Abstract
We consider the problem : Given a planar region \(T_{orb}\) described by one inequality \(g(x,y)\leq c_{0}\), find the potentials \(V=V(x,y)\) which can generate monoparametric families of orbits \(f(x,y)=c\) (also to be found) lying exclusively in the region \(T_{orb}\). We make assumptions on the homogeneity of both the function \(g(x,y)\) describing the boundary of the region \(T_{orb}\) and of the slope function \(\U{3b3} (x,y)=fy/fx\) of the required family. We show that, under certain conditions, the slope function \(\U{3b3} (x,y)\) can be obtained as the common solution of two algebraic equations. The theoretical results are illustrated by an example.
Authors
George Bozis
Department of Physics, University of Thessaloniki, GR-54006, Greece
Mira-Cristiana Anisiu
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania
Keywords
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Paper coordinates
G. Bozis, M.-C. Anisiu, Programmed motion with homogeneity assumptions, Proceedings of the International Conference on the Dynamics of Celestial Bodies, 23-26 June 2008, Litohoro-Olympus, Thessaloniki, Greece, Eds. H. Varvoglis and Z. Knezevic, Beograd 2009, 83-87 (ISBN 978-960-243-664-6)
About this paper
Journal
Publications of the Astronomical Observatory
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DOI
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Online ISSN
0373-3742
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[1] Anisiu, M.-C. : 2003, Analysis and Optimization of Differential Systems, eds. V. Barbu et al., Kluwer Academic Publishers, Boston/Dordrecht/London, 13.
[2] Anisiu, M.-C., Bozis, G. : 2000, Inverse Problems 16, 19.
[3] Bozis, G. : 1995, Inverse Problems 11, 687.
[4] Bozis, G. : 1996, Proceedings of the 2nd Hellenic Astronomical Conference, eds. M. E. Kontadakis et al., Thessaloniki, Greece, 587.
[5] Bozis, G., Anisiu, M.-C. : 2001, Rom. Astron. J. 11, 27.
[6] Bozis, G., Ichtiaroglou, S. : 1994, Celest. Mech. Dyn. Astron. 58, 371.
PROGRAMMED MOTION WITH HOMOGENEITY ASSUMPTIONS
Abstract
We consider the problem : Given a planar region T_("orb ")T_{\text {orb }} described by one inequality g(x,y) <= c_(0)g(x, y) \leq c_{0}, find the potentials V=V(x,y)V=V(x, y) which can generate monoparametric families of orbits f(x,y)=cf(x, y)=c (also to be found) lying exclusively in the region T_("orb ")T_{\text {orb }}. We make assumptions on the homogeneity of both the function g(x,y)g(x, y) describing the boundary of the region T_("orb ")T_{\text {orb }} and of the slope function gamma(x,y)=f_(y)//f_(x)\gamma(x, y)=f_{y} / f_{x} of the required family. We show that, under certain conditions, the slope function gamma(x,y)\gamma(x, y) can be obtained as the common solution of two algebraic equations. The theoretical results are illustrated by an example.
1. INTRODUCTION
Monoparametric families of orbits f(x,y)=cf(x, y)=c, which are produced by a given potential V(x,y)V(x, y) and which have 'slope function' gamma(x,y)=f_(y)//f_(x)\gamma(x, y)=f_{y} / f_{x}, satisfy the second order nonlinear PDE (Bozis 1995)
{:(1)gamma^(2)gamma_(xx)-2gammagamma_(xy)+gamma_(yy)=(-(gammagamma_(x)-gamma_(y)))/(V_(x)+gammaV_(y))(gamma_(x)V_(x)-(2gammagamma_(x)-3gamma_(y))V_(y)-gamma(V_(xx)-V_(yy))-(gamma^(2)-1)V_(xy))",":}\begin{equation*}
\gamma^{2} \gamma_{x x}-2 \gamma \gamma_{x y}+\gamma_{y y}=\frac{-\left(\gamma \gamma_{x}-\gamma_{y}\right)}{V_{x}+\gamma V_{y}}\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{1}
\end{equation*}
where the subscripts denote partial derivatives. Families of straight lines, for which it is gammagamma_(x)-gamma_(y)=0\gamma \gamma_{x}-\gamma_{y}=0 and V_(x)+gammaV_(y)=0V_{x}+\gamma V_{y}=0 (Bozis & Anisiu 2001), are excluded from our study.
