Entire minimalparabolic trajectories:the planar anisotropicKepler problem

Entire minimal parabolic trajectories: the planar anisotropic Kepler problem

Abstract

We continue the variational approach to parabolic trajectories introduced in our previous paper [5], which sees parabolic orbits as minimal phase transitions.

We deepen and complete the analysis in the planar case for homogeneous singular potentials. We characterize all parabolic orbits connecting two minimal central configurations as free-time Morse minimizers (in a given homotopy class of paths). These may occur for at most one value of the homogeneity exponent. In addition, we link this threshold of existence of parabolic trajectories with the absence of collisions for all the minimizers of fixed-ends problems. Also the existence of action minimizing periodic trajectories with nontrivial homotopy type can be related with the same threshold.

1 Introduction and Main Results

For a positive, singular potential , vanishing at infinity, we study the Newtonian system

(1)

searching for parabolic solutions, i.e. entire solutions satisfying the zero-energy relation

(2)

In the Kepler problem () all global zero-energy trajectories are indeed parabola. In this paper we are concerned with -homogeneous potentials, with . Within this class of potentials, parabolic trajectories are homoclinic to infinity, which represents the minimum of the potential.

In celestial mechanics, and more in general in the theory of singular hamiltonian systems, parabolic trajectories play a central role and they are known to carry precious information on the behavior of general solutions near collisions. On the other hand, parabolic trajectories are structurally unstable and therefore are usually considered beyond the range of application of variational or other global methods. In spite of this, in our previous paper [5], we introduced a new variational approach to their existence as minimal phase transitions.

The purpose of the present paper is to deepen and complete the analysis in the planar case : we will succeed in characterizing all parabolic orbits connecting two minimal central configurations as free-time Morse minimizers (in a given homotopy class of paths). In addition, we shall link the threshold of existence of parabolic trajectories with the absence of collisions for all the minimizers of fixed-ends problems. Also the existence of action minimizing periodic trajectories with nontrivial homotopy type will be related with the same threshold.

In the plane we use the polar coordinates (despite the ambiguous notation, it will always be clear from the context wether a pair denotes either cartesian or polar coordinates). Under this notation any -homogeneous potential can be written as

where

The potential is then a generalization of the anisotropic Kepler potential (extensively studied for instance in [16, 17, 21, 22, 23]), which actually corresponds to the value of the parameter and a specific . For such potentials, it is well known that parabolic trajectories admit in/outgoing asymptotic directions which are necessarily critical points of : these are called central configurations. We are mostly interested to parabolic trajectories connecting two minimal central configurations. To be more precise, given

we define the sets of potentials

and, with a slight abuse of notation,

For a given , we introduce the action functional

In our previous paper [5], we introduced the set of Morse parabolic minimizers associated to and having asymptotic directions and . Nonetheless, since is not simply connected, as a peculiar fact in the planar case one can also impose a topological constraint in the form of a homotopy class for the minimizer, for example imposing counterclockwise rotations around the origin. Lifting such a trajectory to the universal covering of , this corresponds to joining with . Motivated by these considerations, we introduce the set

and, given in (or, more in general, central configurations), we define the following class of paths.

Definition 1.1.

We say that is a parabolic trajectory associated with , and , if it satisfies equations (1), (2) and

  • ;

  • , as ;

We say that is a (free time) parabolic Morse minimizer if moreover there holds

  • for every , , and , there holds

    (this last property actually implies (1), (2)). A fixed time minimizer fulfills the above minimality condition only with .

Under the previous definition the following holds.

Theorem 1.2.

Let and , be fixed minimal central configurations; then

  • there exists at most one such that admits a corresponding parabolic trajectory associated with if and only if ;

  • every parabolic trajectory associated with , and is a free time Morse minimizer;

  • if then there exists exactly one such that admits a corresponding parabolic Morse minimizer if and only if .

Let us point out that, if , such a number may or may not exist depending on the properties of .

To proceed with the description of our results, let us extend the function to the whole of the possible triplets by setting its value to zero if there are no parabolic trajectories for any . This exponent can be related to the presence/absence of collisions for both the fixed time and the free time Bolza problems within the sector defined by the angles and .

