\sigma}(z)=V_{\sigma}(\hat{z})Vσ(z)=Vσ(z^) for all z z zzz in a neighbourhood of the trace { α ( t ) : t ∈ I } { α ( t ) : t ∈ I } {alpha(t):t in I}\{\alpha(t): t \in I\}{α(t):t∈I} of α , z ^ α , z ^ alpha, hat(z)\alpha, \hat{z}α,z^ being the reflection of z z zzz in ℓ ℓ ℓ\ellℓ.

3. MAIN RESULTS

From now on we shall consider that the finite Borel measure σ σ sigma\sigmaσ is supported in the unit circle T = { z ∈ C : | z | = 1 } T = { z ∈ C : | z | = 1 } T={z inC:|z|=1}\mathbb{T}=\{z \in \mathbb{C}:|z|=1\}T={z∈C:|z|=1}.
Proposition 1. The form of the logarithmic potential generated by the Lebesgue measure Λ Λ Lambda\LambdaΛ supported in T T T\mathbb{T}T is
V Λ ( z ) = 0 for | z | < 1 (9) V Λ ( z ) = − 2 π log ⁡ | z | for | z | > 1 V Λ ( z ) = 0  for  | z | < 1 (9) V Λ ( z ) = − 2 π log ⁡ | z |  for  | z | > 1 {:[V_(Lambda)(z)=0" for "|z| < 1],[(9)V_(Lambda)(z)=-2pi log |z|" for "|z| > 1]:}\begin{align*} & V_{\Lambda}(z)=0 \text { for }|z|<1 \\ & V_{\Lambda}(z)=-2 \pi \log |z| \text { for }|z|>1 \tag{9} \end{align*}VΛ(z)=0 for |z|<1(9)VΛ(z)=−2πlog⁡|z| for |z|>1
For the Dirac measure concentrated at z 0 ∈ T z 0 ∈ T z_(0)inTz_{0} \in \mathbb{T}z0∈T we obtain
(10) V δ z 0 ( z ) = log ⁡ 1 | z − z 0 | (10) V δ z 0 ( z ) = log ⁡ 1 z − z 0 {:(10)V_(delta_(z_(0)))(z)=log((1)/(|z-z_(0)|)):}\begin{equation*} V_{\delta_{z_{0}}}(z)=\log \frac{1}{\left|z-z_{0}\right|} \tag{10} \end{equation*}(10)Vδz0(z)=log⁡1|z−z0|
Proof. Jensen's formula ([15], p. 307, Theorem 15.18) states that if g g ggg with g ( 0 ) ≠ 0 g ( 0 ) ≠ 0 g(0)!=0g(0) \neq 0g(0)≠0 is holomorphic on a disk centered at 0 and having the radius greater than 1 , and α 1 , … , α N α 1 , … , α N alpha_(1),dots,alpha_(N)\alpha_{1}, \ldots, \alpha_{N}α1,…,αN are the zeros of g g ggg in D ― D ¯ bar(D)\overline{\mathbb{D}}D―, then
| g ( 0 ) | ∏ n = 1 N 1 | α n | = exp ⁡ ( 1 2 π ∫ − π π log ⁡ | g ( e i θ ) | d θ ) . | g ( 0 ) | ∏ n = 1 N   1 α n = exp ⁡ 1 2 π ∫ − π π   log ⁡ g e i θ d θ . |g(0)|prod_(n=1)^(N)(1)/(|alpha_(n)|)=exp((1)/(2pi)int_(-pi)^(pi)log|g(e^(itheta))|dtheta).|g(0)| \prod_{n=1}^{N} \frac{1}{\left|\alpha_{n}\right|}=\exp \left(\frac{1}{2 \pi} \int_{-\pi}^{\pi} \log \left|g\left(\mathrm{e}^{\mathrm{i} \theta}\right)\right| \mathrm{d} \theta\right) .|g(0)|∏n=1N1|αn|=exp⁡(12π∫−ππlog⁡|g(eiθ)|dθ).
