On the Stromgren-Wintner Natural Termination PrincipleAuthor(s): G. Baley PriceSource: American Journal of Mathematics, Vol. 55, No. 1 (1933), pp. 303-308Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2371132 .

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ON THE STROMGREN-WINTNER NATURAL TERMINATION PRINCIPLE.

By G. BALEY PRICE.

Introduction. The subject of analytic continuation of periodic orbits, developed by Poincare,t has been considered in two papers recently by Wintner.t In the first of these he proved the StrPmgren-Wintner Natural Termination Principle for groups of periodic orbits. The only example of the principle which has been given so far is the restricted problem of three bodies, but it is so complicated that the groups of periodic orbits can be studied only by numerical integration of the equations of motion. The present paper furnishes a simple example which can be treated mathematically. Also it gives illustrations of Poincare's theorems on the disappearance of periodic orbits by pairs t and on the change of stability of periodic orbits.?

Wintner shows that Poincare's theorem on the disappearance of periodic orbits by pairs is without significance in the study of groups of periodic orbits. By means of the present example it is pointed out that there are other points of view, or other problems, in dynamics in which the disappear- ance of periodic orbits in Poincare's sense is a real and significant phenomenon.

Finally, this paper adds further information about a class of dynamical systems previously investigated.? In order to make the example as simple and specific as possible, a special case of the general class is studied here, but the results hold for any of the systems for which r0,2 + u02 7&0 [see (8) below], and only obvious modifications in the results are necessary in case this condition does not hold.

1. The equationrs of motion. We shall consider the motion of a heavy particle on a surface of revolution S of genus one. Choose the positive c-axis directed downward [R 1, p. 753 (i. e., reference 1 at the end of this paper)] .

t Poincare, M6thodes Nouvelles de la Mecanique Ce'leste, Vol. 1 (1892), chapter 3. t Wintner, " Beweis des E. Stromgrenschen dynamischen Abschlussprinzips der

periodischen Bahngruppen im restringierten Dreiko6rperproblem," Mathematische Zeit- schrift, Vol. 34 (1931), pp. 321-349; "Sortengenealogie, Hekubakomplex und Gruppen- fortsetzung," Mathematische Zeitschrift, Vol. 34 (1931), pp. 350-402.

? Poincar6, Methodes Nouvelles de la M6canique Celeste, Vol. 3 (1899), pp. 343-351. 1 Price, " A class of dynamical systems on surfaces of revolution," American

Journal of Mathematics, Vol. 54 (1932), pp. 753-768. 303

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304 G. BALEY PRICE.

Then u (x) = gg(x), where g is the acceleration of gravity, and from the first paper [R 1, equations (9), (12), (13) ] we have

(1-) r2y - c, [the integral of areas]

(2) z (c2rT + gr3g)/r3

(3) x=2 [2r2(gC + h) - c2]/r2.

Here h is the energy constant. The functions v and w are [R 1, (14), (15)]

(4) v 2r2(gC + h),

(5) w=- gr34,O/r,, r 7o.

Finally, using (4) and (5), we can write (2) and (35) in the form

(6) x" =r (c2-w)/r3, r# 0

(7) e2_ (V -62)Ir2.

The following relation is important also [R 1, (5)]:

(8) r.2+ tx 2

2. Properties of v and w. A direct computation gives

(9) v= 2 [2rr, (gg + h) + gr2],

which can be written in the form

(10) v ~ 2r,(v-w)/r, r. #5 0.

Now since (8) holds, we see from (9) that vet never vanishes when r, = 0. Then from (10) we obtain the proof of the following lemma.

LEMMA 1. The derivative vw vanishes when and only when (v - w) vanishes.

The following lemma states another important fact.

LEMMA 2. A necessary and sufficient condition that the parallel x x* be a trajectory is that v - C2 o 0, v0, = 0 on this parallel.

A necessary and sufficient condition that x = x* be a trajectory is that x O, x" 0 on x - x8. Then since vt =0 when and only when (v w) =0, the lemma follows from (6) and (7).

Let a parallel x = x* on which vt = 0 be designated by P*. On a parallel P*, re, #5 0 and v = w, and we find that

(11) Vxx = - 2r,wS/r.

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ON THE STROAIGREN-WINTNER NATURAL TERMINATION PRINCIPLE. 305

Hence, on a parallel P*, v has a maximum if r$w& > 0 and a minimum if r$w$ < 0. Using this result, we can prove without difficulty the following lemma. The details are left to the reader.