The problem of the exclusion of collisions for action minimizing trajectories has nowadays a long history, starting from the first elaborations in the late eighties, e.g. [1, 15, 14, 11, 12, 27, 28] up to the extensive researches of the last decade, mostly motivated by the search of new symmetric collisionless periodic solutions to the –body problem (e.g. [9, 10, 6, 18]). Starting from the idea of averaged variation by Marchal [25, 8], later made fully rigorous, extended and refined in [19], a rather complete analysis of the possible singularities of minimizing trajectories has been recently achieved in [3]. In the literature, minimal parabolic trajectories have been studied in connection with the absence of collisions for fixed-endpoints minimizers. More precisely, as remarked by Luz and Maderna in [13], the property to be collisionless for all Bolza minimizers implies the absence of parabolic trajectories which are Morse minimal for the usual –body problem with . On the contrary, minimal parabolic arcs (i.e., defined only on the half line) exist for every starting configuration, as proved by Maderna and Venturelli in [24].

A special attention has been devoted to minimizers subject to topological constraints and to the existence of trajectories having a particular homotopy type (see e.g. [20, 26, 2, 25, 29, 7]). For such constrained minimizers the averaged variation technique is not available, and other devices have to be designed to avoid the occurrence of collisions. Starting from [29], motivated by the search of periodic solutions having prism symmetry, a connection has been established between the apsidal angles of parabolic trajectories and the exclusion of collisions for minimizers with a given rotation angle. In fact we can now draw a complete picture of the role played by the parabolic orbits in the solution of the collision-free minimization problem with fixed ends.

Definition 1.3.

Given a potential , we say that is a fixed-time Bolza minimizer associated to the ends , , if

  • and , ;

  • for every , there holds

If we say that the Bolza minimizer is collisionless.

Theorem 1.4.

Let , , and consider a perturbed potential , with , and

(3)

If then all fixed-time Bolza minimizers associated to and within the sector are collisionless.

It is worthwhile noticing that, if conversely , then there are always some Bolza problems which admit only colliding minimizers. In addition, the very same arguments imply, when , the following statement, which gives a variational generalization of Lambert’s Theorem on the existence of the direct and inverse arcs for the planar Kepler problem ([25, 30]).

Proposition 1.5.

Let , , and be a perturbed potential as in the previous theorem, with . Given any pair of points and in the sector , all fixed-time Bolza minimizers associated to , within the sector , for some , are collisionless.

Some further interesting consequences can be drawn, in the special case when , which connect the parabolic threshold with the existence of non-collision periodic orbits having a prescribed winding number (this is connected with the minimizing property of Kepler ellipses, see [20]).

Theorem 1.6.

Let be such that all its local minima are non-degenerate global ones, and consider the potential . Given any integer and period , if

(4)

then any action minimizer in the class of –periodic trajectories winding times around zero is collisionless.

The outline of the paper is the following: in Section 2 we exploit some results due to Devaney [16, 17] in order to rewrite equations (1), (2) in terms of an equivalent planar first-order system; this allows us to develop a first phase-plane analysis of the dynamical properties of parabolic trajectories. In Section 3 we turn to the variational properties of zero-energy solutions. In Section 4 we prove Theorem 1.2 in the particular case in which . Finally Sections 5 and 6 are devoted to the end of the proof of Theorem 1.2 and to the proofs of Theorems 1.4, 1.6, respectively.

2 Phase Plane Analysis

Following Devaney [16, 17], an appropriate change of variables makes the differential problem (1), (2) equivalent to a planar first order system, for which a phase plane analysis can be carried out. This allows a first investigation of its trajectories from a dynamical (i.e. not variational) point of view.

Let , and let us assume for simplicity that is a Morse function, even though the only important assumption is that , are non-degenerate. Introducing the Cartesian coordinates , and the momentum vector , we write equations (1) and (2) as

and

Since , we have that . As a consequence, for every solution of the previous dynamical system we can find smooth functions and in such a way that , . These functions satisfy

and

This system has a singularity at that can be removed by a change of time scale. Assuming , we introduce the new variable via

in order to rewrite the dynamical system as (here “  ” denotes the derivative with respect to )

(5)

which contains the independent planar system

(6)

It is immediate to see that the systems above enjoy global existence, and that the stationary points of (6) are the points , where and .

