Computing 53, 75-94 (1994)
COI~pI~I~ 9 Springer-Verlag1994 Printed in Austria
S o m e Orbital Test P r o b l e m s J. C. Butcher Received September 17, 1990; revised February 21, 1994
Abstract- Zusammenfassung Some Orbital Test Problems. Problems with periodic solutions are convenient as test problems for differential equation software because of the ease with which the accuracy of computed results can be assessed. Even the motion of a single planet around a heavy sun is useful as a test problem because orbits of varying eccentricity make varying demands on numerical software. The orbits discussed here are based on this same simple problem but with the essential difference that the distance from the planet to the sun is based on the norm II. [1~ rather than the usual Euclidean norm II. II2- Specifically, we explore orbits based on each of the differential equation systems 17
1
and X
X'#__
tlXII3" A feature of both these systems, when the II-II~ norm is used, is the occurrence of discontinuities in the higher derivatives of the solution. This is why they have a potential value as difficult test problems. With this application in mind, some periodic solutions are identified. For an arbitrary choice of norm, the second of the two differential equation systems considered in this paper is shown to possess periodic orbits.
elMS Subject Classification: 65L05 Key words: Periodic orbits, test problems. Einige orbitale Testprobleme. Probleme mit periodischen L6sungen eignen sich besonders als Testprobleme fiir Software zur Behandlung von Differentialgleichungen, da sie es leicht erm6glichen, die Genauigkeit der berechneten Ergebnisse zu beurteilen. Selbst der Lauf eines einzigen Planeten um die schwere Sonne ist niitzlich als Testproblem, da Umlaufbahnen mit sich/indernder Exzentrizit~it wechselnde Anforderungen an die numerische Software stellen. Die Umlaufbahnen, die hier betrachtet werden, griinden sich auf genau diesem einfachen Problem, jedoch mit dem wesentlichen Unterschied, dal3 der Abstand zwischen Planet und Sonne mit der Norm II. I1~ anstelle der tiblichen Euclid'schen Norm II. 112 gemessen wird. Insbesondere untersuchen wir Umlaufbahnen basierend auf jedem der Differentialgleichungssysteme 17
1
und X
n
~
R
m
X
IIxII3'
Ein Merkmal beider dieser Systeme, wenn die Norm II. I1~ verwendet wird, ist das Auftreten von Unstetigkeiten in h6heren Ableitungen der Lfsung. Hierin begriindet sich ihr potentieller Wert als
76
J.C. Butcher
schwierige Testprobleme. Im Hinblick auf diese Anwendung werden einige periodische L6sungen aufgezeigt. Es wird gezeigt, dab das zweite der beiden in diesem Papier betrachteten Differentialgleichungssysteme, bei beliebiger Wahl der Norm, periodische Umlaufbahnen besitzt.
1. Introduction
Many differential equation systems have been proposed as test problems for numerical solvers. For example, the so-called DETEST collection [1] contains a mixture of both contrived and physically-significant initial value problems. Some of these provide routine acceptance tests and some impose severe difficulties for a solver. Within this collection of problems are several which model gravitational motion and within this smaller set are some whose solutions represent periodic orbits. One of the purposes of the present paper is to extend this collection of periodic orbital problems in such a way as to impose a further difficulty for numerical software. The specific test problems to be presented in this paper are constucted so that they combine discontinuous behaviour with periodicity. Amongst the examples that we will present, are various levels of eccentric behaviour, so that the efficiency of variable stepsize mechanisms can be adequately tested. In Section 2, we will show how the new problems arise as modifications of wellknown gravitational equations. Of two types of modifications, one satisfies energyconservation and will be studied in detail in Section 3. In Section 4, examples of periodic orbits arising from this system will be presented. Section 5 is devoted to a second type of modification whose solutions satisfy an angular-momentum conservation law and in Section 6 some example orbits for this problem are presented. For both types of orbit, various points on the solution trajectories are given to 30 decimal places in an Appendix to this paper. In Section 7, further generalizations of the angular-momentum conserving case are considered and a theorem is proved establishing the existence of periodic orbits in a wide variety of circumstances. 2. Modified Gravitational Motion
We will consider only two-dimensional problems of the form
X"(t) = f(X(t))
(2.1)
where f is defined in one of two ways, =V
1
(2.2)
or
1
f(X) =
iI~X.
In other words, we consider inverse-square force laws.
