Index theory for asymptotic motions
We develop an index theory for parabolic and collision solutions to the classical -body problem and we prove sufficient conditions for the finiteness of the spectral index valid in a large class of trajectories ending with a total collapse or expanding with vanishing limiting velocities. Both problems suffer from a lack of compactness and can be brought in a similar form of a Lagrangian System on the half time line by a regularising change of coordinates which preserve the Lagrangian structure. We then introduce a Maslov-type index which is suitable to capture the asymptotic nature of these trajectories as half-clinic orbits: by taking into account the underlying Hamiltonian structure we define the appropriate notion of geometric index for this class of solutions and we develop the relative index theory.
Key words and phrases:Index theory, Maslov index, Spectral flow, Colliding trajectories, Parabolic motions, Homothetic orbits
1991 Mathematics Subject Classification:70F10, 70F15, 70F16, 37B30, 58J30, 53D12, 70G75
Most dynamical systems of interacting particles in Celestial and other areas of Classical Mechanics, are governed by singular forces. In this paper we are concerned with potential of -body type having the form
where . The positive real numbers can be thought as masses and the (real analytic) self-interaction potential function (the opposite to the potential energy) is defined and smooth in
is the space of admissible configurations with vanishing barycenter. This class of potentials includes the case, for instance, of the gravitational potential () governing the motion of point masses interacting according to Newton’s law of gravitation and usually referred to as the (classical) -body problem. Besides the homogeneity of some negative degree, a common feature of all these singular potentials is that their singularities are located on some stratified subset (in general an arrangement of subspaces or more generally submanifolds) of the full configuration space. Both properties play a fundamental role in the study of the dynamics of the system and strongly influence the global orbit structure being responsible, among others, of the presence of chaotic motions as well as of motions becoming unbounded in a finite time.
Associated with the potential we have Newton’s Equations of motion
where denotes the (Euclidean) gradient and is the mass matrix. As the centre of mass has an inertial motion, it is not restrictive to prescribe its position at the origin and the choice of as configuration space is justified. For , the tangent bundle of , the Lagrangian function is given by
where denotes the norm induced by the mass matrix .
We deal with the following class of trajectories having a prescribed asymptotic behaviour.
Given , a -asymptotic solution (or simply asymptotic solution or a.s. in shorthand notation) for the dynamical system (1.2) is a function which pointwise solves (1.2) on and such that the following alternative holds:
if , then and it experiences a total collision at the final instant , namely
if , then and
Such class includes homothetic self-similar motions , where is a central configuration, i.e. a critical point of the potential constrained on the ellipsoid . The study of these two kinds of motions has occupied a quite extensive research in the field and their strong connection already appeared in Devaney’s works [Dev78, Dev81] and it has been clearified recently in [DaLM13] and [BTV13].
For a.s. some asymptotic estimates, at , are available (cf. [Spe70, Sun13] when and [BTV14] when ) and one can prove that total collision and completely parabolic trajectories share the same behaviour: the radial component of the motion is, in both cases, asymptotic to the power (infinite or infinitesimal), while its configuration approaches the set of central configurations of the potential at its limiting level (for the details we refer to Lemma 2.3).
We shall be concerned with asymptotic solutions having a precise limiting normalized central configuration , terming these trajectories -asymptotic solutions. The aim of the present paper is to relate the Morse/Maslov index of the trajectory with its asymptotic properties. This will be the starting point for our long term project of building a Morse Theory, tailored for singular Hamiltonian Systems, which takes into account the contribution of the flow on the collision manifold, defined by McGehee in [MG74] by blowing up the singularity. A major problem in using directly McGehee regularizing coordinates is that the resulting flow looses its Hamiltonian as well as its Lagrangian character. We shall thus use a variant which keeps both the Hamiltonian and Lagrangian structure, while both types of asymptotic orbits, total collision and parabolic, will transform into half-clinic trajectories asymptotic to the collision manifold. By taking into account the underlying Hamiltonian structure we will introduce appropriate notions of spectral index and geometric index for this class of half-clinic orbits and we will relate them through an Index Theorem. These two topological invariants respectively encode, in the index formula, the spectral terms which are computationally inaccessible, and the geometric terms containing analytic information of a fundamental nature which are quite explicit and involve the spectrum of a finite dimensional operator.