The inequality (Bozis & Ichtiaroglou 1994)
{:(2)B(x","y) >= 0:}\begin{equation*}
B(x, y) \geq 0 \tag{2}
\end{equation*}
where
{:(3)B(x","y)=(V_(x)+gammaV_(y))/(-(gammagamma_(x)-gamma_(y)))",":}\begin{equation*}
B(x, y)=\frac{V_{x}+\gamma V_{y}}{-\left(\gamma \gamma_{x}-\gamma_{y}\right)}, \tag{3}
\end{equation*}
determines the region T_("orb ")T_{\text {orb }} of the xyx y plane where the potential V(x,y)V(x, y) creates real orbits or real parts of the orbits belonging to the family gamma(x,y)\gamma(x, y).
Conversely, we can select a specific region T_("orb ")T_{\text {orb }} of the xyx y plane which we want to make the exclusive allowed region for certain unknown families created by an unknown potential.
We restrict ourselves to regions which are described by one inequality, say
{:(4)b(x","y) >= 0",":}\begin{equation*}
b(x, y) \geq 0, \tag{4}
\end{equation*}
and impose the condition that the function B(x,y)B(x, y) (corresponding to the pair (V,gamma)(V, \gamma) ) defines the same region (2) as the inequality (4) does. We interpret this by stating that there must exist a nonvanishing function Theta(x,y)\Theta(x, y), in a region T_(0)T_{0} broader from the region T_("orb ")T_{\text {orb }}, such that
{:[(5)B(x","y)=b(x","y)Theta(x","y)],[(6)Theta(x","y) >= 0" for "(x","y)inT_(0)" and " tilde(Theta)(x","y)!=oo]:}\begin{gather*}
B(x, y)=b(x, y) \Theta(x, y) \tag{5}\\
\Theta(x, y) \geq 0 \text { for }(x, y) \in T_{0} \text { and } \tilde{\Theta}(x, y) \neq \infty \tag{6}
\end{gather*}
where tilde(Theta)(x,y)\tilde{\Theta}(x, y) denotes the (one-variable) function Theta(x,y)\Theta(x, y) evaluated at the points of the curve b(x,y)=0b(x, y)=0.
Bozis (1996) solved the problem of finding the force fields which produce a given family of orbits in a fixed in advance region, and Anisiu & Bozis (2000) considered the conservative case for the families f(x,y)=y-H(x)f(x, y)=y-H(x) and a given region.
2. BASIC PROGRAMMED MOTION PROBLEM
The function BB satisfies the second order linear equation (Bozis 1995, Anisiu 2003)
{:[(7)-B_(xx)+k^(**)B_(xy)+B_(yy)=lambda^(**)B_(x)+mu^(**)B_(y)+nu^(**)B","],[k^(**)=(1-gamma^(2))/(gamma)","quadlambda^(**)=(gamma_(x)+2gammagamma_(y))/(gamma)","],[(8)mu^(**)=(2gammagamma_(x)-3gamma_(y))/(gamma)","quadnu^(**)=(2(gamma_(x)gamma_(y)-gamma_(yy)+gammagamma_(xy)))/(gamma).]:}\begin{align*}
& -B_{x x}+k^{*} B_{x y}+B_{y y}=\lambda^{*} B_{x}+\mu^{*} B_{y}+\nu^{*} B, \tag{7}\\
& k^{*}=\frac{1-\gamma^{2}}{\gamma}, \quad \lambda^{*}=\frac{\gamma_{x}+2 \gamma \gamma_{y}}{\gamma}, \\