When N = 0 N = 0 N=0N=0N=0, the product is considered 1. By taking g ( ζ ) = z − ζ g ( ζ ) = z − ζ g(zeta)=z-zetag(\zeta)=z-\zetag(ζ)=z−ζ, the left hand side equals | z | / | z | = 1 | z | / | z | = 1 |z|//|z|=1|z| /|z|=1|z|/|z|=1 if 0 < | z | < 1 0 < | z | < 1 0 < |z| < 10<|z|<10<|z|<1, and | z | | z | |z||z||z| if | z | > 1 | z | > 1 |z| > 1|z|>1|z|>1 (for z = 0 z = 0 z=0z=0z=0, V σ ( 0 ) = 0 V σ ( 0 ) = 0 V_(sigma)(0)=0V_{\sigma}(0)=0Vσ(0)=0 obviously); therefore (9) follows. The result for the Dirac measure is obtained by an easy calculation.
The next result expresses some properties of the logarithmic potential; a) and b) appear in the proof of the basic Theorem 4 in [4].
Theorem 2. Let V σ V σ V_(sigma)V_{\sigma}Vσ be the logarithmic potential given by (8).
a) If D D DDD is an open disk so that D ∩ T = ∅ D ∩ T = ∅ D nnT=O/D \cap \mathbb{T}=\emptysetD∩T=∅ and
(11) ∂ ∂ θ V σ ( z ) = 0 for each z = r e i θ ∈ D (11) ∂ ∂ θ V σ ( z ) = 0  for each  z = r e i θ ∈ D {:(11)(del)/(del theta)V_(sigma)(z)=0" for each "z=re^(itheta)in D:}\begin{equation*} \frac{\partial}{\partial \theta} V_{\sigma}(z)=0 \text { for each } z=r \mathrm{e}^{\mathrm{i} \theta} \in D \tag{11} \end{equation*}(11)∂∂θVσ(z)=0 for each z=reiθ∈D
then V σ ( z ) = 0 V σ ( z ) = 0 V_(sigma)(z)=0V_{\sigma}(z)=0Vσ(z)=0 for each z z zzz in the unit disk D = { z ∈ C : | z | < 1 } D = { z ∈ C : | z | < 1 } D={z inC:|z| < 1}\mathbb{D}=\{z \in \mathbb{C}:|z|<1\}D={z∈C:|z|<1}.
b) If V σ V σ V_(sigma)V_{\sigma}Vσ is constant in D D D\mathbb{D}D, then the Borel measure σ σ sigma\sigmaσ is a constant multiple of the Lebesgue measure Λ Λ Lambda\LambdaΛ on T T T\mathbb{T}T.
c) If the logarithmic potential V σ V σ V_(sigma)V_{\sigma}Vσ is constant in C ∖ D ― C ∖ D ¯ C\\ bar(D)\mathbb{C} \backslash \overline{\mathbb{D}}C∖D―, it follows that σ = 0 σ = 0 sigma=0\sigma=0σ=0.
Proof. The proof of part a) makes use of the fact that ∂ ∂ θ V σ ∂ ∂ θ V σ (del)/(del theta)V_(sigma)\frac{\partial}{\partial \theta} V_{\sigma}∂∂θVσ can be expressed using the Poisson transform associated to the measure σ σ sigma\sigmaσ. Part b) relies on the uniqueness of the Borel measure used in the representation of a harmonic function. Part c) follows from the fact that σ σ sigma\sigmaσ must be a multiple of the Lebesgue measure, σ = C ⋅ Λ σ = C ⋅ Λ sigma=C*Lambda\sigma=C \cdot \Lambdaσ=C⋅Λ, and from Proposition 1 V σ ( z ) = − 2 π C log ⁡ | z | 1 V σ ( z ) = − 2 π C log ⁡ | z | 1V_(sigma)(z)=-2pi C log |z|1 V_{\sigma}(z)=-2 \pi C \log |z|1Vσ(z)=−2πClog⁡|z| in C ∖ D ― C ∖ D ¯ C\\ bar(D)\mathbb{C} \backslash \overline{\mathbb{D}}C∖D―; therefore if V σ V σ V_(sigma)V_{\sigma}Vσ is constant in C ∖ D ― C ∖ D ¯ C\\ bar(D)\mathbb{C} \backslash \overline{\mathbb{D}}C∖D―, we have C = 0 C = 0 C=0C=0C=0.
