Programmed motion with homogeneity assumptions

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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

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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)

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https://ictp.acad.ro/anisiu/papers/2009-Bozis-Anisiu-ProgrammedLitohoro.pdf

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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.

2009-Bozis-Anisiu-ProgrammedLitohoro

PROGRAMMED MOTION WITH HOMOGENEITY ASSUMPTIONS

G. BOZIS $^{1}$ and M.-C. ANISIU $^{2}$ $^{1}$ Department of Physics, University of Thessaloniki, GR-54006, GreeceE-mail gbozis@auth.gr $^{2}$ T. Popoviciu Institute of Numerical Analysis, Romanian Academy, PO Box 68, 400110 Cluj-Napoca, RomaniaE-mail mira@math.ubbcluj.ro

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 $γ (x, y) = f_{y} / f_{x}$ of the required family. We show that, under certain conditions, the slope function $γ (x, y)$ 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) = c

, which are produced by a given potential

V (x, y)

and which have 'slope function'

γ (x, y) = f_{y} / f_{x}

, satisfy the second order nonlinear PDE (Bozis 1995)

\begin{matrix} (1) & γ^{2} γ_{x x} - 2 γ γ_{x y} + γ_{y y} = \frac{- (γ γ_{x} - γ_{y})}{V_{x} + γ V_{y}} (γ_{x} V_{x} - (2 γ γ_{x} - 3 γ_{y}) V_{y} - γ (V_{x x} - V_{y y}) - (γ^{2} - 1) V_{x y}), \end{matrix}

where the subscripts denote partial derivatives. Families of straight lines, for which it is

γ γ_{x} - γ_{y} = 0

and

V_{x} + γ V_{y} = 0

(Bozis & Anisiu 2001), are excluded from our study.

The inequality (Bozis & Ichtiaroglou 1994)

\begin{matrix} (2) & B (x, y) \geq 0 \end{matrix}

where

\begin{matrix} (3) & B (x, y) = \frac{V_{x} + γ V_{y}}{- (γ γ_{x} - γ_{y})}, \end{matrix}

determines the region

T_{orb}

of the

x y

plane where the potential

V (x, y)

creates real orbits or real parts of the orbits belonging to the family

γ (x, y)

.

Conversely, we can select a specific region

T_{orb}

of the

x 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

\begin{matrix} (4) & b (x, y) \geq 0, \end{matrix}

and impose the condition that the function

B (x, y)

(corresponding to the pair

(V, γ)

) defines the same region (2) as the inequality (4) does. We interpret this by stating that there must exist a nonvanishing function

Θ (x, y)

, in a region

T_{0}

broader from the region

T_{orb}

, such that

\begin{matrix} (5) & B (x, y) = b (x, y) Θ (x, y) \\ (6) & Θ (x, y) \geq 0 for (x, y) \in T_{0} and \tilde{Θ} (x, y) \neq \infty \end{matrix}

where

\tilde{Θ} (x, y)

denotes the (one-variable) function

Θ (x, y)

evaluated at the points of the curve

b (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)

and a given region.

2. BASIC PROGRAMMED MOTION PROBLEM

The function

B

satisfies the second order linear equation (Bozis 1995, Anisiu 2003)

\begin{aligned} (7) & - B_{x x} + k^{*} B_{x y} + B_{y y} = λ^{*} B_{x} + μ^{*} B_{y} + ν^{*} B, \\ k^{*} = \frac{1 - γ^{2}}{γ}, λ^{*} = \frac{γ_{x} + 2 γ γ_{y}}{γ}, \\ (8) & μ^{*} = \frac{2 γ γ_{x} - 3 γ_{y}}{γ}, ν^{*} = \frac{2 (γ_{x} γ_{y} - γ_{y y} + γ γ_{x y})}{γ} . \end{aligned}

The first partial derivatives of

V

are related to

B

by

\begin{matrix} (9) & V_{x} = - B (γ γ_{x} - γ_{y}) + \frac{1}{2} γ (B_{y} - γ B_{x}), V_{y} = - \frac{1}{2} (B_{y} - γ B_{x}) . \end{matrix}

