Bifurcations of quasi-periodic solutions from relative equilibria in the Lennard–Jones 2-body problem

We propose the general method of proving the bifurcation of new solutions from relative equilibria in N-body problems. The method is based on a symmetric version of Lyapunov center theorem. It is applied to study the Lennard–Jones 2-body problem, where we have proved the existence of new periodic or quasi-periodic solutions.


Introduction
N -body problems have been widely studied in celestial mechanics since Kepler and Newton. We study the motion of N particles moving under the action of mutual potential forces. Denote by q 1 , . . . , q N ∈ R d the positions of the particles with the masses m 1 , . . . , m N , respectively, by U i j (q i , q j ) : R d × R d → R the potential between particles q i and q j , and put U (q 1 , . . . , q N ) = 1≤i< j≤N U i j (q i , q j ). Then, denoting q = (q 1 , . . . , q N ), the Newtonian equation of motion can be written as the system: A typical question for the N-body problem is the existence of relative equilibria, i.e., solutions where the system of particles behaves like a rigid body-the mutual distances of the particles remain constant during the motion. In other words, the system of particles moves under the action of rotation group (the center of mass of the system is fixed). Therefore, relative The author was partially supported by the National Science Centre, Poland (Grant No. 2017/25 Meyer et al. (2009).
In this paper, we show how to utilize the existence of relative equilibria to the demonstration of the existence of quasi-periodic solutions that are not relative equilibria. This method can be applied for a variety of problems like gravitational N -body problem, Lennard-Jones N -body problem or N -vortex problem. We apply this method to study the 2-body problem with Lennard-Jones potential.
Lennard-Jones potential describing the potential energy of a pair of molecules is given by where r is a distance of the molecules and the parameters ε and σ can be found experimentally for the specific model. The first term describes the Pauli repulsion (the molecules repel each other at a close distance), the second one-London forces (attraction at a moderate distance). Lennard-Jones potential is used frequently in a chemistry modeling. By the change of units and scale, we can assume ε = σ = 1. Then, the potential for the N -body problem is given by (1.1) Lennard-Jones N -body problem has been studied in many ways, including more general class of potential. In Llibre and Long (2015), the authors have studied the circular periodic solutions and antiperiodic solutions of the generalized problem. The classical case studied here corresponds to the case γ = 0 in Llibre and Long (2015), where the authors have proved the existence of a circle of equilibria.
In Liu et al. (2018), the authors have used variational methods to study the structure of solutions of the generalized Lennard-Jones system. They have proved, among others, the existence of periodic non-circular solution with any period greater than the number τ # 1 , where τ # 1 = 2·11 7 in the classical case. They have also studied the existence of periodic solutions in a rotating frame, giving the period of solution as an integral, see Proposition 5.2 in Liu et al. (2018).
Lennard-Jones (N + 2)-and (N + 3)-body problems with the generalized potential on R 3 have been studied in Liu (2020) where the existence of circular and non-circular homographic solutions has been proven.
The 2-body problem with the potential (1.1) has been studied in Corbera et al. (2004) for the existence of equilibria and relative equilibria. In Pérez-Chavela et al. (2018), the existence of periodic solutions bifurcating from the stationary ones has been proven. Moreover, the detailed study of generalized 2-body Lennard-Jones potential was done in Bȃrbosu et al. (2011).
In this article, we study Lennard-Jones 2-body problem and we prove the existence of connected families of periodic or quasi-periodic solutions bifurcating from the relative equilibria with the moment of inertia in the given interval. The article is organized as follows. In Sect. 2, we reformulate the problem into Hamiltonian equation in rotating frame and we formulate the main tool of this work, Theorem 1. Section 3 is devoted to study the problem in a number of cases depending on the moment of inertia of the relative equilibrium.