LEMMA 3. A necessary and sufficient condition that v have a maximum (minimum) on a parallel P*: x = x* is that x = x* be an interior point of some tnterval in which r,,wx ? 0 (r,w, ? 0).

One further lemma is necessary.

LEMMA 4. A necessary and sufficient condition that v have a point of inflection with a horizontal tangent on a parallel P*: x = x* is that r,w. have opposite signs in sufficiently small intervals on opposite sides of x = x8.

The condition is necessary, for if r_ww has the same sign on the two sides of x*, then v has a maximum or minimum by lemma 3. Also, the condition is sufficient, for by lemma 3 v can have neither a maximum nor a minimum; hence, it has a point of inflection with a horizontal tangent.

Now plot v =v (x) and w =w (x) on the same field of rectangular cooirdinates. Since v and ?V are periodic with period o [R 1, (2)], we may restrict attention to the interval 0 ? x < w. At each zero of r0,, w has a vertical asymptote. Now r has at least one maximum and one minimum, at which r, vanishes and changes sign. Because of (8) then, w' has at least two vertical asymptotes at which wv is asymptotic to one end of the asymptote on one side and to the other end on the other side.

Now iv is fixed by the choice of the surface S and does not vary for a given system. On the other hand, v varies with the energy constant h. But since v is finite for all values of x and h, the curves v and w have a certain number of intersections. -By lemma 1, vx 0 at each point of intersection. The nature of v at the point of intersection is further determined by lemmas 3 and 4.

If v and w intersect at a point where wx :# 0, or at a point where w has a point of inflection with a horizontal tangent, then v has a maximum or minimum by lemma 3. At such a point v and w cross. If v and w intersect at a point where w has a maximum or minimum, then v has a point of in- flection with a horizontal talngent. At such a point v and w do not cross.

3. Groups of periodic orbits. It is possible to choose h uniquely so that v intersects iv at an arbitrary point of w. Furthermore, by lemma 2 any intersection of v and w at which v and w are positive corresponds to a parallel P* which is a closed periodic orbit on S. As h varies, the intersections of v and w vary, and in general analytically. Thus the closed orbits P* can be

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306 G. BALEY PRICE.

continued analytically with h to form what Wintner [R 2] has called a group of periodic orbits.

The points of the curve w for which w > 0 are in one-to-one continuous correspondence with closed periodic orbits P* in (x, y, h) space. Then each connected piece of w! lying in the region w > 0, is in one-to-one continuous correspondence with the orbits of a group. We may therefore refer to a con- nected piece of w in the region w > 0 as the graph of a group.

Let us investigate the manner in which these groups terminate. In the present case, a group terminates in one of two ways. In the first place, the graph may have an end point on the line w = 0. As we approach such an end point along the graph, the period of the corresponding orbit P* becomes infinite [see (1) and lemma 2] with h and the dimensions of the orbit remaining finite. In the second place, the graph may have a vertical asymp- tote. As we approach such an asymptote along the graph of the group, the energy constant h becomes positively infinite for the corresponding orbit P*. The period of the orbit approaches zero, and its dimensions remain finite.

These results are in accord with the Natural Termination Principle. The example does not show a group which closes into itself, nor a group which terminates because the dimenisions of the orbit become infinite.

Let us view this example in the light of Poincare's conclusions [IR 3, Vol. 1, p. 83]. Consider any intersection of v and w. As h varies, this intersection varies and generates what we may call a bran ch of a group of periodic orbits. Start at any point on the graph of a group and continue along the graph in each direction as far as possible without passing a maximum or minimum of w = w(x); the periodic orbits which correspond to any such piece of the graph form what we call a branch of a group. There is at most one periodic orbit in each branch of a group for a given value of h. Poincare's conclusion was that a branch of a group can be continued with increasing and decreasing h unless it combines with a second such branch and disappears. Our example shows exactly how this may happen.

Consider a maximum of w = w (x). For a suitqble value of h, v will intersect w twice in the neighborhood of this maximum; at one intersection v has a maximum and at the other a minimum (see lemma 3). As h in- creases, the maximum and minimum vary until v is finally tangent to w at the maximum of w. The maximum and minimum of v have combined to form a point of inflection with a horizontal tangent. At the same time the two corresponding periodic orbits P* have combined to form a multiple orbit. For still larger values of h, there is no periodic orbit P* in the neighborhood. The orbits have disappeared as described by Poincare.

