Periodic oscillations in a $2N$-body problem

Hip-hop solutions of the $2N$-body problem are solutions satisfying that: (i) The masses of all the bodies are the same. (ii) At every instance of time $t$, $N$ of the bodies are at the vertices of a regular $N$-gon contained in a plane $\Pi_{1}(t)$ and the other $N$ bodies are at the vertices of a second regular $N$-gon that differ from the first $N$-gone by a translation and a rotation of $\frac{2\pi}{N}$ radians and it is contained in a plane $\Pi_{2}(t)$. (iii) For every $t$, the planes $\Pi_{1}(t)$ and $\Pi_{2}(t)$ are parallel and they are equidistant from a fixed plane $\Pi_{0}$. (iv) The center of the $N$-gones are always in a fixed line $l_{0}$ perpendicular to $\Pi_{0}$. In particular when $\Pi_{1}(t)\neq\Pi_{2}(t)$ the $2N$ bodies form an antiprism, and when $\Pi_{1}(t)=\Pi_{2}(t)$ the $2N$ bodies form a regular $2N$-gon. For any hip-hop solution we can define the function $d(t)$ that gives the oriented distance from the plane $\Pi_{0}$ to the plane $\Pi_{1}$ and the function $r(t)$ that gives the distance from the line $l_{0}$ to any of the bodies. We will assume that the line $l_{0}$ is the $z$-axis and the plane $\Pi_{0}$ is the $x$-$y$ plane.

We will call a hip-hop solution, double symmetric, if the function $d(t)$ satisfies that for all $t$, $d(-t)=-d(t)$ and $d(T-t)=d(T+t)$ for some nonzero $T$. We will call a hip-hop solution, single symmetric, if the function $d(t)$ satisfies that $d(-t)=-d(t)$ for all $t$ and there is not a $T$ such that, $d(T-t)=d(T+t)$ for all $t$.

The differential equations for hip-hop solutions were provided by Meyer and Schmidt in 1993 [MR1205666] as a model for braided rings of a planet based on a previous model for the rings of Saturn. They called these solutions alternating solutions and the only difference with the hip-hop solutons is that the alternating solutions consider an additional motionless body in the center of mass of the $2N$ bodies. For the alternating solutions, the body in the center makes the role of the planet while the $2N$ bodies are the small bodies conforming the ring. One of the pioneer works that showed the periodicity of some hip-hop solutions is the work of Chenciner and Venturelli [MR1820355], where the authors used variational methods to prove existence of periodic solutions. Some other papers that show the periodicity of hip-hop solutions by variational methods are [MR1820355, MR2401905, MR2031430]. It is well known that the periodic solutions of the two body problem can be extended to periodic planar solutions of the $n$ body problem, with the trajectory of each body being a rotation of a single ellipse. The link \urlhttps://youtu.be/RjlZpqDFsDM leads to a video showing some of these periodic solutions. Using a Bolzano argument, in [MR2570295] the authors show the periodicity of hip-hop solution with motion close to ellipses with high eccentricity. In [MR2267950] the authors use a Poincaré analytical continuation method to show the existence of double symmetric hip-hop solutions with trajectories close to circles.

Similar to the work in [MR2267950] our solutions emerge from planar periodic solutions where all the bodies move in the same circle. We study both, doubly and single symmetric solutions. The technique that we use is slightly different from the regular Poincaré analytical method. We can say that it is more direct. Let us give the idea behind the method. We consider the functions $r(t)$ and $d(t)$ defined above as functions of one of the initial conditions and the angular momentum, more precisely we consider $R(a,b,t)=r(t)$ and $D(a,b,t)=d(t)$ where $b=\dot{d}(0)$ and $a$ is the angular momentum. All our solutions satisfy $d(0)=0$. For the circular solutions $b=0$ because the solution are planar, $d(t)\equiv 0$, and the value of $a=a_{0}$ can be easily found from the condition that $r(t)$ is constant. In the Euclidean space $(a,b,T)$ with $T$ a fixed value of time, we have that due to the symmetries of the ordinary differential equations that govern the hip-hop solutions, we have that points in the space that satisfy the system of equations $D_{t}(a,b,T)=R_{t}(a,b,T)=0$ produce double symmetric hip-hop solutions with angular momentum $a$ and period $4T$, while points $(a,b,T)$ that satisfy $D(a,b,T)=R_{t}(a,b,T)=0$ produce hip-hop solutions with angular momentum $a$ and period $2T$ that are potentially single symmetric. With this idea in mind we have that double and single symmetric solutions are obtained from studying three surfaces: The surface in the space $(a,b,T)$ that satisfies $R_{t}(a,b,T)=0$, the surface in the space $(a,b,T)$ that satisfies $D_{t}(a,b,T)=0$ and the surface in the space $(a,b,T)$ that satisfies $D(a,b,T)=0$. Recall that the intersection of two surfaces is in general a curve and that if the gradient of the functions that define two surfaces are linearly dependent at a particular point in the intersection, then the solution of the system may be the union of two curves. As an example, Figure 5 show points $(a,b,T)$ that solve the system $D(a,b,T)=R_{t}(a,b,T)=0$. In this paper we notice that the circular solutions of the $2N$ body problem produces a line $\Gamma=\{(a_{0},0,T)\in\mathbb{R}^{3}\}$ in the space $(a,b,T)$ that solve the systems $D(a,b,T)=R_{t}(a,b,T)=0$ and $D_{t}(a,b,T)=R_{t}(a,b,T)=0$. By computing the gradient of the functions $D$, $D_{t}$ and $R_{t}$ we find points in $\Gamma$ that potentially allows nearby points that solve both systems. Half of these (bifurcation) points were found in the paper [MR2267950] using the Poincare analytical method. The other half that we found were those bifurcation points that potentially may produce single symmetric solutions. It turned out that points in the smooth curves $\Omega_{i}$ emanating from the new bifurcation points, did not produced single symmetric hip-hop solutions near the circular solutions but, after doing analytic continuation, we found out that for some values of $N$, $\Omega_{i}$ has a bifurcation point that generates a curve of points $\Phi$ in the space $(a,b,T)$ with the property that each of its point represents a single symmetric solution. We proved that an interesting difference between the single and double symmetric families of hip-hop solutions of the $2N$ body problem is the fact that no single symmetric solution can produce a choreographic while there are infinitely many choreographies in the family of double symmetric hip-hop solutions.

