Continuous Choreographies as Limiting Solutions of $N$-body Type Problems with Weak Interaction

We consider the limit $N\to +\infty$ of $N$-body type problems with weak interaction, equal masses and $-\sigma$-homogeneous potential, $0<\sigma<1$. We obtain the integro-differential equation that the motions must satisfy, with limit choreographic solutions corresponding to travelling waves of this equation. Such equation is the Euler-Lagrange equation of a corresponding limiting action functional. Our main result is that the circle is the absolute minimizer of the action functional among zero mean (travelling wave) loops of class $H^1$.


Introduction
It is well known that the planar regular N -gon relative equilibrium is a solution of the equations of motion for the Newtonian N -body problem with N ≥ 3 and equal mass bodies [1,3,6]. We consider N -body type problems with weak interaction, equal masses in R d , d ≥ 2, and −σhomogeneous potential, 0 < σ < 1, and study the resulting equation of motion when N → +∞, which we will refer to as continuous system, as well as a particular type of solutions which we call continuous choreographies using a variational approach. We can roughly say that a continuous choreography is a limiting configuration of classical choreographies when the number of particles grows without limit and indeed the circle turns out to be the continuous choreography associated to regular N -gons on the plane. G. Buck [2] considered curves which are locally approximated by solutions of the N -body problem. There is an essential difficulty in trying to study a continuous curve of particles interacting under the Newtonian law, since the combined force exerted by the neighbours blows up, except at points where the curvature vanishes. It seems that overcome this difficulty is impossible. Therefore we confined ourselves to consider weak interaction.
The paper is structured as follows: In the introduction we recall the variational approach of the N -body problem, as well as the particular type of solutions that we consider in this paper, the choreographic solutions. In Section 2 we obtain the limit equation of the N -body problem when N → +∞ and look at the particular case of travelling wave type continuous distributions of masses. Then we consider the limiting action functional and its critical points. In Section 3 we first show that the circle is a continuous choreography for 0 < σ < 1, and then prove our main result that it is in fact the absolute minimizer of the action functional among zero mean loops of class H 1 with period 1. This is the continuous distributions of masses counterpart of Theorem 1 in [1] according to which, under the choreographic restriction, the Lagrangian action attains its absolute minimum at the planar regular N -gon relative equilibrium.

Choreographies for the N -body problem
The N -body problem with homogeneous potential of degree σ ∈ (−1, 0), is the study of the dynamics of N positive mass points m 1 , . . . , m N moving in R d , d ≥ 2, interacting according to equations where q i (t) ∈ R d denotes the position of m i at time t. The case σ = 1 is the Newtonian N -body problem.
Equivalently, we have the second-order equation is the potential function (the negative of the potential energy), ∇ is the gradient in R d×N , Since the center of mass describes an uniform motion, we can fix it at the origin, and take According to the variational approach, solving (1.1) is equivalent to seeking the critical points of the Lagrangian action where L σ : T X → [0, +∞] is defined on the tangent bundle Equations (1.1) are the Euler-Lagrange equations associated to the functional (1.3).
There is an important class of solutions of (1.1) known as choreographies. A choreography of (1.1) is a periodic solution q(t) whose orbit is the union of closed curves, each of these is the trajectory of at least two bodies. If the solution consists of only one closed curve, then we call this solution a simple choreography. Definition 1.1 (see [4,5,8]). We say that a solution of the N -body problem (1.1) is a simple choreography if it is T -periodic and all bodies move on the same curve, interchanging their mutual positions after a time fixed, τ = T /N , that is, there exists a function x : R → R d such that: An example of a simple choreography is the relative equilibrium associated to Lagrange's equilateral triangle and this fact was extended in [6] to the case of N equal masses. For details on choreographies we refer the reader to [8] and the references therein.
2 Continuous choreographies as the limit N → ∞ of N -body type problems

The equation of motion
Consider the N -body problem (1.1) with equal masses (m 1 = · · · = m N = 1/N ) on R d and 0 < σ < 1 periodic on the first variable and twice differentiable on the second variable. Let ∆s = 1/N and suppose that the position of the i-th mass at time t is given by So when N → +∞, ∆s → 0, the limit of the right-hand side of equation (2.2) can be written as a Cauchy principal value A natural way to make the positions q i (t) satisfy the coreography condition is to take q(s, t) of travelling wave type, that is where y is a 1-periodic function in C 2 (R, R d ).
Using expression (2.4) in equation (2.3), we obtain that y must satisfy Claim 2.1. Assume y ∈ C 2 1 (R, R d ) defines a regular simple closed curve. Then the right-hand side of (2.5) is well defined.
Proof . There is a continuous function g : Since y is regular, for s fixed, the function can be considered as a continuous function on [−ε, ε] with F (0) =ẏ(s) ẏ(s) −2−σ and differentiable at 0. Thus

Variational approach to continuous choreographies (2.5)
Consider the action functional Proof . For y ∈ C 2 1 (R, R d ) and any z ∈ C 2 1 (R, R d ) we have (2.7) Differentiating and then integrating by parts the first term on the right-hand side of (2.7) we obtain d dε −v 2ÿ (s) · z(s)ds.

Circular choreography as minimizer of the action
In this section we consider circular planar curves as solutions of equation (2.5) and in fact as absolute minimizers of the action functional. This is motivated by papers [6,7], which prove that the planar regular N -gon relative equilibrium is a solution of the N -body problem, and mainly by Theorem 1 in [1]. We naturally follow the ideas from that paper.
Remark 3.3. We write a ∈ C d as a = Re a + i Im a andā = Re a − i Im a. Proof . Observe that the inequality (3.2) holds true for y constant. We can then assume that ξ y never vanishes. Our aim is to show that the functional J(y) := 1 0 ẏ 2 1 0 ∆ µ y · y defined on Λ attains its infimum and that its minimal value is 4π 2 . Since the functional J is homogeneous of degree zero, minimizing J is equivalent to minimizing the coercive functional J(y) = 1 0 ẏ(t) 2 dt on the constraint which is closed with respect to the weak H 1 1 -topology. Let (y n ) n be a minimizing sequence, then 1 0 ẏ n 2 dt = ẏ n L 2 is bounded. By lower semi-continuity of the norm we can select a subsequence (y n k ), weakly convergent to y ∈ M and obtaiñ J(y) ≤ lim inf n kJ (y n k ).
So we can state that the minimum ofJ exist. The minimal value ofJ corresponds to the first eigenvalue λ min for the problem −ÿ = λ∆ µ y, y ∈ H 1 1 R, R d , To study problem (3.3) write a solution as so that the eigenvalues of the problem (3.3) are λ k = 4π 2 k 2 /d k , k ∈ N. Thus λ min = λ 1 = 4π 2 and the minimum is attained if and only if y(t) = ae 2πit +āe −2πit .
Theorem 3.5. The absolute minimum of A σ on Λ, with 0 < σ < 1 and v given by (3.1), is attained at and only at a unit circle

Open problems
It is possible that imposing additional symmetries beyond the choreography condition, could give some other critical points of the action besides the absolute minimizers. One also could impose some topological non-trivial condition to the curve defining the continuous coreography. Connections with other topics such as vortex lines in fluid dynamics might also be of interest.