& \mu^{*}=\frac{2 \gamma \gamma_{x}-3 \gamma_{y}}{\gamma}, \quad \nu^{*}=\frac{2\left(\gamma_{x} \gamma_{y}-\gamma_{y y}+\gamma \gamma_{x y}\right)}{\gamma} . \tag{8}
\end{align*}
The first partial derivatives of VV are related to BB by
{:(9)V_(x)=-B(gammagamma_(x)-gamma_(y))+(1)/(2)gamma(B_(y)-gammaB_(x))","quadV_(y)=-(1)/(2)(B_(y)-gammaB_(x)).:}\begin{equation*}
V_{x}=-B\left(\gamma \gamma_{x}-\gamma_{y}\right)+\frac{1}{2} \gamma\left(B_{y}-\gamma B_{x}\right), \quad V_{y}=-\frac{1}{2}\left(B_{y}-\gamma B_{x}\right) . \tag{9}
\end{equation*}
Remark If gamma\gamma is homogeneous of degree zero, then so is k^(**)k^{*}, whereas lambda^(**),mu^(**)\lambda^{*}, \mu^{*} are of degree -1 and nu^(**)\nu^{*} of degree -2 . If B(x,y)B(x, y) is weighted homogeneous of degrees e.g. n_(1)n_{1} and n_(2)n_{2}, then the entire equation (7) will lead to a weighted homogeneous expression of degrees n_(1)-2n_{1}-2 and n_(2)-2n_{2}-2.
We suppose that a region is given by the unique inequality (4). The basic programmed motion problem is: What families can be created in the given region (4) and which potentials do generate them? We introduce the function B(x,y)B(x, y), as given by (5), into the equation (7) and we obtain the linear in Theta\Theta PDE
{:(10)b(-Theta_(xx)+KTheta_(xy)+Theta_(yy))=LTheta_(x)+MTheta_(y)+N Theta",":}\begin{equation*}
b\left(-\Theta_{x x}+K \Theta_{x y}+\Theta_{y y}\right)=L \Theta_{x}+M \Theta_{y}+N \Theta, \tag{10}
\end{equation*}
where
{:[K=k^(**)","L=lambda^(**)b+2b_(x)-k^(**)b_(y)","M=bmu^(**)-k^(**)b_(x)-2b_(y)],[(11)N=nu^(**)b+lambda^(**)b_(x)+mu^(**)b_(y)+b_(xx)-k^(**)b_(xy)-b_(yy)]:}\begin{align*}
& K=k^{*}, L=\lambda^{*} b+2 b_{x}-k^{*} b_{y}, M=b \mu^{*}-k^{*} b_{x}-2 b_{y} \\
& N=\nu^{*} b+\lambda^{*} b_{x}+\mu^{*} b_{y}+b_{x x}-k^{*} b_{x y}-b_{y y} \tag{11}
\end{align*}
3. HOMOGENEITY ASSUMPTIONS
The remark in the preceding section shows that the problem becomes simpler if the functions are homogeneous, therefore we suppose that:
(i) The allowed region is given by (4), where
(i) The allowed region is given by (4), where
{:(12)b=c_(0)-x^(m)b_(0)(z)","quad z=(y)/(x)","quadb_(0)!=0:}\begin{equation*}
b=c_{0}-x^{m} b_{0}(z), \quad z=\frac{y}{x}, \quad b_{0} \neq 0 \tag{12}
\end{equation*}
(ii) The slope function gamma\gamma is homogeneous of degree zero, i.e.
{:(13)gamma=gamma(z):}\begin{equation*}
\gamma=\gamma(z) \tag{13}
\end{equation*}
(iii) The function Theta\Theta is also homogeneous of degree kk, i.e.