We consider now the case when the potential V σ V σ V_(sigma)V_{\sigma}Vσ given by (8) gives rise to a pencil of lines through z 0 z 0 z_(0)z_{0}z0.
Theorem 3. Let D D DDD be an open disk and z 0 z 0 z_(0)z_{0}z0 a point so that D ∪ { z 0 } ⊆ D D ∪ z 0 ⊆ D D uu{z_(0)}subeDD \cup\left\{z_{0}\right\} \subseteq \mathbb{D}D∪{z0}⊆D or D ∪ { z 0 } ⊆ C ∖ D ― D ∪ z 0 ⊆ C ∖ D ¯ D uu{z_(0)}subeC\\ bar(D)D \cup\left\{z_{0}\right\} \subseteq \mathbb{C} \backslash \overline{\mathbb{D}}D∪{z0}⊆C∖D―. Let
(12) { s p : p ∈ J } (12) s p : p ∈ J {:(12){s_(p):p in J}:}\begin{equation*} \left\{s_{p}: p \in J\right\} \tag{12} \end{equation*}(12){sp:p∈J}
be the family of all chords in D D DDD passing through z 0 z 0 z_(0)z_{0}z0. If V σ V σ V_(sigma)V_{\sigma}Vσ generates the family (12), then V σ V σ V_(sigma)V_{\sigma}Vσ is constant on the connected component containing z 0 z 0 z_(0)z_{0}z0, hence σ = C ⋅ Λ σ = C ⋅ Λ sigma=C*Lambda\sigma=C \cdot \Lambdaσ=C⋅Λ. Furthermore, if z 0 ∈ C ∖ D ― z 0 ∈ C ∖ D ¯ z_(0)inC\\ bar(D)z_{0} \in \mathbb{C} \backslash \overline{\mathbb{D}}z0∈C∖D―, then σ = 0 σ = 0 sigma=0\sigma=0σ=0, i. e. C = 0 C = 0 C=0C=0C=0.
Proof. Using Theorem 1, we obtain that V σ V σ V_(sigma)V_{\sigma}Vσ is locally symmetric with respect to each line supporting s p s p s_(p)s_{p}sp. It results that V σ ( z ) V σ ( z ) V_(sigma)(z)V_{\sigma}(z)Vσ(z) depends only on | z − z 0 | z − z 0 |z-z_(0)|\left|z-z_{0}\right||z−z0|, as it was already shown in the Introduction; V σ V σ V_(sigma)V_{\sigma}Vσ being also harmonic, we have
(13) V σ ( z ) = a log ⁡ 1 | z − z 0 | + b , for each z ∈ D ∖ { z 0 } . (13) V σ ( z ) = a log ⁡ 1 z − z 0 + b ,  for each  z ∈ D ∖ z 0 . {:(13)V_(sigma)(z)=a log((1)/(|z-z_(0)|))+b","" for each "z in D\\{z_(0)}.:}\begin{equation*} V_{\sigma}(z)=a \log \frac{1}{\left|z-z_{0}\right|}+b, \text { for each } z \in D \backslash\left\{z_{0}\right\} . \tag{13} \end{equation*}(13)Vσ(z)=alog⁡1|z−z0|+b, for each z∈D∖{z0}.
The potential V σ V σ V_(sigma)V_{\sigma}Vσ being bounded at z = z 0 z = z 0 z=z_(0)z=z_{0}z=z0, it follows that a = 0 a = 0 a=0a=0a=0, hence V σ V σ V_(sigma)V_{\sigma}Vσ is constant on the connected component containing z 0 z 0 z_(0)z_{0}z0. Applying Theorem 2, we obtain the conclusion.
The case z 0 ∈ T z 0 ∈ T z_(0)inTz_{0} \in \mathbb{T}z0∈T is covered by Theorem 5 from [4], where z 0 z 0 z_(0)z_{0}z0 was chosen equal to 1 (which is possible by means of a rotation). We shall state the theorem for arbitrary z 0 z 0 z_(0)z_{0}z0.
