Abstract
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We nd galactic potentials described by polynomial perturbations of harmonic oscillators, which are compatible with families of orbits \(x^{p} y=const(p\neq0)\). To this aim we apply the techniques of the planar inverse problem of dynamics.
Authors
Mira-Cristiana Anisiu
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania,
Keywords
galactic potentials; inverse problem of dynamics.
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Cite this paper as:
Anisiu Mira-Cristiana, Families of orbits compatible with galactic potentials, AIP Conference Proceedings, vol. 895, 2007, pag. 308-312
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references
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Cluj-Napoca, 2003, ISBN 973-686-466-9 (in Romanian).
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Spatial Families of Orbits in 2D Conservative Fields
Abstract
In the framework of the 3D inverse problem of dynamics, we establish the conditions which must be fulfilled by a spatial family of curves to possibly be described by a unit mass particle under the action of a 2D potential V=v(y,z)V=v(y, z), and give a method to find the potential.
Keywords: inverse problem of Dynamics
PACS: 95.10.Ce
PACS: 95.10.Ce
INTRODUCTION
In the 3D inverse problem of Dynamics, assuming that a two-parametric set of orbits
{:(1)f(x","y","z)=c_(1)","quad g(x","y","z)=c_(2):}\begin{equation*}
f(x, y, z)=c_{1}, \quad g(x, y, z)=c_{2} \tag{1}
\end{equation*}
in the OxyzO x y z space can be traced by a unit mass particle in the presence of an unknown potential V=V(x,y,z)V=V(x, y, z), one aims to finding the potential (see [1] for a review of the results up to 1995).
Bozis and Kotoulas [2] and also Anisiu [3] produced a system of two linear in V(x,y,z)V(x, y, z) PDEs, one of the first and one of the second order, which will be used in what follows.
We shall treat the problem for the special case of potentials V=v(y,z)V=v(y, z).
THE EQUATIONS OF THE PROBLEM
We deal with two-parametric families of orbits written in the form (1), which are in an one-to-one correspondence with a pair ( alpha,beta\alpha, \beta ) of 'slope functions' defined by
{:(2)alpha=(f_(z)g_(x)-f_(x)g_(z))/(f_(y)g_(z)-f_(z)g_(y))","quad beta=(f_(x)g_(y)-f_(y)g_(x))/(f_(y)g_(z)-f_(z)g_(y)):}\begin{equation*}
\alpha=\frac{f_{z} g_{x}-f_{x} g_{z}}{f_{y} g_{z}-f_{z} g_{y}}, \quad \beta=\frac{f_{x} g_{y}-f_{y} g_{x}}{f_{y} g_{z}-f_{z} g_{y}} \tag{2}
\end{equation*}
The indices denote partial derivatives.
Let us assume that alpha_(0)!=0\alpha_{0} \neq 0 and adopt the notation
Let us assume that alpha_(0)!=0\alpha_{0} \neq 0 and adopt the notation
{:(3) bar(epsi)=(1","alpha","beta)","alpha_(0)= bar(epsi)grad alpha","beta_(0)= bar(epsi)grad beta","Theta=1+alpha^(2)+beta^(2)","n=(Theta)/(alpha_(0))","n_(0)= bar(epsi)grad n.:}\begin{equation*}
\bar{\varepsilon}=(1, \alpha, \beta), \alpha_{0}=\bar{\varepsilon} \operatorname{grad} \alpha, \beta_{0}=\bar{\varepsilon} \operatorname{grad} \beta, \Theta=1+\alpha^{2}+\beta^{2}, n=\frac{\Theta}{\alpha_{0}}, n_{0}=\bar{\varepsilon} \operatorname{grad} n . \tag{3}
\end{equation*}
We shall consider exclusively potentials of the form V=v(y,z)V=v(y, z) and families (1) with alpha_(0)!=0\alpha_{0} \neq 0. In this case the system of the two PDEs mentioned in the introduction becomes
{:(4)v_(z)=Gv_(y)","Theta(alpha+beta G)v_(yy)+Psipsi_(y)=0:}\begin{equation*}
v_{z}=G v_{y}, \Theta(\alpha+\beta G) v_{y y}+\Psi \psi_{y}=0 \tag{4}
\end{equation*}
where G=beta_(0)//alpha_(0),Psi=beta ThetaG_(y)+alpha_(0)(n_(0)-2(alpha+beta G))G=\beta_{0} / \alpha_{0}, \Psi=\beta \Theta G_{y}+\alpha_{0}\left(n_{0}-2(\alpha+\beta G)\right).