{:(14)Theta(x","y)=x^(k)Theta_(0)(z)","quadTheta_(0)!=0:}\begin{equation*}
\Theta(x, y)=x^{k} \Theta_{0}(z), \quad \Theta_{0} \neq 0 \tag{14}
\end{equation*}
Then, equation (10) becomes
{:(15)R_(1)x^(k)+R_(2)x^(m+k)=0:}\begin{equation*}
R_{1} x^{k}+R_{2} x^{m+k}=0 \tag{15}
\end{equation*}
Both R_(1)R_{1} and R_(2)R_{2} must vanish identically, resulting in a system of two ODEs of the form
{:[(16)2Theta_(0)(z gamma+1)gamma^(¨)+2Theta_(0)zgamma^(˙)^(2)+k_(1)gamma^(˙)+k_(0)=0],[(17)2b_(0)Theta_(0)(z gamma+1)gamma^(¨)+2b_(0)Theta_(0)zgamma^(˙)^(2)+m_(1)gamma^(˙)+m_(0)=0","]:}\begin{gather*}
2 \Theta_{0}(z \gamma+1) \ddot{\gamma}+2 \Theta_{0} z \dot{\gamma}^{2}+k_{1} \dot{\gamma}+k_{0}=0 \tag{16}\\
2 b_{0} \Theta_{0}(z \gamma+1) \ddot{\gamma}+2 b_{0} \Theta_{0} z \dot{\gamma}^{2}+m_{1} \dot{\gamma}+m_{0}=0, \tag{17}
\end{gather*}
where k_(1),m_(1)k_{1}, m_{1} are linear in Theta_(0)\Theta_{0} and Theta^(˙)_(0)\dot{\Theta}_{0}, and k_(0),m_(0)k_{0}, m_{0} in Theta_(0),Theta^(˙)_(0)\Theta_{0}, \dot{\Theta}_{0} and Theta^(¨)_(0)\ddot{\Theta}_{0}.
Our hypotheses ( b_(0)!=0,Theta_(0)!=0b_{0} \neq 0, \Theta_{0} \neq 0 and straight lines excluded) assure that
Our hypotheses ( b_(0)!=0,Theta_(0)!=0b_{0} \neq 0, \Theta_{0} \neq 0 and straight lines excluded) assure that
{:(18)b_(0)Theta_(0)(1+gamma z)!=0:}\begin{equation*}
b_{0} \Theta_{0}(1+\gamma z) \neq 0 \tag{18}
\end{equation*}
therefore the equations (16) and (17) are equivalent to
{:(19)gamma^(˙)=(Gamma_(2)gamma^(2)+Gamma_(1)gamma+Gamma_(0))/(Delta_(1)gamma+Delta_(0))","quad2(1+gamma z)gamma^(¨)+2zgamma^(˙)^(2)+K_(1)gamma^(˙)+K_(0)=0:}\begin{equation*}
\dot{\gamma}=\frac{\Gamma_{2} \gamma^{2}+\Gamma_{1} \gamma+\Gamma_{0}}{\Delta_{1} \gamma+\Delta_{0}}, \quad 2(1+\gamma z) \ddot{\gamma}+2 z \dot{\gamma}^{2}+K_{1} \dot{\gamma}+K_{0}=0 \tag{19}
\end{equation*}
where
{:[(20)Gamma_(2)=Gamma_(00)+Gamma_(01)w","quadGamma_(1)=Gamma_(10)+Gamma_(11)w","quadGamma_(0)=-Gamma_(2)],[Gamma_(00)=(1-k-m)r+z((r^(˙))+r^(2))","quadGamma_(01)=2zr-m],[(21)Gamma_(10)=m(1-2k-m)-2(1-k-m)zr+(1-z^(2))((r^(˙))+r^(2))],[Gamma_(11)=2(r+mz-rz^(2))","],[(22)Delta_(1)=2(m-2rz)","quadDelta_(0)=rz^(2)-mz-3r;],[(23)K_(1)=K_(11)gamma+K_(10)","quadK_(0)=K_(02)gamma^(2)+K_(01)gamma+K_(00)","],[(24)K_(11)=4zw+2(1-k)","quadK_(10)=-(z^(2)-3)w+kz],[K_(02)=(1-k)w+z((w^(˙))+w^(2))],[(25)K_(01)=k(1-k)-2z(1-k)w+(1-z^(2))((w^(˙))+w^(2))],[K_(00)=-(1-k)w-z((w^(˙))+w^(2))","]:}\begin{align*}