(2.3)
Some Orbital Test Problems
77
The components of X(t) will be denoted by x(t) and y(t) and the polar coordinates by r(t) and O(t). Thus,
Ly(t)J
Lr(t) sin(O(t))_]'
where r(t) > O. For the familiar choice of the Euclidean norm, (2.2) and (2.3) are identical, and we have nothing other than the classical case of a light planet moving under gravitational attraction by a heavy sun. If 11. II is not restricted to the case of the Euclidean norm, at least one general statement can be made in the case of each of the two modifications. In the case of the equation
we have the "principle of conservation of energy". Theorem 2.1. For any solution of (2.4), the value of 89
1
2 + y'(t) 2)
(2.5)
IIx(t)II is constant as t varies. Proof." Differentiate (2.5) and note that the result can be written in the form -V
1
We now consider the case where f is given by (2.3), that is
x"(t) -
1 IIX(t) ll3x(t).
(2.6)
For this problem we have the "principle of conservation of angular momentum". Theorem 2.2. For any solution of(2.6), the value of x(t)y'(t) -- y(t)x'(t)
(2.7)
is constant. That is, "angular momentum is conserved.". Proof: Differentiate (2.7) to obtain x(t)y"(t) - y(t)x"(t) which is found to be zero by substituting the components of X"(t) from (2.6). 9 For (2.6), there is no restriction on the type of norm we may choose for the differential equation system to be well-posed and we will consider alternative norms in Section 7.
78
J.C. Butcher
For the next four sections we will confine ourselves to the case that the norm is used.
II. lib
3. The Energy-Conserving Case
Since (2.5) is conserved, we will set its value at - 1. A negative total energy allows the planet to approach arbitrarily close to the sun but ensures that escape from its influence is impossible. If another negative value is used instead, then a suitable change of scale will immediately convert the problem to this case. If, for a point on a trajectory, the kinetic energy is zero, that is x'(t) = y'(t) = o.
then this point lies on the unit circle (actually a square with sides of length 2) given by max(Ix[, lY[) = 1 and any point on a trajectory where the velocity is non-zero, lies inside the disc bounded by this unit circle. Because of the symmetry relating different quadrants in (2.4), it will be sufficicent to select just one of them, say the quadrant given by IY[ < x so that 0 s [ - n/4, n/4], and to suppose that motion of the trajectory is directed into the quadrant from below. Thus we suppose that at time t = 0, the X coordinates are given by [a, - a ] and that the velocity components, say [u, v], are such that u+v>0, guaranteeing that the travel path is into the quadrant. We will study the behaviour of the path of motion within this quadrant and note where it leaves the quadrant again. By joining together the motions in a sequence of quadrants, a search can be made for periodic orbits. The equations of motion in the quadrant that has been specified are given by 1 x" = - - -
(3.1)
y" = 0.
(3.2)
X2 ,
From (3.1), we obtain the relation lx,2_ 1 2 x
1 b'
(3.3)
1 = where b is a constant given by ~ = 1 + ~v with v the constant value ofy'. Assuming
that when the trajectory enters the quadrant the value of x' is positive but that it has
Some Orbital Test Problems
79
become negative when the trajectory eventually leaves the quadrant, we can interpret b as the maximum value of x which is attained in the quadrant. From (3.3) we find the time at which a given value of x' is attained to be b3/2
i/ '
[b\
(3.4)
where ~b(~) = arctan(~) + - lq-~ 2
(3.5)
and the constant t o can be interpreted as the time at which height b is achieved. To study the complete motion within the quadrant, we must consider whether the trajectory leaves the quadrant at the top or at the bottom. This in turn depends on the initial state when the quadrant is entered. Define
b 3/2 b r = sgn(v~i~((9(uN/~) + 2) -- a ) , then, ifr = 1, the trajectory leaves at the top and, ifr = - 1 it leaves at the bottom. In all cases, we can write a relation between the velocity components at the instant the path enters the quadrant and the components at the instant it leaves the quadrant. The position coordinates are then given as (x, rx), where x is given by the conservation of energy equation (2.5) and the velocity components [ - w, rv] where V
~
U
-t- 0
--
b u 2 nt-
bw 2.
1+~- 1+~-
In searching for periodic orbits, we vary the initial values at which the motion starts and trace its path through as many quadrants as necessary before the initial values are reproduced. We will present some of the orbits resulting from these searches in the next section.