For any compatible pair of potential v(y,z)v(y, z) and orbit ( alpha,beta\alpha, \beta ) real motion is allowed in the region -v_(y)//alpha_(0) >= 0-v_{y} / \alpha_{0} \geq 0 ([2], [3]).
For any compatible pair of potential v(y,z)v(y, z) and orbit ( alpha,beta\alpha, \beta ) real motion is allowed in the region -v_(y)//alpha_(0) >= 0-v_{y} / \alpha_{0} \geq 0 ([2], [3]).
COMPATIBLE POTENTIALS V=v(y,z)V=v(y, z) AND FAMILIES (1)
It is seen from (4a) that the function GG must be independent of xx, i. e.
{:(5)G_(x)=0:}\begin{equation*}
G_{x}=0 \tag{5}
\end{equation*}
We assume that alpha+beta G!=0\alpha+\beta G \neq 0 and we put H=Psi//Theta(alpha+beta G)H=\Psi / \Theta(\alpha+\beta G). For the PDE(4b)\operatorname{PDE}(4 \mathrm{~b}), now written as v_(yy)+Hv_(y)=0v_{y y}+H v_{y}=0, to have a solution of the form v(y,z)v(y, z) it must be
{:(6)H_(x)=0:}\begin{equation*}
H_{x}=0 \tag{6}
\end{equation*}
From (4b) we get
{:(7)v_(y)=D(z)exp(-int^(y)H(u,z)du):}\begin{equation*}
v_{y}=D(z) \exp \left(-\int^{y} H(u, z) \mathrm{d} u\right) \tag{7}
\end{equation*}
and from (4a) we obtain v_(z)v_{z}. From the compatibility condition (v_(yz)=v_(zy))\left(v_{y z}=v_{z y}\right) for v_(y)v_{y} and v_(z)v_{z}, as these are given by (7) and (4a) respectively, there follows the homogeneous linear first order ODE D^(')(z)=JD(z)D^{\prime}(z)=J D(z), where JJ depends merely on zz if and only if
{:(8)G_(yy)-G_(y)H-GH_(y)+H_(z)=0.:}\begin{equation*}
G_{y y}-G_{y} H-G H_{y}+H_{z}=0 . \tag{8}
\end{equation*}
Proposition For alpha_(0)!=0,alpha+beta G!=0\alpha_{0} \neq 0, \alpha+\beta G \neq 0 and for any family ( alpha,beta\alpha, \beta ) satisfying the conditions (5), (6) and (8), there exists a two-dimension compatible potential v=v(y,z)v=v(y, z). The potential is given by the (compatible) equations (7) and (4a). Real motion is allowed in the region defined by the inequality D(z)//alpha_(0) <= 0D(z) / \alpha_{0} \leq 0.
If alpha_(0)!=0,alpha+beta G=0\alpha_{0} \neq 0, \alpha+\beta G=0 and Psi!=0\Psi \neq 0 it follows that v=v= const. If alpha_(0)!=0,alpha+beta G=0\alpha_{0} \neq 0, \alpha+\beta G=0 and Psi=0\Psi=0, equation ( 4 b ) is satisfied identically and the potential is found from ( 4 a ), provided that the condition (5) holds, and it will not be uniquely determined.
Example For the family f(x,y,z)=x^(4)yz^(3),g(x,y,z)=x^(2)yzf(x, y, z)=x^{4} y z^{3}, g(x, y, z)=x^{2} y z, we get alpha_(0)!=0,alpha+beta G!=\alpha_{0} \neq 0, \alpha+\beta G \neq 0 and obtain the compatible potential V(x,y,z)=-(y^(2)+z^(2))^(2)V(x, y, z)=-\left(y^{2}+z^{2}\right)^{2}.
The case alpha_(0)=0\alpha_{0}=0 and that of potentials depending on ( x,yx, y ) or ( x,zx, z ) will be treated elsewhere.
Acknowledgment The work of the first author was financially supported by the scientific program 2CEEX0611-96 of the Romanian Ministry of Education and Research.
REFERENCES
- G. Bozis, Inverse Problems 11, 687-708 (1995).
- G. Bozis, and T. A. Kotoulas, Inverse Problems 21, 343-356 (2005).
- M.-C. Anisiu, Inverse Probl. Sci. Eng. 13, 545-558 (2005).