\Gamma_{2}= & \Gamma_{00}+\Gamma_{01} w, \quad \Gamma_{1}=\Gamma_{10}+\Gamma_{11} w, \quad \Gamma_{0}=-\Gamma_{2} \tag{20}\\
\Gamma_{00}= & (1-k-m) r+z\left(\dot{r}+r^{2}\right), \quad \Gamma_{01}=2 z r-m \\
\Gamma_{10}= & m(1-2 k-m)-2(1-k-m) z r+\left(1-z^{2}\right)\left(\dot{r}+r^{2}\right) \tag{21}\\
\Gamma_{11}= & 2\left(r+m z-r z^{2}\right), \\
& \Delta_{1}=2(m-2 r z), \quad \Delta_{0}=r z^{2}-m z-3 r ; \tag{22}\\
K_{1}= & K_{11} \gamma+K_{10}, \quad K_{0}=K_{02} \gamma^{2}+K_{01} \gamma+K_{00}, \tag{23}\\
K_{11}= & 4 z w+2(1-k), \quad K_{10}=-\left(z^{2}-3\right) w+k z \tag{24}\\
K_{02}= & (1-k) w+z\left(\dot{w}+w^{2}\right) \\
K_{01}= & k(1-k)-2 z(1-k) w+\left(1-z^{2}\right)\left(\dot{w}+w^{2}\right) \tag{25}\\
K_{00}= & -(1-k) w-z\left(\dot{w}+w^{2}\right),
\end{align*}
with
{:(26)Theta^(˙)_(0)=wTheta_(0)","quadb^(˙)_(0)=rb_(0).:}\begin{equation*}
\dot{\Theta}_{0}=w \Theta_{0}, \quad \dot{b}_{0}=r b_{0} . \tag{26}
\end{equation*}
We consider m,r=b^(˙)_(0)//b_(0),c_(0)m, r=\dot{b}_{0} / b_{0}, c_{0} (i.e. the function bb given by (12)) as known and we try to make compatible the two equations (19). In so doing, we prepare gamma^(¨)\ddot{\gamma} from the first of equations (19), insert into the second one and obtain the quintic in gamma\gamma algebraic equation
{:(27)alpha_(5)gamma^(5)+alpha_(4)gamma^(4)+alpha_(3)gamma^(3)+alpha_(2)gamma^(2)+alpha_(1)gamma+alpha_(0)=0",":}\begin{equation*}
\alpha_{5} \gamma^{5}+\alpha_{4} \gamma^{4}+\alpha_{3} \gamma^{3}+\alpha_{2} \gamma^{2}+\alpha_{1} \gamma+\alpha_{0}=0, \tag{27}
\end{equation*}
where the coefficients alpha_(5),alpha_(4),dots,alpha_(0)\alpha_{5}, \alpha_{4}, \ldots, \alpha_{0} are functions of zz, and of ww and its derivative of the first order.
We now differentiate (27) in zz and we obtain gamma^(˙)\dot{\gamma} which we equate to gamma^(˙)\dot{\gamma} given by the first of equations (19), and get
{:(28)beta_(6)gamma^(6)+beta_(5)gamma^(5)+beta_(4)gamma^(4)+beta_(3)gamma^(3)+beta_(2)gamma^(2)+beta_(1)gamma+beta_(0)=0",":}\begin{equation*}
\beta_{6} \gamma^{6}+\beta_{5} \gamma^{5}+\beta_{4} \gamma^{4}+\beta_{3} \gamma^{3}+\beta_{2} \gamma^{2}+\beta_{1} \gamma+\beta_{0}=0, \tag{28}
\end{equation*}
where the coefficients beta_(6),beta_(5),dots,beta_(0)\beta_{6}, \beta_{5}, \ldots, \beta_{0} are functions of zz, and of w,w^(˙),w^(¨)w, \dot{w}, \ddot{w}. We are interested in the common roots of the equations (27) and (28) and this leads us to the eleventh order Sylvester determinant which is an ODE in ww of the second order.
