Entire Parabolic Trajectories
as Minimal Phase Transitions
For the class of anisotropic Kepler problems in with homogeneous potentials, we seek parabolic trajectories having prescribed asymptotic directions at infinity and which, in addition, are Morse minimizing geodesics for the Jacobi metric. Such trajectories correspond to saddle heteroclinics on the collision manifold, are structurally unstable and appear only for a codimension-one submanifold of such potentials. We give them a variational characterization in terms of the behavior of the parameter-free minimizers of an associated obstacle problem. We then give a full characterization of such a codimension-one manifold of potentials and we show how to parameterize it with respect to the degree of homogeneity.
Let us consider the conservative dynamical system
where , the potential is smooth outside –and goes to infinity near– the collision set , and it satisfies the normalization condition
A (global) parabolic trajectory for (1.1) is a collisionless solution which has null energy:
In the Kepler problem () all global zero-energy trajectories are indeed parabola. 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 [3, 4, 5, 16, 19, 14, 21, 22]. Our aim in this paper is to introduce a new variational approach to the existence and characterization of such trajectories for homogeneous potentials.
Let us then assume that
is homogeneous of degree , for some .
In this setting parabolic trajectories can be equivalently defined as solutions satisfying as [14, 5], see also Appendix C. Furthermore, such orbits enjoy asymptotic properties, regarding both and . First of all as ; on the other hand, recalling that a central configuration for is a unitary vector which is a critical point of the restriction of to the sphere , the normalized configuration has infinitesimal distance from the set of central configurations of , as . In particular, whenever this set is discrete, we have that
where are central configurations.
From this point of view, as enlightened by McGehee in , parabolic trajectories can be seen as heteroclinic connections in the collision manifold between two asymptotic configurations at infinity (in time and space). This characterization has been exploited, starting from McGehee, and up to the work of Moeckel , in order to study the motion in the three-body problem near collisions. In the same perspective, an exhaustive study of the planar anisotropic Kepler problem was performed by Devaney. The potential he considers in [8, 9] has two pairs of non degenerate central configurations, corresponding to two minima and two maxima for the restricted potential. In this situation, parabolic trajectories can be classified into different types, depending on the limiting directions: the typical ones, which always exist, connect two central configurations which correspond to maxima; connections minimum-maximum generically exist and are quite stable objects; finally, connections minimum-minimum generically do not exist (in the setting of , these are saddle-saddle heteroclinic connections in the phase plane of the angular variable). Our aim is to provide conditions for the existence of this latter kind of trajectories, in terms of a variational characterization involving a minimization problem.
An interesting interpretation of the existence of parabolic trajectories which are free Morse minimizers for the action, in the special case of two dimensions, can be given in terms of the weak KAM theory (see [10, 11, 12]). Indeed, the existence of one minimal entire, collision free, parabolic trajectory induces a lamination of the plane by minimal trajectories (all its rescaled orbits), all homoclinic to the minimal Aubry set (in the present case, the infinity) and, correspondingly, leads to the existence of an entire solution of the associated Hamilton-Jacobi equation on the punctured plane (see Remark 6.12 and also Remark 1.5 below).
To start with, given any as above, , and belonging to the Sobolev space , let us consider the (possibly infinite) lagrangian action functional with lagrangian :
(of course, the action may be finite even though the path interacts with the singularity of the potential). Given and ingoing and outgoing asymptotic directions, we consider the following class of minimizers.
We say that is a (free) minimizer of of parabolic type, in the sense of Morse, if
, as ;
for every , , and , there holds
In some situations one may be also interested in Morse minimizers in a local sense, for instance imposing some topological constraint. In any case, a parabolic Morse minimizer is of class and, because of Maupertuis’ principle, it satisfies the Euler-Lagrange equation (1.1) and the zero-energy relation (1.2).
We stress the fact that, in general, a potential does not need to admit a parabolic Morse minimizer. To deal with this intrinsic structural instability we need to introduce an auxiliary parameter and look for parabolic orbits as pairs trajectory-parameter. To clarify the role of the additional parameter, it may be helpful to let the potential vary in a class. As a toy model, we will work on a class shaped on a multidimensional version of the case described by Devaney, choosing as parameter the homogeneity exponent .
More precisely, let us fix in and , and let us define the metric spaces
the latter being equipped with the product distance. With some abuse of notation, we will systematically identify any element of with the homogeneous extension of its first component:
in such a way that are non-degenerate, globally minimal central configurations for , which singular set coincides with the origin.
As we will show, the property of a potential to admit parabolic minimizers is related to its behavior with respect to the following fixed-endpoints problem. For any , let us define
it is not difficult to prove that such infimum is indeed a minimum, achieved by a possibly colliding solution. More precisely, recalling that a homothetic motion is a trajectory with constant angular part, the following result holds.