4. Some Energy-Conserving Orbits In orbits numbered A1 and A2 in the Appendix to this paper, xy' - yx' retains a positive sign so that the two orbits depicted under these names are natural generalizations of the elliptic orbits which occur in the case of the usual Euclidean norm. To facilitate the use of these two orbits as test problems, various points on the paths are identified and for these points accurate information on time, position compo-
80
J.c. Butcher
nents and velocity components is given. For Fig. A1 these points are identified as 0, 1 and 2 and in the tables which accompany the figures in the Appendix, the value of t at which these points are reached, starting from time 0 at point 0, is given as well as the values of [x(t), y(t)] and [x'(t), y'(t)]. The value of the period T is also given for this orbit and it can be seen that the value of this is four times the time taken to reach point 2 (or eight times the time to reach point 1). Similarly, for orbit A2, points numbered 0, 1, 2, 3 and 4 are marked and corresponding information are given for these points. Note that the period T in this case is twice the time taken to reach point 4. As a guide to the stability of these two orbits, the value of R equal to the spectral radius of S is given, where S is the matrix of partial derivatives of the velocity components after time T as functions of the initial velocity components. That is
Ox'(T) r
1
ax'(O) ~6~1
S=
8y'(T)
ay'(T)|"
ax'(O) ay'(O)J It is found that, for orbit A1, R = 1, showing optimal stability. However, for orbit A2, the value R = 1.4204886595, showing that initial errors will tend to increase by about this factor during each circuit of the orbit. The Appendix also contains additional orbits satisfying the same differential equation and numbered A3 to A18 and, for each of these, similar information is given. Note that some of the points on these trajectories, for which time and position are given, there is no serial number attached. The reason for this is that the points are easily identified by symmetry, or because one of the components vanishes. Marking these additional points on the figures would not be convenient because of the limited space available. The value of R is very large for some of these orbits, indicating that they represent very severe tests of a numerical method. They can be used as less demanding tests by restricting the interval of integration to only part of a complete period. Note also that for obits A10 through to A18, the path is retraced a second time, but in the reverse direction, for a complete orbit. The values of T and R are given for this doubled orbit but, as test problems, a half orbit would be convenient and satisfactory. In using these orbits as test problems, it might be regarded as inappropriate to start the integration at a point where a discontinuity occurs. Hence, the initial values should be given at point number 1 rather than at point number 0 in orbits A1 and A2 and this is one of the reasons that accurate values of these points are given as well as for points numbered 0.
S o m e O r b i t a l Test P r o b l e m s
81
5. The Angular Momentum-Conserving Case In this section we consider solutions to (2.6). Since angular m o m e n t u m x y ' - y x ' is constant in any solution, we choose to normalize this quantity to 1. As for solutions of (2.4), we consider motion in the single quadrant defined by 0 e [ - n/4, n/4]. Thus (2.6) takes the form 1 x" = - - - -
(5.1)
y" = - Y - -
(5.2)
X2 '
X3"
Let Uo, vo be the velocity components when the quadrant is entered at the bottom. We will assume that u o + vo > 0 to ensure that the motion is into the quadrant. In this case the point of entry is (Uo + Vo)-1 l-l, - 1] r. Define b by
1
1 2
= Uo + Vo - ~ U o ,
so that
1,2
1
~ x (t) - x(t)
1
b'
(5.3)
while the trajectory remains within the quadrant. 1 2 We will assume, for the moment, that Uo + Vo - ~Uo > 0. This will guarantee that x(t) is bounded by b and that no escape from the influence of the attracting force takes place within the quadrant.
With the known value of b, we find from (3.4) that t = to - (2Uo + 2Vo - u 2 ) - 3 / 2 ~
{2Uo + ~ - -
u2) ~/2 '
(5.4)
where ~ was defined by (3.5). We can also find expressions for y and y' at this time as well as for x. These are x(t) =
y(t) =
1
1 ~ Uo + Vo - ~Uo + 89 Uo -
(5.5)
2'
1 - x'(t)
1 2 uo + vo - ~uo + 89
1 2
(5.6)
2
y'(t) = u o + v o -- ~u o + x ' ( t ) ( u o
1 -- 89
(5.7)
82
J.C. Butcher
At the top of the quadrant
x(t) = y(t)
so that, when this point is reached,
x ' ( t ) = Uo -
2
and, from (5.7), y'(t) = %.
To follow the motion from quadrant to quadrant, write ui, v~as velocity components as the trajectory enters quadrant number i = 0, 1, 2 , . . . . We use the convention that ui is measured in the direction of increasing norm and that v~ is in the direction rotated n/2 further. If it is assumed for i = 1, 2 . . . . that u,
+ v.
- ~1 u2,
> 0,
(5.8)
it follows by generalizing from the case i = 1 that Ui
=
vi :
/)i--1 ,
2 -
u,.
The assumption (5.8) ensures that the trajectory passes into the next succeeding quadrant and does not instead escape within this quadrant. Under the assumption of no escape, we find in turn that
[::] /)3
UO
::]
jut r ol v4
LVo]
implying that the orbit is periodic after four quadrants. The criteria for not escaping in any of the four quadrants can now be written in terms of Uo and Vo as 211 - vol + (1 - Uo)2 < 3, 211 - Uol + (1 - Vo)2 < 3. In the next section, a collection of example orbits is presented.