We have to analyze also the case when
{:(29)Delta_(1)gamma+Delta_(0)=0.:}\begin{equation*}
\Delta_{1} \gamma+\Delta_{0}=0 . \tag{29}
\end{equation*}
If Gamma_(2)gamma^(2)+Gamma_(1)gamma+Gamma_(0)!=0\Gamma_{2} \gamma^{2}+\Gamma_{1} \gamma+\Gamma_{0} \neq 0, the first of equations (19), hence the considered problem, has no solution. If Gamma_(2)gamma^(2)+Gamma_(1)gamma+Gamma_(0)=0\Gamma_{2} \gamma^{2}+\Gamma_{1} \gamma+\Gamma_{0}=0, we express gamma\gamma from (29) and substitute it in the second equation in (19). We obtain a solution for our problem if we can find a function ww which gives a suitable Theta\Theta.
4. EXAMPLE
Let us try to find families of orbits and the corresponding potentials creating them in the region
{:(30)y <= 1.:}\begin{equation*}
y \leq 1 . \tag{30}
\end{equation*}
We can write
{:(31)b(x","y)=1-y",":}\begin{equation*}
b(x, y)=1-y, \tag{31}
\end{equation*}
hence
{:(32)m=1","b_(0)(z)=z" and "c_(0)=1.:}\begin{equation*}
m=1, b_{0}(z)=z \text { and } c_{0}=1 . \tag{32}
\end{equation*}
We can now verify that, with k=2k=2, the Sylvester determinant of (27) and (28) (which for the case at hand are of degree four and five) admits of a solution Theta_(0)(z)=z^(2)//2\Theta_{0}(z)=z^{2} / 2, which gives
{:(33)Theta(x","y)=y^(2)//2.:}\begin{equation*}
\Theta(x, y)=y^{2} / 2 . \tag{33}
\end{equation*}
According to (5), (31) and (33), we find
{:(34)B(x","y)=8y^(2)(1-y).:}\begin{equation*}
B(x, y)=8 y^{2}(1-y) . \tag{34}
\end{equation*}
Equations (27) and (28) have the common solution gamma=1//(2z)\gamma=1 /(2 z), and from (9) we get the Hénon-Heiles type potential
{:(35)V(x","y)=-x^(2)-4y^(2)+4y^(3).:}\begin{equation*}
V(x, y)=-x^{2}-4 y^{2}+4 y^{3} . \tag{35}
\end{equation*}
The potential (35) generates the family of curves f(x,y)=x^(2)yf(x, y)=x^{2} y, traced in the region (30).
Figure 1: Curves of the family x^(2)y=cx^{2} y=c in the region (30) for c_(1)=0.002,c_(2)=0.0045c_{1}=0.002, c_{2}=0.0045 and c_(3)=0.008c_{3}=0.008
References
Anisiu, M.-C. : 2003, Analysis and Optimization of Differential Systems, eds. V. Barbu et al., Kluwer Academic Publishers, Boston/Dordrecht/London, 13.
Anisiu, M.-C., Bozis, G. : 2000, Inverse Problems 16, 19.
Bozis, G. : 1995, Inverse Problems 11, 687.
Bozis, G. : 1996, Proceedings of the 2nd Hellenic Astronomical Conference, eds. M. E. Kontadakis et al., Thessaloniki, Greece, 587.
Bozis, G., Anisiu, M.-C. : 2001, Rom. Astron. J. 11, 27.
Bozis, G., Ichtiaroglou, S. : 1994, Celest. Mech. Dyn. Astron. 58, 371.
Anisiu, M.-C., Bozis, G. : 2000, Inverse Problems 16, 19.
Bozis, G. : 1995, Inverse Problems 11, 687.
Bozis, G. : 1996, Proceedings of the 2nd Hellenic Astronomical Conference, eds. M. E. Kontadakis et al., Thessaloniki, Greece, 587.
Bozis, G., Anisiu, M.-C. : 2001, Rom. Astron. J. 11, 27.
Bozis, G., Ichtiaroglou, S. : 1994, Celest. Mech. Dyn. Astron. 58, 371.