Theorem 2.1 (Devaney [17]).

The path satisfies (1), (2) if and only if satisfies (6) (and satisfies (5)).

The function

is non-decreasing on the solutions of (6), which correspond to

  • saddle-type equilibria , , and ;

  • sink/source-type equilibria , where , and ;

  • heteroclinic trajectories connecting two of the previous equilibria.

To every trajectory of (6) there corresponds infinitely many trajectories of (5), all equivalent through a radial homotheticity.

The corresponding solutions of (1), (2) satisfy the following:

  • if

    as , (7)

    then is globally defined and unbounded in the future/past (in );

  • if as , then and as .

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Figure 1: the figure sketches the phase portrait of (6) when . The dynamical system reads , which critical points satisfy , . Trajectories lie on the bundle , , and, recalling condition (7), we deduce that parabolic solutions coincide with heteroclinic connections departing from points on and ending on , for some . For instance, when , we obtain heteroclinics connecting to , for some . Going back to the original dynamical system, this implies that parabolic motions exists only when the angle between the ingoing and outgoing asymptotic directions is ; let us emphasize that such angle is always greater than . When , i.e. in the classical Kepler problem, this angle is : the heteroclinic between and actually describes a parabola whose axis form an angle with the horizontal line.

In Figure 1 we describe the phase plane for the dynamical system (6) when is isotropic and in particular for the Kepler problem. On the other hand, if we take into account an anisotropic potential in the class and a homogeneous extension , , then we can deduce the following result (by time reversibility, it is not restrictive to assume that ).

Corollary 2.2.

Let belong to and let be an associated parabolic Morse minimizer for . Then (a suitable choice of) the corresponding is an heteroclinic connection between the saddles

Moreover is strictly increasing between and .

Proof.

Since are minima for we have that connects the two saddles (say)

in such a way that

Since is globally defined, condition (7) holds, yielding and , that is is odd while is even. Since is non-decreasing, we have that

Now we observe that

(8)

hence strictly increases. Then , therefore also is strictly monotone. Since we obtain that increases. But this finally implies that , for every . Summing up all the information we deduce that

Motivated by the previous result we devote the rest of the section to study the properties of the stable and unstable trajectories associated to the saddle points of (6), in dependence of the parameter . To start with, using equation (8), we provide a necessary condition for the existence of saddle-saddle connections.

Lemma 2.3.

Let us assume that for some there exists a saddle-saddle connection for (6) between and . Then

where , for every .

Proof.

Let be such an heteroclinic. Reasoning as in the proof of the previous corollary, one can deduce that both and are (strictly) monotone in . It is then possible to write obtaining that

With this notation we can write

Integrating on , we obtain on one hand

(9)

and on the other hand

(10)

Using the previous arguments, together with standard results in structural stability, it is already possible, for appropriate values of , to show the existence of saddle-saddle heteroclinic connections (see Figure 2). In any case, if in principle saddle-saddle connections occur only for particular values of , on the other hand, whenever are minima for , for every they correspond to saddle points. The above techniques allow us to study the dependence of their stable and unstable manifolds on .

Figure 2: the two pictures represent the phase portrait of the dynamical system (6) with , when (at left) or (at right). We focus our attention on the saddles and (that satisfy condition (7)): from the mutual positions of the heteroclinic departing from and the one ending in we deduce that the two vector fields are not topologically equivalent. By structural stability we infer the existence, for some , of a saddle connection between and .
Lemma 2.4.

Let denote the (unique, apart from time translations) unstable trajectory emanating from with increasing . Then it intersects the line in an unique point with first coordinate . Moreover is strictly increasing on and on the same interval can be expressed as a function of . Finally,

and on (see also Figure 3).

Proof.

To start with we observe that, for any , the linearized matrix for (6) at is

where . The eigendirection correspondent to the heteroclinic emanating from is , where is the positive eigenvalue of ; hence

On one hand, we have that

implying that, for different values of , the corresponding unstable trajectories are ordered as claimed near . On the other hand, since , we have that the trajectory is contained in the strip for large negative times.