Let ; then one of the following alternatives is satisfied (see Figure 1):
is achieved by the juxtaposition of two homothetic motions, the first connecting to the origin and the second the origin to ;
, and it is achieved by trajectories which are uniformly bounded away from the origin.
Following the previous proposition, we distinguish potentials with “inner” minimizers (i.e. minimizers which pass through the origin) from potential with “outer” ones:
It is easy to see that these two sets are disjoint and their union is the whole ; moreover we will show that the first one is closed while the second is open. We are interested in their common boundary, that is
The separating property of the common boundary is underlined by the following lemma.
There exists an open nonempty set , and a continuous function such that
Furthermore, we will provide explicit criteria in order to establish whether a potential belongs to the domain of the function .
The main result of this paper states that the above graph coincides with the set of potentials admitting parabolic Morse minimizers.
admits a parabolic Morse minimizer if and only if .
Of course, due to the invariance by homotheticity of the problem, such Morse minimizing parabolic trajectories always come in one-parameter families and give rise to a 2-dimensional Lagrangian submanifold having boundary corresponding to the two homothetic solutions (see Figure 2). In the planar case, minimal orbits can be considered in a given homotopy class of paths with values in , and the same can be done in any non simply connected target. With this variant in mind, it is meaningful to have equal ingoing and outgoing asymptotic directions. In this case we find the aforementioned lamination of the configuration space giving rise to a solution of the Hamilton-Jacobi equation associated with (1.1), which is on the double covering of .
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 , the property to be collisionless for all Bolza minimizers implies the absence of parabolic trajectories which are Morse minimal. In particular, as they point out, this is the case for the -body problem, when no topological constraints are imposed. 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 . Up to our knowledge, the present paper is the first with positive results about the existence of globally defined parabolic minimizers.
The paper is structured as follows. In Section 2, we give an account of the planar case , relating minimal parabolic trajectories with the aforementioned minima connections in Devaney’s work. Next, to construct global-in-time Morse minimizers in higher dimension, we first consider problems on bounded intervals (Sections 3, 4), and then pass to the limit (Section 5). This procedure may fail for two main reasons: sequences of approximating trajectories may either converge to the singularity, or escape to infinity. This naturally leads to introduce some constraint in our construction, and to define constrained Morse minimizers, satisfying (see Definition 5.1). The study of the interaction of such minimizers with the constraint leads to the definition of the position-jump and of the velocity-jump of a trajectory, see Figure 5. Under this perspective, the crucial fact is that such quantities do not depend on the minimizer, but only on the potential (they are indeed related to the corresponding apsidal angle). In Section 6 we give the full details of the relations between parabolic minimizers, constrained Morse minimizers, position- and velocity-jumps, and the separating interface , obtaining as a byproduct the proof of Theorem 1.4. Finally, for the reader’s convenience, in the appendices we collect the proof of some rather known results for which we could not find an appropriate reference.
Throughout the paper we will often use polar coordinates, that corresponds to writing , where
here denotes the projection of on the tangent space :
Finally we will denote with and the extrema of , with
and with any (positive) constant we do not need to specify.
2 The Planar Case
As a guideline for our higher dimensional studies, in this section we consider the planar anisotropic Kepler problem. Indeed, following Devaney [8, 9], when dealing with zero-energy solutions this problem is equivalent, after some suitable change of variables, to a bi-dimensional autonomous dynamical system, for which explicit calculations can be carried out.
We briefly sketch Devaney’s procedure. Introducing the standard polar coordinates , the potential is a -periodic function in ; for a clear-cut notation we define:
and we then deal with the extended -homogeneous potential
The singularity at can be removed by a change of time scale
This allows to rewrite the system as (here “ ” denotes the derivative with respect to )
which solutions are globally defined in . Let us concentrate on the first equation: on one hand we have that can never vanish; on the other hand, to ensure that is unbounded both in the past and in the future, it is sufficient to check that
|is bounded away from zero and positive (resp. negative)||(2.1)|
as (resp. ). Furthermore this also implies that as does. Keeping in mind the above condition, the study of (not necessarily minimal) parabolic orbits reduces to the one of the planar system
To start with, let us take into account the situation when the potential is isotropic, for instance ; in this case, every is a minimal central configuration, and the dynamical system above reads
which critical points satisfy , . Furthermore, trajectories lie on the bundle , . Recalling condition (2.1), we infer 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 . Dealing with the Kepler problem () this angle is , hence the heteroclinic between and describes a parabola whose axis form an angle with the horizontal line. It is worthwhile noticing that, despite connecting minimal configurations, these parabolic trajectories are not globally minimal in the sense of Definition 1.1 (it can be shown that they are minimal in their homotopy class).
