Periodic Solutions for the Generalized Anisotropic Lennard-Jones Hamiltonian

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Introduction and Statement of the Main Results
Frequently mathematical simple models are used in molecular dynamics and computational chemistry to describe the interaction between a pair of molecules or atoms, see for instance [5,9].
One of the most used empirical potentials in molecular dynamics is the Lennard-Jones potential, see [8], which models the interaction between two neutral atoms or molecules, under two different forces in the limit of small and large separation. These forces are: a repelling force at short distances (coming from overlapping electron orbitals, related to the Pauli's exclusion principle), and an attractive force at long distances (coming from the van der Waals force, or the dispersion force).
The Lennard-Jones potential where a/4 is the depth of the potential energy, σ is the finite distance at which the interparticle potential vanishes, ||r 1 − r 2 || is the distance between the two particles localized at the positions r 1 and r 2 in R n . The values of these parameters are chosen in order to reproduce experimental data, or deduced from accurate quantum chemistry computations; [3] is a good reference for these considerations.
We rescale the unit of length and the unit of mass in such a way that the constant σ and a become 1, then the Lennard-Jones potencial becomes When one of the atoms or molecules is at the origin of coordinates and the position of the other atom or molecule is x = (x 1 , . . . , x n ), then the Lennard-Jones Hamiltonian writes where |x| = n k=1 x 2 k .
In the coordinates (x, p x ) the generalized Lennard-Jones potential is central, and consequently it is integrable with the independent first integrals given by the angular momentum C = x ∧ p x , where ∧ is the exterior product of the vectors x and p x . The norm of the angular momentum C on a solution of the Hamiltonian system (2) is denoted by c, and of course it is also a first integral.
For stating our first result on the circular periodic solutions of the generalized Lennard-Jones Hamiltonian system we need some notation. We define The next two proposition characterize the circular periodic solutions of the generalized Lennard-Jones Hamiltonian system. (a) For c = −γ the Hamiltonian system (2) has only one retrograde circular periodic solution centered at the origin of coordinates of radius ρ 2 . The period of this orbit is 2πρ 2 2 /γ . (b) For each c ∈ (−γ, 0) the Hamiltonian system (2) has exactly two retrograde circular periodic solutions centered at the origin of coordinates, one with radius this period tends to 2πρ 2 2 /γ when r 1 (c) ρ 2 and tends to ∞ when r 1 (c) ρ 1 ; and the other periodic solution with radius r 2 (c) in the interval (ρ 2 , ∞) of period this period tends to 2πρ 2 2 /γ when r 2 (c) ρ 2 and tends to ∞ when r 2 (c) ∞.
(c) For c = 0 the Hamiltonian system (2) has a circle of equilibra centered at the origin of coordinates and of radius ρ 1 . (d) For each c ∈ (0, γ ) the Hamiltonian system (2) has exactly two direct circular periodic solutions centered at the origin of coordinates, one with radius r 1 (c) in the interval (ρ 1 , ρ 2 ), and the other with radius r 2 (c) in the interval (ρ 2 , ∞). The periods of these two orbits have the behavior described in statement (b). (e) For c = γ the Hamiltonian system (2) has only one direct circular periodic solution centered at the origin of coordinates of radius ρ 2 . The period of this orbit is the same than in statement (a). (a) For each c ∈ (−γ, 0) the Hamiltonian system (2) has exactly one retrograde circular periodic solution centered at the origin of coordinates with radius r 1 (c) in the interval (ρ 1 , +∞) of period this period tends to +∞ when r 1 (c) ∞ and when r 1 (c) ρ 1 . (b) For c = 0 the Hamiltonian system (2) has a circle of equilibra centered at the origin of coordinates and of radius ρ 1 . (c) For each c ∈ (0, γ ) the Hamiltonian system (2) has exactly one direct circular periodic solution centered at the origin of coordinates with radius r 1 (c) in the interval (ρ 1 , ∞). The period of this orbit has the behavior described in statement (a).
(b) For c = 0 the Hamiltonian system (2) has a circle of equilibra centered at the origin of coordinates and of radius ρ 1 . (c) For each c ∈ (0, ∞) the Hamiltonian system (2) has exactly one direct circular periodic solution centered at the origin of coordinates with radius r 1 (c) in the interval (ρ 1 , ∞). The period of this orbit has the behavior described in statement (a).
Our first main goal is to characterize which of the circular periodic solutions described in Proposition 1 can be continued to the generalized anisotropic Lennard-Jones Hamiltonian system defined by the Hamiltonian for a given integer m and a given ε such that 1 < m < n and |ε| is sufficiently small. Therefore the corresponding Hamiltonian system iṡ We define Our first main result characterize the circular periodic solutions of the generalized Lennard-Jones Hamiltonian system (2), given in Proposition 1, which can be continued into the generalized anisotropic Lennard-Jones Hamiltonian system (5) for small values of |ε|.
(a) If c ∈ (−γ, 0) and R is not an integer, then at every 2-dimensional plane P through the origin of coordinates the retrograde circular orbit of radius r 1 (c) and angular momentum c of the generalized Lennard-Jones Hamiltonian system (2) can be continued into the generalized anisotropic Lennard-Jones Hamiltonian system (5) for small values of |ε|. Theorem 4 is proved in Sect. 4. Following exactly the arguments of the proof of Theorem 4 we could be able to characterize the circular periodic solutions of the generalized Lennard-Jones Hamiltonian system (2), given in Propositions 2 and 3, which can be continued into the generalized anisotropic Lennard-Jones Hamiltonian system (5) for small values of |ε|, but we do not state them here in order to avoid a length article.
In what follows we shall characterize the periods of antiperiodic solutions of the Lennard-Jones Hamiltonian system on whether there exist such solutions.
For a given τ > 0 we study the τ -periodic solutions of the Lennard-Jones Hamiltonian system (2), which now we rewrite into the form where U ∈ C 1 (R 2n \{0}, R) is defined by where we suppose 0 < α < β, a > 0 and b ∈ R.
Firstly, for a given τ > 0 we plug the τ -periodic circular motion x = x(t) into the system (6) with the potential function U = U (x) of (7), and try to see which circular motion can become solution of (6).
Proposition 5 is proved in Sect. 5 below. Note that Proposition 5 provides results on the circular periodic solutions of the generalized Lennard-Jones Hamiltonian system (2) for a > 0 and b ∈ R, while Propositions 1-3 only for a > 0 and b > 0.
By a similar proof which is left to the readers, we have the following result for the higher dimensional case.
for any x ∈ R 2n \{0}, we can look for τ/2antiperiodic solutions of (6), i.e. those solutions x satisfying Note that circular solutions of (6) found by Proposition 5 are all τ/2-antiperiodic solutions. It is well known that τ/2-antiperiodic solutions of (6) are critical points of the functional defined on the space where S τ = R/(τ Z).
Motivated by the method of [11], we obtain our second main result below. Here this theorem characterizes the period τ > 0 for which (6) possesses no, at least one or more τ/2-antiperiodic solutions.
Theorem 7 is proved in Sect. 6 below. See [1,4,14] and the references therein for other works on the periodic orbits of the Lennard-Jones potential, or related with these periodic orbits.

