Abstract
This paper deals with a class of perturbation planar Keplerian Hamiltonian systems, by exploiting the nondegeneracy properties of the circular solutions of the planar Keplerian Hamiltonian systems, and by applying the implicit function theorem, we show that noncollision periodic solutions of such perturbed system bifurcate from the manifold of circular solutions for the Keplerian Hamiltonian system.
1. Introduction
In this paper, we study a class of Hamiltonian systems obtained as perturbation of Keplerian Hamiltonian . More precisely, we consider the Hamiltonians of the following form:where is a perturbation parameter, is a skew-symmetric matrix , and is even in . The corresponding Hamiltonian system is the following:
For , equation (2) becomes
Our aim is to seek noncollision periodic solutions of (2), and we wish to connect them to the circular solutions of the unperturbed system (3). These types of perturbation systems have been the focus of interest by a number of authors and the references therein [1–5]. We mention in particular the works of Poincaré, regarding the three-body problem (these orbits were called “first-view sort solutions”) [6], and the studies by Ambrosetti et al. [7] and Celletti et al. [8] which showed the existence of a skew-T/2 periodic solution of the following problem:
The essential difficulty in studying this problem is the free action of the -group acting on equation (2) (if is a solution of (2), then is also a solution of (2), for all ). To overcome this difficulty, we seek solutions of equation (2) near the circular orbits of the Keplerian system (3). These circular orbits are the more stable solutions and by exploiting their nondegeneracy property, we neutralize the free action of the -group ([9, 10]). The degenerate solutions of the Keplerian problem are the least stable solutions (KAM theory [11]); we cannot dominate the invariance of the problem under the action of the group in the neighborhood of these solutions.
The proofs rely on the implicit function theorem and the nondegeneracy of the circular orbits for (3) in the space of the skew- periodic functions. Let and satisfy the following:
For all, .
We consider the following perturbed system of ordinary differential equations:where is a skew-symmetric matrix , , and is a fixed period.
The unperturbed system corresponding to (2) is the following:
By a noncollision orbit of (6), we mean a solution of (6) such that for all . We will say that is skew- periodic if for every . The following result holds.
Theorem 1 (see [7, 8, 12]). There exists such that , the perturbed system (6), has at least one noncollision symmetric (skew- periodic) orbit near the circular orbit of (6).
1.1. Bifurcation in the Nondegenerate Case
Actually, we wish to relate the skew- periodic solutions of (6) to the -periodic circular solutions of (7). Letand
We consider the open subset of defined by
On , we define a functional by setting
Lemma 1. The functions belongs to and for all , we have
Proof. The proof is left to the reader.
We now show the following lemma.
Lemma 2. The following statements are equivalent:(i)(ii)(6)
Proof. We prove the lemma in two steps:
Step 1. . The equation, , means for every ,Therefore,LetwhereandHence,It is clear thatTherefore,This implies and .
Step 2. is a noncollision skew periodic solution. Since , it is skew- periodic.
Denote by the orthogonal subspace in to andFrom (13), it follows thatUsing , one finds thatHence,and thus, ,Then,By substituting in , we obtainMoreover,This completes the proof.
1.2. Finding Critical Points for
Let , andwhereis a manifold of critical points for (that is, , for every ). Using Lemma 2, we find thatis a manifold of circular solutions for the unperturbed system,
We wish to investigate the situation around by applying the inverse function theorem. For this, we need to know more about the derivative of and .
Lemma 3. is the Fredholm operator of index zero, .
Proof. Letting , for all and in , we haveThen,Hence,where is a linear operator from into and defined byIt is easy to verify that is a compact operator. is the Fredholm operator of index zero.
Thus, the proof is complete.
The preceding lemma impliesWe deduce that, cannot be an onto function. The ultimate reason for this lies in the fact that the function is invariant by the action which sends into and this induces degeneracy in the derivatives. We can estimate by relating to the linearized equation (32) around . This is done as in the following lemma.
Lemma 4. Let . The following two conditions are equivalent:(a).(b) is a skew periodic solution ofwhere
Proof. Let , such that . By reasoning as in the proof of Lemma 2, we obtainandSetBy substituting (42) in (41), we obtain,This completes the proof.
More precisely, we have the following lemma.
Lemma 5.
Proof. According to the above lemma, the dimension of is equal to that of the set of the solutions for equation (38). Let be the solution of (38), and by using the fact that , equation (38) reduces toSetsince , then . By substituting, we findfrom which we deduce the following equation for , where ,It is easy to see that conditions define the tangent manifold to . According to the definition of a nondegenerate critical manifold, this means that is a nondegenerate critical:Therefore,This completes the proof.
We denoteThis is a fact that gives us a considerable simplification.
Lemma 6. For close to , the following statements are equivalent:
Proof. is spanned by . The equation ,Multiplying both sides by and integrating, we get and if we integrate the first and second terms on the left by parts, we get zero. In the last term, we recognize the time derivative of , which integrates away to zero. Finally, we get,If is close to , in , the integral is strictly positive, and, hence, must be zero.
We now state our main result.
Theorem 2. Let . Ifthen there are positive numbers , a neighborhood of the path in , and a map,such thatand for any and such the curve,is a skew- periodic solution of equation (6). Conversely, whenever is a skew periodic solution of (6), with , , and remaining in for all , then some can be found such that
Proof. Set . It is known that is a Fredholm map of index zero. Split into ; then, is an isomorphism of onto itself. By the implicit function theorem, the equationdetermines in terms of the remaining variables. These are and the component of in . By this, we meanBy Lemma 6, this means that the equation can be solved in as follows in neighborhood of ,We now replace by more convenient variables.
For any , set . If , we also have . We, thus, have an -action which leaves our equations invariants, and we wish to find a coordinate system adapted to this group-invariance. For near , the complex number,has a well-defined argument , called the phase of with respect to . In the subsequent paragraphs, we will verify thatI now claim that we can use as a local coordinate system for near . Computing the Jacobian at this point givesSinceso the Jacobian does not vanish. The equation now becomesUsing Lemma 4 to translate in terms of and , we get the desired result. is at least a map from into the space ; it will then have a Taylor expansion.
2. Conclusion
We can conclude that the class of planar perturbations are Keplerian Hamiltonian systems, as we have shown that the noncollision periodic solutions of this perturbed system radiate from the complex of circular solutions of the Keplerian Hamiltonian system. We have studied a class of Hamiltonian systems obtained as perturbation in the Keplerian Hamiltonian
Our goal was to search for noncollision periodic solutions of (2), and we wish to relate them to circular solutions for the nonperturbed system (3). These kinds of systems were restless focus of a number of authors and the references therein [1–5]. We mention in particular, the work of Poincaré, on the three-body problem (these orbitals are called “first-view sort solutions”).
In an effort to organize another piece of work into a paper, we determine the coefficients of the Taylor expansion to the second order of the noncollision periodic solutions for the perturbed planar Keplerian Hamiltonian system, which is connected to Kepler Hamiltonian systems by a perturbation parameter. This Taylor expansion is made with respect to a perturbation term and the period of the solution.
Data Availability
The authors claim that this work is a theoretical result, and there are no available data source.
Conflicts of Interest
The authors declare that they have no conflicts of interest.