The theoretical background
Consider a planar N -body problem with all masses equal to 1 described by the equation the configuration space and U : → R is of the class C 2 . We treat q and gradient as column vectors. Denote by K (θ ) the matrix of planar rotation, i.e., K (θ ) = cos θ − sin θ sin θ cos θ ∈ SO (2) and by R(θ ) = K (θ ) ⊕ . . . ⊕ K (θ ) -the 2N × 2N block matrix with 2 × 2 matrices K (θ ) on its diagonal. We recall that the rotation group SO(2) acts on R 2N as SO (2) Suppose that for some q 0 ∈ , the mapq(t) = R(ωt)q 0 , q 0 ∈ is a solution of the equation (2.1). A solution of this form is called a relative equilibrium generated by q 0 , and the point q 0 is called a central configuration. A relative equilibriumq(t) can be viewed as an equilibrium point in a rotating frame with rotation rate ω. One can show thatq(t) is a solution of (2.1) if and only if ∇U (q 0 ) = ω 2 q 0 (see also Corbera et al. (2004)).
We are going to put the system into a rotating frame. Firstly, we translate the Newtonian system (2.1) to the Hamiltonian one. Let's introduce the variable p = ( p 1 , . . . , p N ) =q. The problem can be written in the form for H (q, p) = 1 2 p 2 + U (q). Now, we introduce the new variables in the frame rotating with velocity ω: In these variables, the equation of motion can be written in the general form where Hamiltonian function H is given by H(Q, P) = 1 2 P 2 + ωQ T J N P + U (Q) and To prove the existence of family of solutions of the problem (2.3) emanating from the stationary solution (Q 0 , P 0 ), we can apply the following theorem. The theorem is a generalization of the famous Lyapunov center theorem, see Lyapunov (1895); Moser (1976); Weinstein (1973). Since we are going to apply it to 2 · (2N )-dimensional system, it is given in this case.
Theorem 1 ( Strzelecki (2020)) Assume that a compact Lie group acts unitary on R 4N . Under the following assumptions: there exists a connected family of non-stationary periodic solutions of the systemż(t) = J ∇H(z(t)) emanating from the stationary solution z 0 (i.e., with amplitude tending to 0) such that minimal periods of solutions in a small neighborhood of z 0 are close to 2π/β j 0 .

Remark 1
The assumption (A6) is needed to be studied if the period of the new solution has to be minimal. If we are interested in the existence of solutions only, we do not have to verify it.
, and as a consequence, the Brouwer degree , 0 equals ±1 for sufficiently small . Therefore, in this case the assumption (A5) is satisfied.
for λ → ∞ is a singular matrix; however, for large values of λ we can estimate implies the existence of a purely imaginary eigenvalues of J ∇ 2 H(z 0 ) (the assumption (A4)). To summarize, when we are able to verify the stronger assumption then the assumptions (A4) and (A7) are satisfied. However, then we do not know the level λ where the assumption (A7) holds true, and therefore, we cannot give the minimal period of new solutions. The assumption (A6) has not to be verified.
See Strzelecki (2020) for more details.
Theorem 1 provides the existence of periodic solutions of the problem in the rotating frame whose trajectories lie arbitrarily close to the stationary solution (Q 0 , P 0 ). In the original frame these solutions, we call quasi-periodic as a composition of two periodic motions with not necessarily resonant frequencies. Their trajectories lie arbitrarily close to the trajectory of the relative equilibrium R(ωt)q 0 .
One can ask whether we can apply classical Lyapunov center theorem to study the problem. Note that in the rotating frame, the Hamiltonian function H still has symmetries of the group SO(2). It implies that critical points of H (i.e., stationary solutions of the problem in the rotating frame) are not isolated and therefore they are degenerate-Lyapunov theorem cannot be applied. It is possible to reduce the system to the space of orbits of the SO(2)-action (reduced Hamiltonian system). However, if dim ker ∇ 2 H(z 0 ) ≥ 2, the relative equilibrium is degenerate stationary solution of the reduced system and Lyapunov theorem is not applicable, while Theorem 1 could be applied.
Even when we research non-degenerate case, an application of Theorem 1 is direct and does not require the study of the properties of the orbit space which is a manifold. Note that in the case of general unitary action of the compact Lie group on the problem (as is stated in Theorem 1), the orbit space does not have to have a structure of manifold, while Theorem 1 is still applicable.

The results for the Lennard-Jones 2-body problem
In the paper Corbera et al. (2004) Corbera, Llibre and Pérez-Chavela have described equilibria and central configurations for Lennard-Jones 2-and 3-body problems with equal masses. We focus on 2-body problem where for each value of the moment of inertia I ∈ ( 1 4 , ∞) there exists a relative equilibrium generated by the central configurations in the set CC = {(q 1 , q 2 ) ∈ R 2 × R 2 : q 2 = −q 1 , |q 1 − q 2 | = 2 √ I }-a closed trajectory of this relative equilibrium. The period of this relative equilibrium equals T = 2π/ω I , where ω I = √ 384I 3 −6 64 √ I 7 . Moreover, the period function T (I ) has a minimum at the point I 0 = 1 4 7 4 1/3 , see Corbera et al. (2004).
Applying the method described in the previous section and Theorem 1, we are going to prove the bifurcation of periodic solutions from this relative equilibrium for various values of the moment of inertia.
The symbolic computations in this section were performed by using Maple. One can find the Maple file under the following link: https://mat.umk.pl/~danio/LJ2BP.html.
Fix I > 1 4 . We consider the problem in the frame rotating with the angular velocity ω I as described in the previous section. In this frame, the Hamiltonian of the motion has the form Since the Lennard-Jones potential is SO(2)-invariant, the Hamiltonian H is invariant under the diagonal action of SO(2). Note that for any point z = 0 ∈ R 8 , the isotropy group SO(2) z is trivial.