To the authors knowledge, there are very few results in literature that investigate the contribution of collisions to the Morse index of a variational solution when the potential is homogeneous and weakly singular. In a very recent paper G. Yu (inspired by some paper by Tanaka, we cite for all [Tan93]) proved that the Morse index of a solution gives an upper bound on the number of binary collision of the -body problem (cf. [Yu17]). The only partial results that takes into account collisions involving more then two bodies are contained in the pioneering papers [BS08, HO16], in which the authors, using the precise asymptotic estimates near collisions, prove that the collision Morse index is infinite whenever the smallest eigenvalue of is less than a threshold depending on the homogeneity and itself.
In the present paper the following spectral condition, naturally associated with the -asymptotic solution, plays a central role
We will refer to this condition as the -condition. An analogue spectral condition has been recently used in [Hua11] to prove non-minimality of a class of collision motions in the planar Newtonian three body problem, expressed using Moser coordinates. It is worthwhile noticing that the -condition has an important dynamical interpretation, marking the threshold for hyperbolicity of as a rest point on the flow restricted to the collision manifold.
In one of the main result of the present paper, Theorem 3.8, we prove that when the central configuration satisfies the -condition then the Morse Index of any -a.s. is finite. A direct consequence of this theorem is that the condition given by authors in [BS08] is sharp for the infiniteness of the (collision) Morse index. Actually the infiniteness of the Morse index is paradigmatic of a more general situation of boundary value problems for systems of ordinary differential equations on the half-line (cf. [RS05a, RS05b]). In fact, we shall prove in Theorem 3.14 that the -condition gives the threshold for the linearized operator belonging to the Fredholm class. We observe that in the particular case in which the smallest eigenvalue of is equal to , then the Morse index could be finite even if the associated index form is not Fredholm.
The class of Fredholm operators has been the natural environment, since their debut, for index theories, whose principal protagonist is represented by the spectral flow for paths of selfadjoint Fredholm operators. This celebrated topological invariant was introduced by Atiyah, Patodi and Singer in the seminal paper [APS76] and since then it has reappeared in connection with several other phenomena like odd Chern characters, gauge anomalies, Floer homology, the distribution of the eigenvalues of the Dirac operators. In the finite-dimensional context the spectral flow probably dates even back to Morse and his index theorem while, in the one-dimensional case, it traces back to Sturm. There are several different definitions of the spectral flow that appeared in the literature during the last decades.
The last section of the paper is devoted to prove a Morse type Index Theorem for -asymptotic solutions. This results states that, when -condition is satisfied, then the spectral (or Morse) index and the geometric (or Maslov) index of an -a-s. coincide. The equality between these two topological invariants allows us to mirror the problem of computing the Morse index of a -a.s. (integer associated to an unbounded Fredholm operator in an infinite dimensional separable Hilbert space) into an intersection index between a curve of Lagrangian subspaces and a finite dimensional transversally oriented variety. The key of this result relies on the fact that the spectral index can be related to the spectral flow of a path of Fredholm quadric forms. The interest for this spectral flow formula is twofold. On the one hand, it could be useful for the computation of the Morse index since the computation is confined on a finite dimensional objects; on the other hand, from a theoretical point of view, it could be interesting to prove a Sturm oscillation theorem for singular systems and to relate the geometrical index with the Weyl classification of the singular boundary condition (cf. [Zet05] and references therein). To the authors knowledge, very few spectral flow formulas have been (recently) proved in the case of Lagrangian and Hamiltonian systems defined on an unbounded interval and in particular on half-line. (We refer the interested reader to [CH07],[HP17] and references therein). For this reason, we think also that, the interests of the spectral flow formula proved in Theorem 4.1 goes even beyond the framework of the present paper.
We finally remark that in the case of collisionless periodic solutions, in the last few years the Morse Index of some special equivariant orbits has been computed via Index Theory: in [HS10] and [HLS14] the authors studied the Morse Index of the family of elliptic Lagrangian solutions of the planar classical three body problem, while in [BJP16, BJP14] the authors restrict to the circular Lagrangian motions, taking into account different homogeneity parameter of the potential. The first step in the rigorous analysis of the relation between the symmetry of the orbit and its Morse Index (still via Index Theory) have been performed by Hu and Sun in [HS09] in which the authors computed the Morse Index and studied the stability of the figure-eight orbit for the planar 3-body problem by means of symplectic techniques.
