Unifying the Hyperbolic and Spherical 2-Body Problem with Biquaternions

Home , Complexification, Hyperboloid model, Unit hyperbola

Philip Arathoon December 2020

Abstract The 2-body problem on the sphere and hyperbolic space are both real forms of holomorphic Hamiltonian systems deﬁned on the complex sphere. This admits a natural description in terms of biquaternions and allows us to address questions concerning the hyperbolic system by complexifying it and treating it as the complexiﬁcation of a spherical system. In this way, results for the 2-body problem on the sphere are readily translated to the hyperbolic case. For instance, we implement this idea to completely classify the relative equilibria for the 2-body problem on hyperbolic 3-space for a strictly attractive potential.

Background and outline

The case of the 2-body problem on the 3-sphere has recently been considered by the author in [1]. This treatment takes advantage of the fact that S3 is a group and that the action of SO(4) on S3 is generated by the left and right multiplication of S3 on itself. This allows for a reduction in stages, first reducing by the left multiplication, and then reducing an intermediate space by the residual right-action. An advantage of this reduction-by-stages is that it allows for a fairly straightforward derivation of the relative equilibria solutions: the relative equilibria may first be classified in the intermediate reduced space and then reconstructed on the original phase space. For the 2-body problem on hyperbolic space the same idea does not apply. Hyperbolic 3-space H3 cannot be endowed with an isometric group structure and the symmetry group SO(1, 3) does not arise as a direct product of two groups. This prevents us from reducing in stages. Nevertheless, despite these differences, the sphere and hyperbolic space are both two sides 3 arXiv:2012.12166v1 [math-ph] 22 Dec 2020 of the same coin. The sphere S is the real affine variety

u2 + v2 + w2 + z2 = 1, (0.1)

whereas H3 is a connected component of

t2 x2 y2 z2 = 1. (0.2) − − − 3 If we complexify the variables these each deﬁne the same complex 3-sphere CS in C4. We call 3 3 CS the complexiﬁcation of S3 and H3 and refer to S3 and H3 as real forms of CS . This is

1 analogous to the notion of real forms and complexifications for vector spaces and Lie groups. To extend this analogy further, one can complexify an analytic Hamiltonian system on a real form to obtain a holomorphic Hamiltonian system on the complexification. The original real system appears as an invariant sub-system in the complexified phase space. We shall consider the 2-body problem on H3 and complexify this to obtain a holomorphic 3 Hamiltonian system for the complexified 2-body problem on CS . The group of complex sym- metries is the complexification SO4C of SO(1, 3). Pleasingly, this puts us back into the same regime that we had earlier with the SO(4)-symmetry. In particular, we may take advantage of the SL2C SL2C-double-cover over SO4C and reduce in stages. We can then apply the exact same analysis× for the spherical problem as performed in [1], except now all variables are taken to be complex and one must remember to restrict attention to the invariant hyperbolic real form. This trick allows us to classify the relative equilibria solutions for the hyperbolic problem using the same methods which are used for the spherical case. One-parameter subgroups of SO(1, 3) come in four types: elliptic, hyperbolic, loxodromic, and parabolic. The classification of relative equilibria for the 2-body problem on the Lobachevsky plane H2 was carried out in [4, 6, 3]. For each configuration of the two bodies there exists precisely one elliptic and hyperbolic solution up to time-reversal symmetry. We extend this result and show that for each configuration of two bodies in H3 there exists, up to conjugacy, a circle of loxodromic relative equilibria, including the elliptic and hyperbolic solutions from the H2 case. There are no parabolic solutions for a strictly attractive potential. Furthermore, we piggy-back on the stability analysis performed in [3] and apply a continuity argument to derive complete results for the reduced stability of the relative equilibria. The paper is arranged as follows. We first provide a short review of the theory of real forms for holomorphic Hamiltonian systems. The material is distilled from the more complete treatment given in [2], but is provided to ensure that the paper remains for the most part self- contained. The complexified 2-body problem is then formulated and conveniently presented in terms of biquaternions. Next, we consider in closer detail the hyperbolic real form. We discuss the hyperbolic symmetry, momentum, and one-parameter subgroups. Finally, we emulate the work in [1] and completely classify the hyperbolic relative equilibria by working on an intermediate reduced space for the complexified system. We then derive stability results in the full reduced space.