Proofs of Propositions 1 and 2
Since the notion of angular momentum is defined in any dimension by using the exterior product in R n , one would guess that central force problems in any dimension are completely integrable, as it is well known for n = 3. This was proved explicitly in [7], by constructing n first integrals independent and involution: the energy and some combinations of the angular momentum components. It is shown that the motion of these central problems are always reduced to a 2-dimensional plane through the origin of coordinates.
The Lennard-Jones Hamiltonian (1), restricted to a 2-dimensional plane P through the origin of coordinates with initial position and momenta in P, in polar coordinates in P becomes where is the norm of the angular momentum restricted to P in polar coordinates. Therefore its corresponding Hamiltonian system writeṡ We fix the value of = c. On a circular periodic solution in P we haveṙ = R = 0. ThereforeṘ Hence the value of the angular momentum over the circular periodic solution of radius r in P is Proof of Proposition 1 The radius of a circular periodic orbit must satisfy r ≥ ρ 1 , see the graphic of the function c(r ) in Fig. 1. The maximum and the minimum of the function c(r ) takes place when r = ρ 2 and c(ρ 2 ) = ±γ . So the value of the angular momentum on the circular periodic solutions in P run in the interval [−γ, γ ], as it is stated in Proposition 1.
On a circular periodic solution we have that R = 0, and from (15) its angular velocity isθ = ω = c/r 1 (c) 2 . Therefore its period is Note that the period is the necessary time in order that θ increases or decreases by 2π . Now the statements of Proposition 1 follows easily from Fig. 1.  Since β > α > 0, the period T as a function of r 2 can be written as Clearly (β + 2)/2 > (β − α)/2, so T → +∞ as r 2 → ∞. Fig. 3 the proof is completely similar to the proofs of Propositions 1 and 2.