We conclude the introduction, by observing that we have only scratched the surface of the implication of the Maslov index for halfclinic trajectories in Celestial Mechanics or more generally in singular Hamiltonian systems. It is reasonable to conjecture that the index theory constructed in this paper for -a.s. could be carried over for more general class of asymptotic motions arising in a natural way in Celestial Mechanics; more precisely, motions in which the limit of the normalized central configuration does not hold, but the linearized system along the motion experiences an exponential dichotomy as well as for orbits experiencing only partial collisions. Both these problems are promising research directions that should be investigated in order to obtain a deeper understanding of the intricate dynamics governed by singular potentials.
For the sake of the reader, in the following list we collected some mathematical symbols and constants that will appeared in the paper.
is the homogeneity of the potential
is the mass matrix
is the inertia ellipsoid with respect to the mass metric
is a normalized central configuration
The paper is organised as follows:
- 1 Introduction
- 2 Variational setting and regularised action functional
- 3 Spectral index and geometrical index for asymptotic motions
- 4 The Index Theorem
- A Maslov index and Spectral flow
2. Variational setting and regularised action functional
In this section we settle the variational framework of the problem. In Subsection 2.1 we introduce the main definitions and the basic notation which we need in the sequel as well as we define a suitable class of asymptotic motions both collision and parabolic. Subsection 2.2 is devoted to fix the variational setting. In Subsections 2.3 and 2.4, by means of the Lagrangian version of McGehee’s transformation, we compute the second variation of the corresponding Lagrangian action functional in these new coordinates.
2.1. Asymptotic motions: total collisions and parabolic trajectories
For any integer , let be positive real numbers (that can be thought as the masses of point particles) and let be the diagonal block matrix defined as with , where is the -dimensional identity matrix and . Being the Euclidean scalar product in , we indicate with
respectively the Riemannian metric and the norm induced by the mass matrix. For brevity we shall refer to them respectively as the mass scalar product and the mass norm.
Let denotes the configuration space of the point particles with masses and centre of mass in
Thus is a -dimensional (real) vector space, where . For each pair of indices let be the collision set of the -th and -th particles and let
be the collision set in . It turns out that is a cone whose vertex is the point ; it corresponds to the total collision or to the total collapse of the system (being the centre of mass fixed at ). The space of collision free configurations is denoted by
For any real number and any pair , we consider the smooth function given by
We define the (real analytic) self-interaction potential function on (the opposite of the potential energy), as follows
Newton’s Equations of motion associated to are
where denotes the gradient with respect to the Euclidean metric. It turns out that, since the centre of mass has an inertial motion, it is not restrictive to fix it at the origin and the choice of as configuration space is justified. Denoting by the tangent bundle of , whose elements are denoted by with and a tangent vector at , the Lagrangian function of the system is
where the first term is the kinetic energy of the system.
We will deal with -asymptotic solutions as introduced in Definition 1.1. Concerning the definition of completely parabolic motions let us recall that, by using the concavity of the second derivative of the moment of inertia, one can prove that completely parabolic motions have necessarely zero energy (cf. [Che98, Definition 1 and Lemma 2]). For this reason, from now on we will name this kind of asymptotic motions simply parabolic trajectories.
The behaviour of asymptotic motions as approaches to is well-known (cf. [Sun13, Spe70, BFT08, BTV14] and references therein) and in order to describe it, we perform a polar change of coordinates in the configuration space with respect to the mass scalar product. Let and be respectively the radial and angular variables associated to a configuration :
is the inertia ellipsoid, namely the unitary sphere in the mass scalar product.
In this remark we aim to motivate the fact that the assumption in Definition 1.1
is not too restrictive. Let us start considering a time- total collision trajectories; since the centre of mass of the system has been fixed at the origin, assumption given in Formula (2.7) implies the following condition on the radial component, ,
This fact is indeed a consequence of the Lagrange-Jacobi inequality (which actually shows the convexity of in a neighbourhood of the total collision). Furthermore, the authors in [BFT08] prove that a total collision is indeed isolated among collisions (not only total, but also partial). For these reasons the class of total collision motions we choose is a natural set of total collision motions.
When is a (completely) parabolic motion, since the kinetic energy goes to 0 and the total energy is conserved then, as , each mutual distance between pairs of bodies is bounded away from 0. Being an open set, also in this case assumption (2.7) is not restrictive.