Remark If

γ

is homogeneous of degree zero, then so is

k^{*}

, whereas

λ^{*}, μ^{*}

are of degree -1 and

ν^{*}

of degree -2 . If

B (x, y)

is weighted homogeneous of degrees e.g.

n_{1}

and

n_{2}

, then the entire equation (7) will lead to a weighted homogeneous expression of degrees

n_{1} - 2

and

n_{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)

, as given by (5), into the equation (7) and we obtain the linear in

Θ

PDE

\begin{matrix} (10) & b (- Θ_{x x} + K Θ_{x y} + Θ_{y y}) = L Θ_{x} + M Θ_{y} + N Θ, \end{matrix}

where

\begin{aligned} K = k^{*}, L = λ^{*} b + 2 b_{x} - k^{*} b_{y}, M = b μ^{*} - k^{*} b_{x} - 2 b_{y} \\ (11) & N = ν^{*} b + λ^{*} b_{x} + μ^{*} b_{y} + b_{x x} - k^{*} b_{x y} - b_{y y} \end{aligned}

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

\begin{matrix} (12) & b = c_{0} - x^{m} b_{0} (z), z = \frac{y}{x}, b_{0} \neq 0 \end{matrix}

(ii) The slope function

γ

is homogeneous of degree zero, i.e.

\begin{matrix} (13) & γ = γ (z) \end{matrix}

(iii) The function

Θ

is also homogeneous of degree

k

, i.e.

\begin{matrix} (14) & Θ (x, y) = x^{k} Θ_{0} (z), Θ_{0} \neq 0 \end{matrix}

Then, equation (10) becomes

\begin{matrix} (15) & R_{1} x^{k} + R_{2} x^{m + k} = 0 \end{matrix}

Both

R_{1}

and

R_{2}

must vanish identically, resulting in a system of two ODEs of the form

\begin{matrix} (16) & 2 Θ_{0} (z γ + 1) \ddot{γ} + 2 Θ_{0} z {\dot{γ}}^{2} + k_{1} \dot{γ} + k_{0} = 0 \\ (17) & 2 b_{0} Θ_{0} (z γ + 1) \ddot{γ} + 2 b_{0} Θ_{0} z {\dot{γ}}^{2} + m_{1} \dot{γ} + m_{0} = 0, \end{matrix}

where

k_{1}, m_{1}

are linear in

Θ_{0}

and

{\dot{Θ}}_{0}

, and

k_{0}, m_{0}

in

Θ_{0}, {\dot{Θ}}_{0}

and

{\ddot{Θ}}_{0}

.
Our hypotheses (

b_{0} \neq 0, Θ_{0} \neq 0

and straight lines excluded) assure that

\begin{matrix} (18) & b_{0} Θ_{0} (1 + γ z) \neq 0 \end{matrix}

therefore the equations (16) and (17) are equivalent to

\begin{matrix} (19) & \dot{γ} = \frac{Γ_{2} γ^{2} + Γ_{1} γ + Γ_{0}}{Δ_{1} γ + Δ_{0}}, 2 (1 + γ z) \ddot{γ} + 2 z {\dot{γ}}^{2} + K_{1} \dot{γ} + K_{0} = 0 \end{matrix}

where

\begin{aligned} (20) & Γ_{2} = & Γ_{00} + Γ_{01} w, Γ_{1} = Γ_{10} + Γ_{11} w, Γ_{0} = - Γ_{2} \\ Γ_{00} = & (1 - k - m) r + z (\dot{r} + r^{2}), Γ_{01} = 2 z r - m \\ (21) & Γ_{10} = & m (1 - 2 k - m) - 2 (1 - k - m) z r + (1 - z^{2}) (\dot{r} + r^{2}) \\ Γ_{11} = & 2 (r + m z - r z^{2}), \\ (22) & Δ_{1} = 2 (m - 2 r z), Δ_{0} = r z^{2} - m z - 3 r; \\ (23) & K_{1} = & K_{11} γ + K_{10}, K_{0} = K_{02} γ^{2} + K_{01} γ + K_{00}, \\ (24) & K_{11} = & 4 z w + 2 (1 - k), K_{10} = - (z^{2} - 3) w + k z \\ K_{02} = & (1 - k) w + z (\dot{w} + w^{2}) \\ (25) & K_{01} = & k (1 - k) - 2 z (1 - k) w + (1 - z^{2}) (\dot{w} + w^{2}) \\ K_{00} = & - (1 - k) w - z (\dot{w} + w^{2}), \end{aligned}