6. Some Angular Momentum-Conserving Orbits We present just six orbits selected by appropriate choices of Uo and vo. These are shown as orbits numbers B1 to B6 in the Appendix. As for the example orbits discussed in Section 4, we have identified a number of points on each orbit and shown in tabular form the elapsed time and the position and velocity coordinates for these and some additional points. The orbit numbered B1 is the appropriate generalization of circular motion while B2 and B3 are eccentric in each of two different ways. Finally, orbits B4, B5 and B6
Some Orbital Test Problems
83
exhibit skewness as well as eccentricity. The scale of distance, which differs amongst these six orbits, is indicated by the superposition on each diagram of the square IIXL = 1, shown with broken lines. The numerical difficulties associated with this collection of orbits are, of course, not nearly as intense as for the orbits presented for the energy-conserving problem, because the formulas for the acceleration components are now continuous. The central force problem, in fact, satisfies a Lipschitz condition but it can still be made quite difficult if very eccentric orbits are selected for testing purposes. In Section 7, we consider a further generalization of the inverse-square law problem and we arrive at a surprising fact.
7. Arbitrary Norms We consider solutions to (2.6) in which the norm II. II can now be chosen in any way. Numerical experiements suggest that periodicity for non-escaping orbits, such as occurred for the infinity norm, also occurs in other cases. The remainder of the paper investigates this phenomena and shows that this behaviour is the general rule. It will also be shown that for any choice of norm, there exists initial values for which non-escaping, and therefore periodic, orbits, actually occur. We consider some preliminary lemmas.
Lemma 7.1. Let g denote a continuous periodic function with period ~. Then every solution to the differential equation s"(O)+s(O)=9(O),
(7.1)
is periodic with period 2~. Proof" Let z(O) = s(O + ~) - s(0), so that z satisfies the differential equation z" + z = 0 for which any solution is of the form z(O)= A cos 0 + B sin0. Hence, z(O + ~) = -z(O) implying that s(0 + 2zc) - s(O + ~) = -(s(O + ~) - s(O)) and leading to the result s(O + 2re) = s(O). 9 Lemma 7.2. Let g satisfy the requirements of Lemma 7.1 and, in addition, g(O) > 0 for all O. Then there is a solution of (7.1) 9iven by
s(O) = ~
g(O + (k) sin ~bd~
Proof" Write s given by (7.2) in the form 1
r O+rc
s(O) = ~ Jo
g(~b)sin(~ - O) d~
(7.2)
84
J.C. Butcher
and evaluate in turn the first two derivatives as
s'(O) =
1 ff+~
g ( O cos(
-
0)
and
1 I ~ g(ff)sin(~ - O)d~
s"(O) = g(O) - ~ J0 from which the result follows.
We note that the solution given by (7.2) takes only positive values. We are now in a position to state and prove the main theorem of this section. Theorem 7.3. Consider Eq. (2.6), where 11. II denotes a norm. Consider a solution for which the initial values at time t o are such that the angular momentum L = xy' - yx' is positive. Then, either (case i) r(t) ~ oo as t ~ oo or (case ii) there is a number T, such that, for all t, X ( t + T) = X(t) and such that, for t e (to, to + r), O(t) # 0(t0). Furthermore, there exists initial values for which case ii occurs.
Proof." Write (2.6) in polar coordinates as follows d dt (r(t)20'(t) - L) = 0,
(7.3)
L3 -r(t) r"(t) -- r ~ - IIX(t)l[ s"
(7.4)
From (7.3) and the given initial value, it follows that O'(t) = L/r(t) 2. Let s = L2/r and consider the differential equation relating the value of s to 0. This equation is found to be
d2s dO--g+ s = g(0),
(7.5)
where
( r ( t ) ~3 g(O(t)) = \llX(t)llJ and an application of Lemma 7.1 shows that every solution of (7.5) is periodic. If s(O) becomes 0, this corresponds to case i. On the other hand, if s(O) is bounded above 0, r is bounded and case ii holds with
T = L 3 ~2~ dO Jo s(O) 2" The existence of such case ii solutions follows from Lemma 7.2.
9
S o m e Orbital Test P r o b l e m s
85
Appendix \
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J.C. Butcher
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J.C. Butcher: Some Orbital Test Problems References
[1] Hull, T. E., Enright, W. H., Fellen, B. W., Sedgwick, A. E.: Comparing numerical methods for ordinary differential equations. SIAM J. Numer. Anal. 8, 603-637 (1972). J. C. Butcher Department of Mathematics University of Auckland Auckland New Zealand