The asymptotic behaviour both of total collision and parabolic motions is described by the forthcoming Lemma 2.3. For its proof we refer the interested reader respectively to [BFT08, Theorem 4.18] and [BTV14, Theorem 7.7]. We recall the following definition.
A configuration is a central configuration for the potential if it is a critical point for . Any central configuration for , homogeneous of degree , satisfies the central configuration equation
The previous lemma motivates the choice to gather in only one definition total collision motions and parabolic ones: their asymptotic behaviour is really similar when approaches to . In particular let us pause on assertion (iii) of Lemma 2.3: it states that the angular part of these trajectories tends to a set , but it does not necessarily admit a limit. The absence (or presence) of spin in the angular part of an asymptotic motion is indeed still an open problem in Celestial Mechanics, known as the infinite spin problem (cf. [SH81, Saa84]). Our studies will focus on a.s. admitting a limiting central configuration: we are interested in collision/parabolic trajectories whose angular part tends exactly to an assigned central configuration .
Given a central configuration for the potential and an a.s. , we say that is an -asymptotic solution (-a.s. for short) if
Remark 2.5 (on -homothetic motions).
Given a central configurations for , , the set of -a.s. is not empty. It indeed contains the -homothetic motions: these self-similar trajectories has the form where solves the 1-dimensional Kepler problem. If the constant energy, , is negative these solutions are bounded, ending in a collision as (actually these motions can be extended to ). When they are unbounded motions starting or ending with a total collision. When we find an -homothetic parabolic motion, : the radial part of these trajectories can be computed explicitely and there holds
where the constant and the function have been introduced in Lemma 2.3. When an -homothetic parabolic motion borns in a collision and becomes unbounded as ; when the trajectory ends in a total collision and can be extended to the maximal definition interval , becoming unbonded as . Hence Lemma 2.3 states that -asymptotic solutions are indeed, as , perturbations of -homothetic parabolic motions.
Whenever is a minimal configuration, existence of -asymptotic arcs of minimal parabolic trajectories departing from arbitrary configurations has been proved in [MV09]. The set of -a.s. also contains this kind of trajectories.
2.2. The variational framework
Collision trajectories. Let us now take ; it is indeed well known that classical solutions to Newton’s equations (2.4) (i.e. trajectories in ) can be found as critical points of the Lagrangian action functional associated to the Lagrangian (introduced in Eq. (2.5))
in the smooth Hilbert manifold of all -paths in with some boundary conditions (fixed end points or periodic). Let us then introduce
with scalar product pointwise induced by the mass scalar product. Since we are interested in solutions satisfying some boundary conditions, for any , we consider the closed linear submanifold of
Given , we will denote by the Hilbert space of all -vector fields along :
It is well-known that the tangent space at to can be identified with .
When , then the tangent space at to is the subspace of defined by
As before, there exists an identification between with . This space admits as a dense subset the space of smooth functions .
In order to characterize asymptotic solutions as critical points of , we need to work in the general setting of non-smooth critical point theory, for this reason we recall the following Definition.
Let be a Hilbert space, a continuous functional on and let be a dense subspace of . If
the directional derivative of exists for all in all direction (i.e. exists for all and ), we say that is Gâteaux -differentiable and we will denote the Gâteaux -differential with . If is Gâteaux -differentiable, a point is said to be a Gâteaux -critical point if
there exists a bounded linear functional such that
for all , we say that is Fréchet -differentiable and we denote the -Fréchet differential at , , by . If is Fréchet -differentiable, a point is said to be a -critical point if
In the next result we characterise weak solutions of Equation (2.4) (with fixed ends) as Gâteaux -critical points (where = ) of the Lagrangian action functional.
(i) As a direct consequence of item (i) in Lemma 2.3, the following pointwise asymptotic behaviour in the neighbourhood of the final instant holds:
By using item (ii) in Lemma 2.3, we obtain .
Combining the first estimate with the continuity of on , we
deduce that . Furthermore, by taking into account that the
image of the function defined by
is contained in the interval , it
follows that, the integral along any time- total collision solutions
converges. This proves that is finite.
Equation (2.9) is the weak formulation of Equation (2.4) for the solution .