If is not constant, stationary points of (2.2) are such that , and . By linearizing it is easy to see that non-degenerate minima (resp. maxima) for correspond to critical points which are saddles (resp. sinks/sources). Accordingly, taking into account condition (2.1), let us assume that the system admits a pair of saddles , , such that and .
Let us define the function
By direct computations, we obtain that is non-decreasing on the solutions of (2.2). Let now and be defined as above. A parabolic trajectory between projects, on the plane , on an increasing graph connecting and (we refer to Figure 3).
Generically (see also Theorem 4.10 in ), the unstable manifold at falls directly into a sink, while the stable manifold at emanates from a source: we claim that, for some values of the parameter , such two points can be connected. Let us focus on the heteroclinic from . Since the derivative of can be written as
strictly increases whenever ; in such monotonicity intervals we have that , hence also is strictly monotone. As a consequence we can read as a function of (inverting )). Our aim is to show that, for some and some , there exists a solution of the dynamical system such that
integrating on and , we obtain on one hand
and on the other hand
From the first inequality we deduce that, as becomes very small, does not exceed ; from the second one we infer that, as tends to 2, diverges to . It is possible to conclude that, for any (large) there exists such that and , see Figure 4. More results in this direction are contained in .
3 Bolza Minimizers
Now we turn to the general case of dimension . In this section we investigate constrained fixed-endpoints problems for lagrangians with a potential , under the assumptions that is -homogeneous and that is positive and smooth. In particular all the results will hold if , even though here the assumptions about do not play any role.
Let us fix , , and . We introduce the sets of constrained paths
and their union
This section is devoted to study the Bolza minimization problem
Let us remark that, for a unified treatment, we let and/or belong to the constraint . To avoid degenerate situations we suppose that implies , excluding the trivial case.
As we noticed in the introduction, we minimize with respect to both trajectory and time length . The reason is that such procedure will provide zero-energy motions (see Appendix A). To exploit this property we define, for any , the Maupertuis’ functional
If achieves , then , , achieves
On the other hand, if achieves the infimum above, then , achieves where
For any , there exists a one-to-one correspondence between the sets and : if and only if . Taking into account this fact, the lemma follows by arguing as in the proof of Lemma A.1. ∎
The reformulation of problem (3.1) in terms of allows to easily prove the existence of a minimizer.
The infimum of on is achieved.
Let be a minimizing sequence. We claim that is bounded away from zero. If this is true then the lemma follows in a standard way: indeed, as a consequence, is uniformly bounded and hence weakly convergent in , is weakly closed, and is weakly lower semi-continuous. To prove the claim, let us assume by contradiction that ; then there exists such that and then, by homogeneity, . By Hölder’s inequality we have, as becomes large,
Hence and, since , this contradicts the fact that is a minimizing sequence. ∎
is achieved in . For every minimizer, the function is continuous. In particular (1.2) holds for every .
On every interval with , satisfies the Euler-Lagrange equation, which implies the Lagrange-Jacobi identity
Therefore is a convex function; in particular this implies that if there exist , such that , then for every , and the corollary follows. ∎
Let and be as above. Then
(here denotes the tangential part of , defined in equation (1.4)).
By definition, minimizes with the pointwise constraint. Applying Lagrange multipliers rule we obtain that
We can compute multiplying by , and recalling that, since , then and . ∎
From the previous discussion it follows that a minimizer may be not regular only in and . Our last aim is to study the behavior of in these points.
Let achieve , and , be defined as in Corollary 3.4. Then one of the following three situations occurs:
and ; in such a case undergoes a radial reflection, that is
We prove the proposition in the case ; the general one follows straightforwardly. We recall the definition of the Kelvin transform:
We have that is a conformal map, , ,
Hence, whenever and we have that
this means that is the reflection matrix with respect to the hyperplane orthogonal to .
Let be the definition interval of ; let , be the path
Using the homogeneity of we obtain
The function is then a minimizer for
on the set , without any other constraint. Since is Lipschitz continuous with respect to , one can prove, by standard arguments in the Calculus of Variations, that (see for instance ).
We now go back to the path . From Corollary 3.4 we deduce that only two different situations can occur: in the first case , in the second one , with .
Let us focus on the first situation; being of class we have
As previously remarked, is the reflection matrix with respect to the hyperplane orthogonal to , hence if then , and (case (b)); otherwise if then . In this case we can deduce the radial reflection of case (c); indeed, since and , we have
while the component of the velocity orthogonal to is conserved, that is:
Let us now consider the second situation, when the minimizer remains on for a nontrivial time interval. Since is of class , both vectors and are tangent to and, still using the properties of , we have that (case (a)). ∎
The previous proposition suggests to classify minimizers with respect to the discontinuity of the quantities and on the constraint.
Let be a constrained Bolza minimizer, and , as above. Then we can define the following quantities (see Figure 5):