with

\begin{matrix} (26) & {\dot{Θ}}_{0} = w Θ_{0}, {\dot{b}}_{0} = r b_{0} . \end{matrix}

We consider

m, r = {\dot{b}}_{0} / b_{0}, c_{0}

(i.e. the function

b

given by (12)) as known and we try to make compatible the two equations (19). In so doing, we prepare

\ddot{γ}

from the first of equations (19), insert into the second one and obtain the quintic in

γ

algebraic equation

\begin{matrix} (27) & α_{5} γ^{5} + α_{4} γ^{4} + α_{3} γ^{3} + α_{2} γ^{2} + α_{1} γ + α_{0} = 0, \end{matrix}

where the coefficients

α_{5}, α_{4}, \dots, α_{0}

are functions of

z

, and of

w

and its derivative of the first order.

We now differentiate (27) in

z

and we obtain

\dot{γ}

which we equate to

\dot{γ}

given by the first of equations (19), and get

\begin{matrix} (28) & β_{6} γ^{6} + β_{5} γ^{5} + β_{4} γ^{4} + β_{3} γ^{3} + β_{2} γ^{2} + β_{1} γ + β_{0} = 0, \end{matrix}

where the coefficients

β_{6}, β_{5}, \dots, β_{0}

are functions of

z

, and of

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

w

of the second order.

We have to analyze also the case when

\begin{matrix} (29) & Δ_{1} γ + Δ_{0} = 0 . \end{matrix}

If

Γ_{2} γ^{2} + Γ_{1} γ + Γ_{0} \neq 0

, the first of equations (19), hence the considered problem, has no solution. If

Γ_{2} γ^{2} + Γ_{1} γ + Γ_{0} = 0

, we express

γ

from (29) and substitute it in the second equation in (19). We obtain a solution for our problem if we can find a function

w

which gives a suitable

Θ

.

4. EXAMPLE

Let us try to find families of orbits and the corresponding potentials creating them in the region

\begin{matrix} (30) & y \leq 1 . \end{matrix}

We can write

\begin{matrix} (31) & b (x, y) = 1 - y, \end{matrix}

hence

\begin{matrix} (32) & m = 1, b_{0} (z) = z and c_{0} = 1 . \end{matrix}

We can now verify that, with

k = 2

, the Sylvester determinant of (27) and (28) (which for the case at hand are of degree four and five) admits of a solution

Θ_{0} (z) = z^{2} / 2

, which gives

\begin{matrix} (33) & Θ (x, y) = y^{2} / 2 . \end{matrix}

According to (5), (31) and (33), we find

\begin{matrix} (34) & B (x, y) = 8 y^{2} (1 - y) . \end{matrix}

Equations (27) and (28) have the common solution

γ = 1 / (2 z)

, and from (9) we get the Hénon-Heiles type potential

\begin{matrix} (35) & V (x, y) = - x^{2} - 4 y^{2} + 4 y^{3} . \end{matrix}

The potential (35) generates the family of curves

f (x, y) = x^{2} y

, traced in the region (30).

Figure 1: Curves of the family

x^{2} y = c

in the region (30) for

c_{1} = 0.002, c_{2} = 0.0045

and

c_{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.

2009

Programmed motion with homogeneity assumptions

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Authors

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About this paper

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PROGRAMMED MOTION WITH HOMOGENEITY ASSUMPTIONS

Abstract

1. INTRODUCTION

2. BASIC PROGRAMMED MOTION PROBLEM

3. HOMOGENEITY ASSUMPTIONS

4. EXAMPLE

References

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