Now recall that, as above, and that both and are strictly increasing whenever is smaller than . We deduce that there exists exactly one such that

As a consequence the value is well defined and, reasoning as in Corollary 2.2 and in Lemma 2.3, we can invert on . We deduce that we can write

(11)

To conclude the proof we have to show that, if , then where they are defined. To this aim, let by contradiction be such that on , and . But the above differential equation implies

a contradiction. ∎

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Figure 3: the unstable (resp. stable) manifold emanating from (resp. entering in ), and its dependence on , according to Lemma 2.4 (resp. Lemma 2.5). Here .

Arguing exactly as above one can prove analogous properties for the stable manifolds.

Lemma 2.5.

Let denote the (unique, apart from time translations) stable trajectory entering in with increasing . Then it intersects the line in an unique point with first coordinate . Moreover is strictly increasing on and on the same interval can be expressed as a function of . Finally,

and on .

By uniqueness, the above unstable/stable trajectories can not be crossed by any other orbit. To be more precise, we have the following.

Corollary 2.6.

Let

be any central configuration, and be a trajectory of system (6) emanating from and intersecting the set

Then, if exits from , it must cross either the union of the segments

or the vertical lines . Analogously, for a trajectory asymptotic (in the future) to , the entering set in

is the union of the segments

and of the vertical lines .

Proof.

We prove only the first part. If then , the unique unstable trajectory emanating from the corresponding saddle point with increasing considered in Lemma 2.4; but then it exits from through the point . In the same way, if then , the unique unstable trajectory emanating from the corresponding saddle point with decreasing (recall that, if solves (6), then also does); in such a case the exit point is . Finally, if , then must lie above and below , and the assertion follows. ∎

The angles defined above represent the (oriented) parabolic apsidal angles swept by the parabolic arc from the infinity up to the pericenter. As a consequence of the previous arguments, the appearance of a parabolic trajectory associated with the asymptotic directions , or, equivalently, the existence of a heteroclinic connection between and can be expressed in terms of the corresponding apsidal angles. Summing up, we have proved the following.

Proposition 2.7.

Let , , and the monotone functions , be defined as in Lemmata 2.4, 2.5, respectively. Then system (6) admits a heteroclinic connection between and for some value if and only if

In particular, if such a value exists, then it is unique.

The function can be extended to all the possible triplets and as follows:

Definition 2.8.

For any triplet , , we define the function

If we define .

In this way, the previous proposition proves the first point of Theorem 1.2.

As a final remark, let us notice that the apsidal angles defined above, and the corresponding stable/unstable trajectories, act as a “barrier” for any heteroclinic traveling in the strip and corresponding to a (not necessarily minimal) parabolic trajectory. Such kind of arguments will turn out to be useful in the proof of Theorems 1.4 and 1.6.

Proposition 2.9.

Let , , and let us assume that

Then does not admit any (not necessarily minimal) parabolic trajectory completely contained in the sector .

Proof.

By Theorem 2.1 (and in particular condition (7)) such a parabolic trajectory would correspond to an heteroclinic connection for system (6), joining an equilibrium (say) to another one , with integer. We want to prove that such a trajectory, under the above assumptions, can not be completely contained in the strip .

To start with, we observe that must be equal to either or . Indeed, the function is non-decreasing along any trajectory, and whenever , integer. W.l.o.g we can assume , so that the trajectory we are considering joins to . Let us assume by contradiction that it is completely contained in the strip ; but then, using the notations of Corollary 2.6, it must both exit and enter , across a single point belonging to the line and the strip. This immediately provides a contradiction with the selfsame corollary, since

3 Minimality Properties near Equilibria

The purpose of this section is to develop a first investigation about the minimality properties of zero energy solutions of (1) with respect to the Maupertuis’ functional

where . Indeed let us recall that

for every (fixed) , indeed is invariant under reparameterizations (see [1]). As a consequence, every parabolic trajectory is a critical point of , at least when restricted on suitably small bounded intervals.

In particular, we want to evaluate the second differential of along zero-energy critical points. In order to do this, we first perform a change of time-scale essentially equivalent to the Devaney’s one we exploited in Section 2. In polar coordinates reads as

introducing the time-variable

we obtain (noting with a prime “ ” the derivative with respect to )

where and depends now on , and

We introduce the change of variables

in order to obtain the Maupertuis’ functional depending on , i.e.