International Journal of Differential Equations
Volume 2023, Issue 1 3575701
Research Article
Open Access

Perturbed Keplerian Hamiltonian Systems

Riadh Chteoui, 

Corresponding Author

Riadh Chteoui

Laboratory of Algebra, Number Theory and Nonlinear Analysis, Department of Mathematics, Faculty of Sciences, University of Monastir, Monastir 5019, Tunisia um.rnu.tn

Department of Mathematics, Faculty of Sciences, University of Tabuk, Saudi Arabia ut.edu.sa

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First published: 09 January 2023
https://doi.org/10.1155/2023/3575701
Academic Editor: Sining Zheng

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 K(p, q) = 1/2‖p‖2 − ‖q‖−1. More precisely, we consider the Hamiltonians of the following form:
()
where p, q ∈ ℝ2, ε > 0 is a perturbation parameter, A is a skew-symmetric matrix A⋆ = −A, and V is even in q. The corresponding Hamiltonian system is the following:
()
For ε = 0, 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 S1-group acting on equation (2) (if q is a solution of (2), then qs = q(.+s) is also a solution of (2), for all s ∈ S1). 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 S1-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 S1 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-T/2 periodic functions. Let Ω = ℝ2/{0} and V : ℝ × Ω⟶ℝ satisfy the following:
()

For all, q ∈ Ω, ε ∈ ℝ.

We consider the following perturbed system of ordinary differential equations:
()
where A is a skew-symmetric matrix (A⋆ = −A), ε ∈ ℝ, and T > 0 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 q(t) = 0 for all t. We will say that q ∈ H1(ℝ/Tℤ, ℝ2) is skew-T/2 periodic if q(t + T/2) = −q(t) for every t. The following result holds.

Theorem 1 (see [7], [8], [12].)There exists ε0 > 0 such that ∀|ε| < ε0, the perturbed system (6), has at least one noncollision symmetric (skew-T/2 periodic) orbit near the circular orbit of (6).

1.1. Bifurcation in the Nondegenerate Case

Actually, we wish to relate the skew-T/2 periodic solutions of (6) to the T-periodic circular solutions of (7). Let
()
and
()
We consider the open subset of E1 defined by
()
On , we define a functional Φ by setting
()

Lemma 1. The functions Φ belongs to and for all h, k ∈ E1, 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)

    q(t) = Tu(tT−1)  is a noncollision skew − T/2  periodic solution of (6)

Proof. We prove the lemma in two steps:

Step 1. q ∈ C2(ST, ℝ2). The equation, , means for every h ∈ E1,

()

Therefore,

()

Let

()
where
()
and
()

Hence,

()

It is clear that

()

Therefore,

()

This implies q ∈ H2(sT, ℝ2) and q ∈ C2(ST, ℝ2).

Step 2. q is a noncollision skew −1/2 periodic solution. Since u ∈ Λ1, it is skew-T/2 periodic.

Denote by the orthogonal subspace in to E1 and

()

From (13), it follows that

()

Using (V2), one finds that

()

Hence,

()
and thus, φ = 0,
()

Then,

()

By substituting in ϕ = 0, we obtain

()

Moreover,

()

This completes the proof.

1.2. Finding Critical Points for ϕ(·, ε, T)

Let T0 > 0, and
()
where
()
is a manifold of critical points for ϕ(·, 0, T0) (that is, , for every ). Using Lemma 2, we find that
()
is a manifold of circular solutions for the unperturbed system,
()

We wish to investigate the situation around (q0, 0, T0), q0 ∈ ℤ0 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 h and k in E1, we have

()

Then,

()

Hence,

()
where ℑ is a linear operator from E1 into E1 and defined by
()

It is easy to verify that ℑ is a compact operator. is the Fredholm operator of index zero.

Thus, the proof is complete.

The preceding lemma implies

()

We deduce that, cannot be an onto function. The ultimate reason for this lies in the fact that the function ϕ is invariant by the S1− action which sends u(t) into u(t + θ) ∈ S1 and this induces degeneracy in the derivatives. We can estimate by relating to the linearized equation (32) around u. This is done as in the following lemma.

Lemma 4. Let . The following two conditions are equivalent:

  • (a)

    .

  • (b)

    h(t) = T0k(tT−1) is a skew −T0/2 periodic solution of

()
where
()

Proof. Let k ∈ E1, such that . By reasoning as in the proof of Lemma 2, we obtain

()
and
()

Set

()

By 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 h be the solution of (38), and by using the fact that ‖q0‖ = r, equation (38) reduces to

()

Set

()
since k ∈ E1, then c2k = 0, ∀k ∈ ℤ. By substituting, we find
()
from which we deduce the following equation for ck, where k ∈ ℤ,
()

It is easy to see that conditions (a) − (c) define the tangent manifold to . According to the definition of a nondegenerate critical manifold, this means that ℤ0 is a nondegenerate critical:

()

Therefore,

()

This completes the proof.

We denote

()

This is a fact that gives us a considerable simplification.

Lemma 6. For (u, ε, T) close to (u0, 0, T0), 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 V(ε, Tu), which integrates away to zero. Finally, we get,

()

If u is close to u0, in E1, the integral is strictly positive, and, hence, α must be zero.

We now state our main result.

Theorem 2. Let q0 ∈ ℤ0. If

()
then there are positive numbers r0, ε, a neighborhood V of the path q0 in ℝ2, and a ℂ2 map,
()
such that
()
and for any ε ∈ [−ε0, ε0] and T such |T − T0‖ < r0 the curve,
()
is a skew- T/2 periodic solution of equation (6). Conversely, whenever q is a skew −T/2 periodic solution of (6), with ε ∈ [−ε, ε], |T − T0| < r0, and q(t) remaining in V for all t, then some θ ∈ ℝ can be found such that
()

Proof. Set . It is known that ϕuu(u0, 0, T0) is a Fredholm map of index zero. Split E1 into ; then, is an isomorphism of onto itself. By the implicit function theorem, the equation

()
determines in terms of the remaining variables. These are (ε, T) and the component α of u in . By this, we mean
()

By Lemma 6, this means that the equation can be solved in E1 as follows in neighborhood of (u0, 0, T0),

()

We now replace α by more convenient variables.

For any s ∈ S1, set us(t) = u(t + s). If , we also have . We, thus, have an S1-action which leaves our equations invariants, and we wish to find a coordinate system adapted to this group-invariance. For u near u0 ∈ E1, the complex number,

()
has a well-defined argument θ(u), called the phase of u with respect to u0. In the subsequent paragraphs, we will verify that
()

I now claim that we can use (θ, ε, T) as a local coordinate system for near (uo, 0, To). Computing the Jacobian at this point gives

()

Since

()
so the Jacobian does not vanish. The equation now becomes
()

Using Lemma 4 to translate in terms of q and q0, we get the desired result. v is at least a C2 map from into the space C2(S1, ℝ2); 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 up 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 ε and the period T of the solution.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Data Availability

The authors claim that this work is a theoretical result, and there are no available data source.

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