Theorem 4. Let D D DDD be an open disk and consider the family (12) of all chords in D D DDD passing through z 0 ∈ T z 0 ∈ T z_(0)inTz_{0} \in \mathbb{T}z0∈T, generated by the logarithmic potential V σ V σ V_(sigma)V_{\sigma}Vσ.
If D ⊂ D D ⊂ D D subDD \subset \mathbb{D}D⊂D, then σ = C 1 ⋅ Λ + C 2 ⋅ δ z 0 σ = C 1 ⋅ Λ + C 2 ⋅ δ z 0 sigma=C_(1)*Lambda+C_(2)*delta_(z_(0))\sigma=C_{1} \cdot \Lambda+C_{2} \cdot \delta_{z_{0}}σ=C1⋅Λ+C2⋅δz0, where C 1 C 1 C_(1)C_{1}C1 and C 2 C 2 C_(2)C_{2}C2 are constants, and δ z 0 δ z 0 delta_(z_(0))\delta_{z_{0}}δz0 is the Dirac measure concentrated at z 0 z 0 z_(0)z_{0}z0.
If D ⊆ C ∖ D ― D ⊆ C ∖ D ¯ D subeC\\ bar(D)D \subseteq \mathbb{C} \backslash \overline{\mathbb{D}}D⊆C∖D―, then σ = C 3 ⋅ δ z 0 σ = C 3 ⋅ δ z 0 sigma=C_(3)*delta_(z_(0))\sigma=C_{3} \cdot \delta_{z_{0}}σ=C3⋅δz0, where C 3 C 3 C_(3)C_{3}C3 is a constant.
Remark 1. This type of inverse problem, treated here for pencils of lines, can be considered for various families of functions. In [4] it was proved that if a logarithmic potential V σ V σ V_(sigma)V_{\sigma}Vσ generates each circular arc in a disk D ⊂ C ∖ T D ⊂ C ∖ T D subC\\TD \subset \mathbb{C} \backslash \mathbb{T}D⊂C∖T, then σ = C ⋅ Λ σ = C ⋅ Λ sigma=C*Lambda\sigma=C \cdot \Lambdaσ=C⋅Λ, where C C CCC is a constant. We mention that if D ⊂ D D ⊂ D D subDD \subset \mathbb{D}D⊂D, then from Proposition 1 it follows that V σ ( z ) = 0 V σ ( z ) = 0 V_(sigma)(z)=0V_{\sigma}(z)=0Vσ(z)=0 in D D D\mathbb{D}D, and this potential does not produce any circle.
The problem of finding all logarithmic potentials which give rise to families of parallel lines is still open.

REFERENCES

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[12] HéNon, M. and Heiles, C., The applicability of the third integral of motion: some numerical experiments, Astronom. J., 69 (1964), 73-79.
[13] van der Merwe, P. du T., Solvable forms of a generalized Hénon-Heiles system, Physics Letters A, 156 (1991), 216-220.
[14] Ransford, T., Potential Theory in the Complex Plane, Cambridge University Press, 1995.
[15] Rudin, W., Real and Complex Analysis, Third Edition, McGraw-Hill, 1987.
[16] Szebehely, V., On the determination of the potential by satellite observation, Proceedings of the International Meeting on Earth's Rotations by Satellite Observations, University of Cagliari, Italy, ed. E. Proverbio, 1974, 31-35.
Received November 20, 2005

Institutul de Calcul Tiberiu PopoviciuC. P. 68, Cluj-NapocaE-mail: mira@math.ubbcluj.roUniversitatea „Babeş-Bolyai"Str. Kogălniceanu nr. 1400084 Cluj-Napoca, RomâniaE-mail: anisiu@math.ubbcluj.ro

2006

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