Basic Results on the Continuation of Periodic Solutions
We deal with autonomous differential systemṡ with ε 0 > 0 is an interval where the parameter ε takes values, and as usual the dot denotes the derivative with respect to the time t. We denote its general solution as Consider the T -periodic solution φ(t, x 0 ; 0). A continuation of this periodic solution is a pair of smooth functions, u(ε), τ (ε), defined for |ε| sufficiently small such that u(0) = x 0 , τ (0) = T and φ(t, u(ε); ε) is τ (ε)-periodic. One also says that the periodic solution can be continued. This means that the solution persists when the parameter ε varies, and the periodic solution does not change very much with the parameter.
The variational equation associated to the T -periodic solution φ(t, where M is a m × m matrix. Note that the matrix f x (x; ε) is the Jacobian matrix of the vector field f (x; ε).
The monodromy matrix associated to the T -periodic solution φ(t, x 0 ; ε) is the solution M(T, x 0 ; ε) of (19) satisfying that M(0, x 0 ; ε) is the identity matrix of R m . The eigenvalues of the monodromy matrix associated to the periodic solution φ(t, x 0 ; ε) are called the multipliers of the periodic orbit.
Let φ(t, x 0 ; ε) be a T -periodic orbit of the C 2 differential system (18). The eigenvector tangent to the periodic orbit has associated an eigenvalue equal to 1. So the periodic orbit has at least one multiplier equal to 1, for more details see for instance Proposition 1 in [10].
Let F : U → R be a locally non-constant function of class C 1 such that Then F is called a first integral of system (18), because F is constant on the solutions of this system. Here the dot · indicates the usual inner product of R m , and the gradient of F is defined as We say that k first integrals F j : U → R for j = 1, . . . , k are linearly independent if their gradients are independent in all the points of U except perhaps in a set of Lebesgue measure zero.
Let F j : U → R a first integral for j = 1, . . . , k with k < m. Assume that F 1 , . . . , F k are linearly independent in U . Let γ be a T -periodic orbit of the vector field f (x; ε) such that at every point x ∈ γ the vectors ∇ F 1 (x), . . . , ∇ F k (x) and f (x; ε) are linearly independent. Then 1 is a multiplier of the periodic orbit γ with multiplicity at least k + 1, see for instance Theorem 2 of [10]. If the differential system (18) has k independent first integrals, we say that a periodic solution φ(t, x 0 ; ε) is non-degenerate if 1 is an eigenvalue of the monodromy matrix M(T, x 0 ; ε) with multiplicity k + 1. The following result goes back to Poincaré, for a proof see for instance the proof of Proposition 9.1.1 of [12].
Proposition 8 A non-degenerate periodic solution of a differential system (18) with ε = 0 and k independent first integrals can be continued to differential systems (18) with |ε| sufficiently small.