(ii) Let , we compute
Since , , whenever has a small absolute value, then still belong to the open set and, by dominated convergence
Hence is a bounded linear functional when regarded on the densely immersed subspace of , is Gâteaux--differentiable and is its Gâteaux differential at . In particular for any means that is a Gâteaux -critical point in the sense specified in Definition 2.7. We conclude by standard elliptic regularity arguments. ∎
Parabolic motions. First of all we remark that, even if parabolic motions do not interact with the singular set (cf. Remark 2.1), they are unbounded motions, they miss the integrability properties at infinity, hence they don’t lie in the set but just in . Although the functional is infinite along a parabolic motion , it can be Gâteaux -differentiable at : indeed the difference in Equation (2.10) turns out to be finite for every . In a more abstract way, it is possible to define the functional on the Hilbert manifold . Hence we can argue as in Lemma 2.8 to deduce the next result.
2.3. A variational version of McGehee regularisation
which is defined on , where
and corresponds to the singular set in the new variables, that is
Let us now consider an -a.s. for the dynamical system (2.4) and let and be its radial and angular components; inspired by the behaviour of when , (cf. Lemma 2.3, (i)), we introduce the new time-variable
The function is strictly monotone increasing and ; furthermore since as
, when (i.e. is a time- total collision motion),
when (i.e. is parabolic motion),
then in both cases the variable varies on the half line, indeed
We now define the variable as
and, denoting by the derivative with respect to we have
Using Lemma 2.3 (i) and (iii), we deduce that in the new time variable , both classes of -a.s., collision (resp. parabolic), share the same asymptotic behavior: the variable has an exponential decay (resp. growth) while the speed of the angular part tends to 0. More precisely
where the constant is
Remark 2.10 (on -homothetic-parabolic motions).
When we consider an -homothetic parabolic motion (cf. Remark 2.5) and its corresponding radial variable , then the following quantity remains constant along the trajectory
so that has an exponential behavior: decays if is finite, increases when . Let us recall that in both cases the new time variable varies on .
In these new coordinates , the Lagrangian transforms into and reads as
We note that, when we have fixed a finite (in the case of total collision motions), the time scaling given in Eq. (2.13) imposes an infinite dimensional constraint given by
Dealing with the Lagrangian one have to take into account also the pointwise holonomic constraint . In order to prevent the presence of this second constraint, it is convenient to define
that actually satisfies the relations
The Lagrangian function in the variable becomes
Let us now consider .
By using the generalization in Calculus of Variations of the classical Lagrangian multipliers method (cf. [GF63] for further details) with respect to integral constraints (for us constraint (2.17)) and by virtue of the exponential decay of the variable stated in Equation (2.14), we can conclude that if is a time- total collision solution for (2.4) then is a (weak) extremal for the functional
In order to compute the Lagrange multiplier we write the Euler-Lagrange equations associated to Equation (2.19)
and the expression of the first integral of the energy (in the variable )
We can now easily deduce that
Finally we conclude that, given a time- total collision solution with energy then the corresponding is a (weak) extremal for
Let us now consider .
In this case the function corresponding to the parabolic motion does not belong anymore to . Here we do not have to impose the integral constraints (2.17), since follows from the exponential growth of . Nevertheless arguing precisely as in Lemma 2.9, it turns out that is a weak -critical point for
although such functional is not finite at .
From now on, with a slight abuse of notation, both when is finite or , we will say that the trajectory is an asymptotic solution whenever it corresponds (through the variable changes given in Equations (2.11) and (2.12)) to an a.s. . The discussion carried out in this section ensures that an -a.s. is a Gateaux -critical point for
In the special case , that is when we consider an homothetic motion (recall Remark 2.5), then .
2.4. Second Variation of the McGehee functional
The aim of this subsection is to compute the second variation and the associated quadratic form for the functional defined in (2.24) along an -a.s.. The following computation lemma will play an import role.
Let be a central configuration for and be an homothetic motion (with a total collision or parabolic). Then the following relation holds
where , .
By a direct calculation, the -directional derivative of the function defined in (2.25) is given by
By a similar calculation we obtain
Thus, since is homogeneous of degree , its first and second derivatives are respectively of degree and ; choosing we infer
Being a central configuration it holds (cf. Definition 2.2) and by the previous computation we immediately obtain
The thesis follows by taking into account that the last member in the previous formula is nothing but (for the details of this computation cf. [BS08, Formula (8), Section 2]). ∎
Let be an -a.s. with energy (possibly 0). Then the quadratic form associated to the second variation of at is