The anti-prism $2N$-body problem will be characterized in the following concrete framework: Consider $Q_{1},Q_{2},\ldots,Q_{2N}$ bodies of equal mass $m>0$, located on the vertices of a regular anti-prism. If $\mathbf{r}_{j}(t)$ is the position of the body $Q_{j},j=1,\ldots,2N$ at each instant $t$ and satisfy

where $\mathcal{R}$ is a rotation/reflection matrix given by

Introducing cylindrical coordinates $(r,\theta,d)$ it is shown that the equations of motions of the bodies are given by

where $\displaystyle{a}$ is the angular momentum of the system and

From now on, for $r_{0},m$ and $N$ fixed, we denote by

the solutions of the system (1) with initial conditions

It is clear that $r(t)$, $d(t)$ solves the reduced problem

then $r(t)$, $d(t)$ and $\theta(t)$ solves (1) if

Notice that if for some $a,b\in\mathds{R}$ the functions $R(a,b,t)$ and $D(a,b,t)$ have the same period $T>0$ then, they provide a periodic hip-hop solution of (1) if and only if $\theta(a,b,T)$ is equal to $p\,\pi/q$ with $p$ and $q$ whole numbers. In general, $T$-periodic solutions of the systems (3) define reduced-periodic solutions of the $2N$-body problem (1). That is, solutions with the property that every $T$ units of time, the positions and velocities of the $2N$-bodies only differ by an rigid motion in $\mathds{R}^{3}.$

Let us present some useful results on the existence of symmetric periodic solutions of $(\ref{cilindricas})$. To this end, let $r_{0},m\in\mathds{R}^{+}$ and consider the following initial value problem

with

Recall that, we are denoting the solutions of (4) as $t\to R(a,b,t)$ and $t\to D(a,b,t)$. Moreover, it is easy to check the following symmetries:

For a given $N>1$ define the parameters

In this section we state and prove the main theoretical result of this document.

Due to Remark 1 we have the following result:

In this section we use analytic continuation to extend the branches emanating form the bifurcations points

obtained in Theorem 3 and Theorem 4.

Notice that the solutions from $p_{0}$, $q_{0}$ are essentially the same, all of them have double symmetry. With the purpose of searching for solutions of type (II) that are not of type (I) we extend the branch starting at the point $q_{0}$ by applying analytic continuation and a method similar to the one presented in [MR2267950] to system (II). Taking $N=3$, $r_{0}=2$ and $m=1$ we have

The analytic continuation method give us a table that we labeled DSP because, as explained above, near the point $q_{0}$, the solutions satisfy the double symmetry property. We found a bifurcation point –that we labeled $B$– along this branch that gave us a new branch with single symmetric hip-hop solutions. Recall from the introduction that a solution is called single symmetric if $d(t)$ is even and there is not $T\neq 0$ such that $d(T-t)=d(T+t)$ for all $t$. Before we address the existence of this bifurcation point $B$, let us elaborate on some properties of the DSP branch.

Figure 3 shows the table $DSP=\{Q_{1}=q_{0},Q_{2},\dots,Q_{5806}=q_{f}\}.$ It is a list of points in the $a,b,T$ space that satisfies the conditions

For the solutions along the DSP branch the system of particles reaches the maximum vertical expansion (that is expansion in the $z$-direction) at the same time that it reaches a maximum contraction towards the $z$-axis. In other words, when the particles are at their maximum height they are also the closest they can be to the $z$-axis. This is illustrated by the joint graph of $d(t)$ and $r(t)$ for the points $Q_{210}$ and $Q_{4225}$ shown in Figure 2.

The name for the set $DSP$ stands for “double symmetry points” because each one of these points represent a solution with period $2T$ with $d$ and $r$ even with respect to $t=T/2$, $r$ even with respect to $t=T$ and $d$ odd with respect to $t=T$.