Continuation of Gerver's supereight choreography

In [6] we developed a continuation technique for periodic orbits in reversible systems having some first integrals and corresponding symmetries. One of the applications was the continuation of Gerver’s supereight choreography when one or several of the masses are varied. In this note we give a more complete description of the families of periodic orbits which can be obtained in this way. 1 Symmetries of the N–body problem Symmetries form one of the key ingredients in understanding the dynamics of the N -body problem. In this section we briefly describe these symmetries; since it is our aim to do continuation under a change of the masses we also include symmetries in which the mass parameters are involved. We denote by x = (q1, . . . ,qN ,p1, . . . ,pN) and m = (m1, . . . , mN) the state space vector and the mass vector, respectively; qj ∈ R n is the position for the j-th body, pj ∈ R n its momentum, and mj its mass. The equations of motion of the N -body problem take the form qj = ∂Hm ∂pj (x), ṗj = − ∂Hm ∂qj (x), j = 1, . . . , N, (1) where the Hamiltonian Hm is given by Hm(x) = N ∑ j=1 1 2mj ‖pj‖ 2 − ∑ 1≤i 0 the transformation Φμ(x) := (μ q1, . . . , μ qN , μ p1, . . . , μ pN); it has the property that if x(t) is a solution of system Xm, then Φμ(x(t)) is a solution of system Xμm. Finally we can define an action of the symmetric group SN (that is the group of all permutations of the set {1, . . . , N}) on the phase space by Σσ(x) := (qσ(1), . . . ,qσ(N),pσ(1), . . . ,pσ(N)), ∀σ ∈ SN . If x(t) is a solution of system Xm, then Σσ(x(t)) is a solution of system Xσ(m), with σ(m) = (mσ(1), . . . , mσ(N)). We also recall the notation for cycles: if {a1, . . . , ar} is any subset of {1, . . . , N}, then we denote by (a1, . . . , ar) the permutation σ given by σ(aj) = aj+1 for j = 1, . . . , r− 1, σ(ar) = a1 and σ(k) = k for all k 6∈ {a1, . . . , ar}. Every permutation can be written as a composition of cycles. All the symmetry operators ΨQ,0, Λλ, Φμ and Σσ commute with each other. 2 Theoretical results about periodic orbits In the past many different numerical methods have been used to calculate families of periodic orbits of the N -body problem. One of the main problems comes from the fact that by applying the symmetry operators above to a periodic orbit one obtains a new periodic orbit; therefore, periodic orbits typically belong to multi-parameter families of such orbits (compare with the cylinder theorem — see [3]), and it is not always easy to devellop a good strategy to calculate appropriate representants from such families. In [4] and [6] the authors, in collaboration with E.J. Doedel, have worked out an approach which in combination with boundary value packages such as Auto (see [1]) appears to be very effective in handling this type of problem. 96 The basic concept in [4] is that of a normal periodic orbit, with normality defined by a geometric condition involving the monodromy matrix and the first integrals. The main result then says that normal periodic orbits belong to families which can be obtained (both theoretically and numerically) by solving an appropriate regular boundary value problem (BVP); this BVP involves, next to certain phase conditions, an adapted set of system equations obtained from the original one by adding artificial parameters multiplied by the gradients of the first integrals (see [4] for more details). In [6] this approach was adapted to take advantage of reversibility properties of the system. A reversor is a linear operator R on the phase space which anti-commutes with the vectorfield. For example, Σσ ◦ ΨQ,0 ◦ Λ−1 is a reversor for (1) on condition that σ(m) = m. An R-symmetric solution is then a solution whose orbit is invariant under R; an orbit is R-symmetric and periodic if and only if it has exactly two intersection points with the fixed point subspace Fix(R) = {x | R(x) = x}. Again, under an appropriate normality condition, a regular BVP can be set up to calculate families of R-symmetric periodic orbits. Examples of such BVP can be found in [5] and [6]. When at a particular orbit along a family the normality condition is not satisfied then bifurcation can occur; in such case Auto allows to switch branches and to resume continuation of the bifurcating family. 3 Gerver’s supereight Gerver’s supereight is a planar choreographic solution for 4 bodies with equal masses, i.e. a solution where the 4 bodies follow the same curve at equal time intervals. More precisely, Σ(1 2 3 4)(x(t)) = x(t + T/4), where T is the period of the solution. Figure 1 shows the orbit and the position of the bodies at two different times, namely t = 0 and t = T/8. A closer analysis shows that the corresponding periodic orbit is invariant with respect to several reversors. To be more precise, let {e1, e2} be the canonical basis in R (from now on we restrict to n = 2), and let S be the reflection in R with respect to the e1-axis: Se1 = e1 and Se2 = −e2. For each σ ∈ S4, let R + σ := Σσ ◦ ΨS,0 ◦ Λ−1 and R σ := Σσ ◦ Ψ−S,0 ◦ Λ−1. It is not hard to see that for the supereight as shown in Fig. 1 we have x(0) ∈ Fix(R (1 3)) ∩ Fix(R − (2 4)), x(T/8) ∈ Fix(R + (1 2)(3 4)) ∩ Fix(R − (1 4)(2 3)), x(T/4) ∈ Fix(R (2 4))∩Fix(R − (1 3)) and x(3T/8) ∈ Fix(R + (1 4)(2 3))∩Fix(R − (1 2)(3 4)). At t = T/2 we have again x(T/2) ∈ Fix(R (1 3)) ∩ Fix(R − (2 4)), and then the sequence repeats. This implies that Gerver’s supereight can be considered as an R-symmetric periodic orbit, with R any of the reversors R σ appearing in the foregoing.