Proof of Theorem 4
We shall work in a fixed 2-dimensional plane P through the origin of coordinates in the space of positions. In fact, from [7] we know that we can find 2n − 3 independent first integrals, such that 2n − 4 fix the motion on the plane P, and the additional first integral is the restriction of the Hamiltonian of the system to the invariant plane P. From section 3 it follows that a circular periodic solution in the plane P is nondegenerate if it has 2n − 2 multipliers equal to 1, and the remainder two are different from 1. Since we shall work with the differential system (18) restricted to the invariant position plane P, in order to see that a circular periodic solution contained in P is non-degenerate it is sufficient to prove that their multipliers are 1 with multiplicity two, and two other multipliers different from 1.
The Jacobian matrix of the Hamiltonian vector field corresponding to the Hamiltonian system (15) is ⎛ When we evaluate this matrix on the circular periodic solution of radius r 1 (c) and c ∈ (0, γ ) with recall (16), we obtain the matrix

Now the variational equation (19) becomeṡ
where M is a 4 × 4 matrix, and the solution M(t) of this differential equation such that M(0) is the identity matrix of R 4 , evaluated at the period (17) of the circular periodic orbit of radius where r 1 = r 1 (c), and Of course, by definition this last matrix is the monodromy matrix of the circular periodic solution of radius r 1 (c). Its eigenvalues are the multipliers of this periodic solution, namely Since r 1 ∈ (ρ 1 , ρ 2 ) we have that AB < 0, and consequently

Finding τ/2-Antiperiodic Circular Solutions of System (6)
Proof of Proposition 5 By definition (7) of U , we obtain Let x ±,r (t) = r cos 2π t τ , ±r sin 2π t τ for some r > 0 to be determined later. Then we obtainẍ Thus we havë That is, x is a solution of (6) if and only if r > 0 is a root of ϕ τ (r ).
This completes the proof of Proposition 5.

Proof of Theorem 7
In order to prove Theorem 7 we need the following two inequalities given in the next two lemmas.
Lemma 9 (Wirtinger's inequality, cf. Theorem 258 of [6]) For real numbers a < b, and the equality holds if and only if for some constant c 1 and c 2 ∈ R.
Lemma 10 (Jensen's inequality, cf. Theorem 204 of [6]) For real numbers a < b, let φ = φ(t) satisfying φ (t) > 0 and be finite for all t ∈ (a, b), and f and p be integrable on [a, b] and satisfying with m and M may be infinite, and f (t) is almost always different from m and M. Then .

Here equality holds if and only if f (t) is a constant function on [a, b].
Proof of Theorem 7 We carry out the proof in two steps.
Here we used the condition 2 < α.
Now other claims of Theorem 7 follow from Propositions 5 and 6. The proof of Theorem 7 is complete.
Remark 1 (i) By our above study, it is natural to ask whether for every τ ≥ τ * , the system (6) possesses any τ/2-antiperiodic solutions which are not circular motions. (ii) It is not clear so far whether τ * = τ * * holds, as well as whether there exists any τ/2-antiperiodic solutions for τ ∈ (τ * , τ * * ) if τ * < τ * * holds. (iii) Based on results in Propositions 3 and 4, it is natural to ask whether the Theorem 5 continues to hold when the potential function is a weak force, i.e., 0 < α ≤ 2 < β. (iv) Here we would like to draw readers attentions to a remarkable result of Ambrosetti and Coti Zelati, i.e., Theorem 9.1 of [1] in 1993, in which they proved the existence of at least one τ -periodic solution of the system (6) for every τ > 0. In their proof, they constructed a mountain pass structure which depends on a set of suitable functions with non-zero mean integral values. Their τ -periodic solutions are not τ/2-antiperiodic. For 0 < τ < τ * , this conclusion follows from our Theorem 7. Here a natural task of the future study on the system (6) is to understand the global structures of the sets of its τ/2-antiperiodic solutions and τ -periodic solutions respectively for prescribed suitable τ > 0 with the potential function being strong force or weak force.