1 Real forms for holomorphic Hamiltonian systems

1.1 Real-symplectic forms A holomorphic symplectic manifold is a complex manifold M equipped with a closed, non- degenerate holomorphic 2-form Ω. Examples of such manifolds include cotangent bundles of complex manifolds, and coadjoint orbits of complex Lie algebras. Exactly as with real Hamiltonian dynamics we can define a Hamiltonian vector field for a holomorphic function on M and consider the dynamical system generated by its flow. A real structure R on a complex manifold M is an involution whose derivative is conjugate- linear everywhere. If (M, Ω) is a holomorphic symplectic manifold then a real structure R is called a real-symplectic structure if it satisfies

R∗Ω = Ω.

2 If the ﬁxed-point set M R is non-empty, then the restriction of Ω to M R deﬁnes a real symplectic form ωbR. A real structure r on a complex manifold C can be lifted to a real-symplectic structure R on the cotangent bundle by setting

R(η),X = η, r∗X (1.1) h i h i ∗ r ∗ R for η Tx C and for all X Tr(x)C. If C is non-empty, then (T C) is canonically symplec- tomorphic∈ to the T ∗Cr. ∈ Theorem 1.1 ([2]). Let f be a holomorphic function on a holomorphic symplectic manifold (M, Ω) equipped with a real-symplectic structure R with non-empty fixed-point set M R. If f is purely real on M R then the Hamiltonian flow generated by f leaves M R invariant. Moreover, R R the flow on M is identically the Hamiltonian flow on (M , ωbR) generated by the restriction u of f.

R In the situation described by this theorem we may refer to (M , ωbR, u) as a ‘real form’ of the holomorphic Hamiltonian system (M, Ω, f).

1.2 Compatible group actions Let M be a complex manifold and G a complex Lie group acting holomorphically on M. If M is equipped with a real structure R then the action is called R-compatible if there exists a real Lie group structure ρ on G which satisﬁes

R(g x) = ρ(g) R(x) (1.2) · · for all x in M and g in G. Recall that a real Lie group structure is a real structure on G which is also a group homomorphism. For such a compatible group action the real subgroup Gρ = Fix ρ acts on M R. One can show (see for instance [11, Proposition 2.3]) that for x in M R

T (G x M R) = T (Gρ x). (1.3) x · ∩ x · This fact is signiﬁcant to the study of relative equilibria solutions for a Hamiltonian system. Consider a Hamiltonian system (M, Ω, f), either real or complex, and suppose it admits a symplectic group action by G which preserves the Hamiltonian. The tuple (M, Ω, f, G) is a Hamiltonian G-system. A solution is called a relative equilibria (RE) if it is contained to a G-orbit. Equivalently, the solution is the orbit of a one-parameter subgroup of G [8]. If we combine this with (1.3) and Theorem 1.1 we have the following. Proposition 1.2. Let (M, Ω, f, G) be a holomorphic Hamiltonian G-system and suppose the action of G is R-compatible with respect to a real-symplectic structure R with non-empty ﬁxed- point set M R. If f is real on M R then there is a one-to-one correspondence between RE of R ρ (M, Ω, f, G) which are contained to N and RE of the real form (M , ωbR, u, G ).

1.3 Holomorphic momentum maps A holomorphic group action of a complex group G on a holomorphic symplectic manifold (M, Ω) is Hamiltonian if it admits a holomorphic momentum map J : M g∗ satisfying J(x), ξ = → h i Hξ(x) for each x in M and for all ξ in the Lie algebra g of G. Here Hξ is a holomorphic function whose Hamiltonian vector ﬁeld is that generated by ξ. If ρ is a real Lie group structure on G

3 and R a real-symplectic structure on M then the momentum map J will be called R-compatible with respect to ρ if J R = ρ∗ J. (1.4) ◦ ◦ ∗ The map ρ is the conjugate-adjoint to ρ∗ = Dρe deﬁned by

∗ ρ η, ξ = η, ρ∗ξ (1.5) h i h i for each η in g∗ and for all ξ in g. If G is connected, the R-compatibility of J implies the action of G on M is also R-compatible in the sense of (1.2). In this case, the symplectic action of Gρ on M R is also Hamiltonian. Indeed, observe that J(x) belongs to Fix ρ∗ for x in M R. This set consists of all complex-linear ρ ∗ forms on g which are real on g = Fix ρ∗. We therefore have an identiﬁcation of Fix ρ with (gρ)∗. In this way the restriction

R ∗ ρ ∗ Jb := J R : M Fix ρ = (g ) (1.6) |M −→ ∼ ρ R is the momentum map for the action of G on (M , ωbR).