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ON THIE STROMGREN-WINTNER NATURAL TERMINATION PRINCIPLE. 307

Furthermore, it should be observed that of the two orbits which combine and disappear, one is stable and the other is unstable [see (11) and R 1, (20)]. This result is in agreement with one of Poincar6's well known theorems [R13, Vol. 3, pp. 343-351]. We see [(11) above and R11, (20)] that the characteristic exponents which Poincare calls a and - a are zero when and only when wx = 0. As we follow along the graph of a group, there is no change in stability when we pass h point of inflection with a horizontal tan- gent. On the other hand, there is a change in stability whenever we pass a maximum or minimum on the graph of the group. From Poincare's point of view, a stable and an unstable orbit combine and disappear with proper variation of h at these points.

Whenever we consider the totality of orbits for a given value of the energy constant h [it has been customary to study dynamical systems from this point of view in the past; see R4, p. 270], Poincare's theorem on the disappearance of periodic orbits by pairs will be meaningful and significant. In particular, suppose the present problem is being studied by means of a surface of section [R 1, ? 4]. In setting up a surface of section we must consider the totality of orbits for a given value of h. The members of a group of periodic orbits which exist for the given value of h give rise to fixed points in the surface transformation on the surface of section. As h varies, these fixed points vary and sometimes appear and disappear by pairs. Whenever two branches of a group combine and disappear with variation of h, two fixed points combine and disappear. The fixed points of a surface transformation are highly significant features of the transformation. Thus, although Poincare's theorem on the disappearance of periodic orbits by pairs is without significance in the study of groups, it is both meaningful and significant in certain other problems.

The group as considered by Wintner is obtained as follows. Take any branch of a group; it may be that at one end, or both, this branch joins to other branches. Join on these branches, and then join any branches that have an end in common with these branches, anid so on. In the present problem we are stopped only when we reach a branch which terminates on the line w = 0, or which has a vertical asymptote. The group appears as the totality of branches that can be reached by continuation from a single branch.

The development of the problem of analytic continuation of periodic orbits can be sketched briefly as follows: Poincar6 considered the branches of a group rather than the group itself, and showed that a branch might terminate by combining with a second branch [R 3, Vol. 1, p. 83]. He con- sidered no other possibilities, however. Later he recognized that a branch might terminate because the period becomes infinite [R 5, p. 258]. Birkhoff

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308 G. BALEY PRICE.

states [R 5, p. 258]: "To make possible an extension to a preassigned interval ,o

, 1 _ i it is necessary to prove that the period of the varying periodic

orbit does not become infinite." The question of the sufficiency of the con- dition is not considered explicitly. Finally, Str6mgren and Wintner have emphasized the point of view of the group, and Wintner has proved that a group does not terminate so long as the period, energy, and dimensions of the orbit remain finite.

Finally, we may emphasize that the Natural Termination Principle does not prove that there is at least one member of a given group for every value of the parameter h. In the above example it is true that there exists a periodic orbit PP in each of at least two groups for all values of h for which motion over the entire surface is possible, but this is not true for every group. We shall show a group to illustrate this fact.

Choose the surf ace S so that

r. > 0,~ X1 - x-:5X2

gX (X1) - ' g (X2) ' 0,

gO < O, X1 < X < X2.

Then w(x1) = W (x2) = Oj and w is positive and finite for x1 < x < X2. We thus have a group which terminates at both ends because the period becomes infinite. For suitable small values of h, there are certain periodic orbits of this group in the system for the corresponding value of h. As h increases, these orbits combine and disappear by pairs. For h sufficiently large, there is no periodic orbit of the group in the system.

UNION COLLEGE, SCHENECTADY, NEW YORK.

REFERENCES

'Price, "A class of dynamical systems on surfaces of revolution," American Journal of Mathematics, Vol. 54 (1932), pp. 753-768.

2 Wintner, " Beweis des E. Strimgrenschen dynamischen Abschlussprinzips der periodischen Bahngruppen im restringierten DreikUrperproblem," Mathematische Zeit- schrift, Vol. 34 (1931), pp. 321-349.

Poincare, Me'thodes Nouvelles de la M6canique Celeste, Vol. 1 (1892), Vol. 3 (1899).

4'Birkhoff, "The restricted problem of three bodies," Rendiconti del Circolo Mate- matico di Palermo, Vol. 39 (1915), pp. 265-334.

5 Birkhoff, " Dynamical systems with two degrees of freedom," Transactions of the American Mathematical Society, Vol. 18 (1917), pp. 199-300.

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