The case a < a 0
By the results above, in this case the stronger assumption (A7.1) is satisfied, and therefore, we do not have to verify the assumptions (A4), (A6) and (A7), but we will obtain a weaker theorem-without an information about the minimal period, see Remark 3. Moreover, in this case dim ker ∇ 2 H(z 0 ) = 1 = dim SO(2), i.e., the assumption (A5) is satisfied, see Remark 2.
To summarize, we can formulate the following result as a consequence of Theorem 1.
When we are interested in the minimal period of solutions in this case, we might study the assumption (A7) in its general formulation as it is done in the next cases where the assumption (A7.1) is not satisfied. However, in the next sections the proofs are much complicated. Therefore, we utilized the condition (A7.1) in the case a < a 0 .

The case a = a 0
The case a = a 0 = √ I 0 is a critical one, where the assumption (A5) has to be studied in its general version and it is more complicated; therefore, the problem is far from being solved. However, in this case it is not hard to verify the assumption (A7).
Note that −64a 6 + 3 and 11 − 320a 6 are positive on the interval (a 0 , γ ). Therefore, the polynomial v i (a) has the same zeros and the same sign on this interval as w i (a) function, for i = 0, . . . , 16.
We are interested in the signs of the values of v i (a); therefore, using Sturm method we verify how many distinct roots in the interval (a 0 , γ ) they have.
Denote the root of v i (a) in the given interval as r i (if it exists). We can perform the Sturm method on smaller intervals to order the roots r i : r 11 < r 9 < r 14 < r 7 < r 12 < r 10 < r 1 < r 3 < r 8 < r 2 < r 15 .
In the next table, we test the signs of the polynomials on the intervals determined by their roots. For example, the symbol + → − denotes that a polynomial with a root ζ is positive for a ∈ (a 0 , ζ ) and negative for (ζ, δ). Now, we are well prepared to study the sequence of signs of (v 0 (a), v 1 (a), . . . , v 15 (a), v 16 (a)) on the subintervals of (a 0 , γ ): Note that in the endpoint r i the sign of v i equals 0, but 0 has no effect on the number of changes of signs, therefore, this number of changes in r i is the same as on the interval (r j , r i ); therefore, we study right-closed intervals.
To summarize, we have proved that for any a ∈ (a 0 , γ ), there are 6 changes of signs in the sequence of coefficients of the characteristic polynomial of the matrix K λ * . Hence, there are at most 6 positive roots of the characteristic polynomial. Moreover, the matrix K λ * is symmetric and non-singular. Therefore, m − (K λ * ) ≥ 10 and the assumption (A7) holds true. In fact, m − (K λ * ) = 10, by the study of characteristic polynomial of the negative variable.
To summarize, in this case the assumptions of Theorem 1 with β j 0 = β 2 are satisfied and we can formulate the following result.
We have verified that for any a ∈ (γ , δ), there are 6 changes of signs in the sequence of coefficients of the characteristic polynomial of K λ * . Therefore, by Descartes' rule of signs there are at most 6 positive eigenvalues of this matrix, and at least 10 negative roots. The assumption (A7) of Theorem 1 is satisfied with β j 0 = β 2 < β 1 . It remains to check the assumption (A6).
To summarize, we can formulate the following result.

The case ı ≤ a
In this unbounded case, there is no way to find a general behavior of the Morse index m − (K λ ). It could be possible to study small intervals of a or specific values of a by Descartes' rule of signs. For example, for δ ≤ a < 1 the Morse index equals 8 and the assumption (A7) is not satisfied. But we are not able to formulate any general result.
The case a = 6 7 256 , where it is hard to verify the assumption (A5), remains to be solved. The method proposed in this paper could be used to a variety of N -body problems where the relative equilibria exist.
Code availability The files of computations in Maple are available at: https://mat.umk.pl/~danio/LJ2BP.html Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.