2 The complexiﬁed 2-body problem

2.1 Biquaternions Let 1, I, J, K denote the standard basis for the real algebra of quaternions. A biquaternion is a{ linear combination} q = u1 + vI + wJ + zK (2.1) where u, v, w, z each belong to C. As a complex algebra this admits a representation by iden- tifying q with the matrix u + iv w + iz Q = . (2.2) w + iz u iv − − The biquaternions are a composition algebra over C, meaning there exists a complex quadratic 2 2 2 2 form which satisﬁes q1q2 = q q2 for all biquaternions q1 and q2. With respect to the|| matrix · || representation|| this quadratic|| || || || form|| corresponds to the determinant. Observe that the determinant of Q is u2 + v2 + w2 + z2.

We may therefore identify the biquaternions with matrices Q in M2(C) whose quadratic form is det Q, or with vectors q = (u, v, w, z)T in C4 whose quadratic form is the standard qT q. We will 4 not always specify whether a biquaternion q is understood to mean an element of M2(C) or C . In most cases it should be clear from the context or not make any difference. In any case, using the polarization identity we shall denote the symmetric, complex bilinear form which gives rise to 2 by , . ||The · || set ofh biquaternionsi q with q 2 = 1 forms a group under multiplication. With respect || || to the matrix representation this corresponds to the special linear group SL2C of matrices with det Q = 1. On the other hand, it also corresponds to the affine variety in C4 of vectors satisfying 3 qT q = 1. This is the complex 3-sphere CS defined in (0.1). Multiplication on the left and right by SL2C on M2(C) preserves the determinant. This provides an action of SL2C SL2C on × M2(C) which we shall denote by (g , g ) Q = g Qg−1. (2.3) 1 2 · 1 2 4 4 This action preserves the complex quadratic form on C ∼= M2(C), and consequently, establishes the well-known double cover of SL2C SL2C over SO4C. We note that this action preserves 3 3 × CS . Indeed, since CS may be identified with the group SL2C, this action is just left and right multiplication of the group on itself.

2.2 The hyperbolic real structure

3 We shall be interested in the following real structure r defined on CS r :(u, v, w, z) (u, v, w, z). (2.4) 7−→ − − − According to the matrix representation in (2.2) this real structure corresponds to taking the conjugate-transpose of Q. The fixed-point set is therefore the set of Hermitian 2 2 matrices with unit determinant. Alternatively, the fixed-point set consists of the points× (t, ix, iy, iz) for t, x, y, z real numbers satisfying (0.2). The solution set has two connected components corresponding to t 1 and t 1. We shall only be interested in the t 1 component and write this as H3, noting≥ that this≤ − is the hyperboloid model of hyperbolic-3≥ space. The action of SL2C SL2C in (2.3) is r-compatible with respect to the real Lie group structure × −† −† ρ(g1, g2) = (g2 , g1 ). (2.5)

The ﬁxed-point set of ρ is a conjugate-diagonal copy of SL2C which acts by g Q = gQg†. (2.6) · The hyperboloid H3 R4 is invariant with respect to this action. Therefore, the action preserves the indeﬁnite⊂ orthogonal form of signature (1, 3). Incidentally, this establishes the double cover of SL2C over SO(1, 3).

2.3 The problem setting

3 Consider the holomorphic symplectic manifold T ∗CS . Thanks to the complex bilinear form 3 on C4 the complex tangent spaces of CS may be identiﬁed with their duals. In this way the 3 cotangent bundle T ∗CS may be identiﬁed with the set

4 4 2 (q, p) C C q, p = 0, q = 1 . (2.7) ∈ × | h i || || 3 3 The phase space for the complexiﬁed 2-body problem is T ∗CS T ∗CS . Strictly speaking the diagonal collision set should be removed, however we will not reﬂect× this in our notation. We consider a holomorphic Hamiltonian

2 2 p1 p1 H(q1, p1, q2, p2) = || || || || + V ( q1, q2 ) (2.8) − 2m1 − 2m1 h i for V : C C some holomorphic function which we call the potential. → 3 3 We now take the product of the real structure r on CS CS and lift this to a real- symplectic structure R on the cotangent bundle. The hyperbolic× phase space T ∗H3 T ∗H3 may be identiﬁed with a connected component of Fix R. Restricting the Hamiltonian× to this component gives 2 2 p1 p1 H(q1, p1, q2, p2) = | | + | | + V (cosh ψ) . (2.9) 2m1 2m1

5 2 ∗ 3 Here p denotes the modulus of the covector p Tq H with respect to the hyperbolic metric on H3|.| Recall that this metric is inherited by the∈ negative of the ambient Minkowski metric of signature (1, 3) on R4. This explains the rather unfortunate presence of what appears to be negative kinetic energy in the holomorphic Hamiltonian. 3 4 For q1 and q2 belonging to H C the inner product q1, q2 is equal to cosh ψ where ψ is ⊂ h i the hyperbolic distance between q1 and q2. For the case of gravitational attraction we choose z V (z) = m1m2 (2.10) − √z2 1 − which corresponds to the potential m1m2 coth ψ. In this case the Hamiltonian H is purely real on the real-symplectic form Fix−R and so we may apply Theorem 1.1 to the holomorphic 3 3 Hamiltonian system on T ∗CS T ∗CS . It follows that the Hamiltonian system for the 2-body problem on hyperbolic 3-space× occurs as a real form of the holomorphic Hamiltonian system described above.

2.4 Symmetry and momentum The momentum map for the cotangent lift of left-multiplication on a group is right-translation 3 of a covector to the identity [9]. Since CS may be identiﬁed with the group SL2C it follows that the momentum map for the action of left-multiplication on phase space is the total left-momenta

Ltot = L1 + L2 (2.11)

−1 th where Lk = pkqk denotes the left-momentum of the k -particle. Likewise, for the case of right-multiplication the momentum is left-translation to the identity. The momentum map for right-multiplication on phase space is the total right-momenta

Rtot = R1 + R2 (2.12)

−1 th where Rk = qk pk denotes the right-momentum of the k -particle. The momentum map for the full group action of SL2C SL2C is therefore × ∗ 3 ∗ 3 ∗ ∗ J : T CS T CS sl2C sl2C ;(q1, p2, q2, p2) (Ltot, Rtot). (2.13) × −→ × 7−→ − Notice that in the second factor we have negated the total right-momenta. This is because for the action in (2.3) we have the inverse of right-multiplication. By once again using the complex bilinear form on the biquaternions to identify sl2C M2(C) with its dual, we may express the conjugate-adjoint ρ∗ of the real Lie group structure⊂ρ as

ρ∗(L, R) = ( R†, L†). (2.14) − − 3 The cotangent lift of r to T ∗CS is given by (q, p) (q†, p†), where q and p are here representing 7→ matrices in M2(C). It follows that the momentum map J is R-compatible with respect to ρ, and hence, the group action is R-compatible. Furthermore, according to (1.6) the momentum ∗ 3 ∗ 3 † Ctot for the action of SL2C on T H T H may be identiﬁed with Ltot = Rtot. ×

6 3 Symmetry and reduction

3.1 One-parameter subgroups

A non-zero element of the Lie algebra sl2C is conjugate up to the adjoint action to either a semisimple or nilpotent matrix of the form η 0 0 1 S = or N = (3.1) 0 η 0 0 − for η a non-zero complex number. The one-parameter subgroup generated by an element conjugate to S is called elliptic for η imaginary, hyperbolic for η real, and loxodromic for η a general complex number. A one-parameter subgroup generated by an element conjugate to N is called parabolic. This terminology is inherited from the group PSL2C of Möbius transformations. Recall that (t, ix, iy, iz) in H3 corresponds via (2.2) to the Hermitian matrix t x iy z . (3.2) iy− z t +−x − − By taking the exponential of the elements S and N we may use (2.6) to compute the action of the one-parameter subgroups on H3. To help visualise these orbits we can use the Poincaré ball model of hyperbolic 3-space to identify H3 with the open ball in R3 via the map which sends (t, ix, iy, iz) to x y z (X,Y,Z) = , , . 1 + t 1 + t 1 + t In Figure 1 we include an illustration of these orbits on the Poincaréball. In these diagrams the X-axis is the vertical line through the north and south poles.

3.2 Geodesics and the Lobachevsky plane Let H2 denote the submanifold of H3 given by setting y = 0. Note from (3.2) that this submanifold equivalently corresponds to those purely real, symmetric matrices with unit determinant. It also corresponds to the open disk in the Poincar´eball by setting Y = 0. More generally, we shall refer to any 2-dimensional submanifold of H3 as a Lobachevsky plane if it can be 2 transformed to H by the action of SL2C. If we set y = 0 and x = 0 we obtain the curve H1. In terms of the general form of a biquaternion given in (2.1) such elements of H1 may be written as eψj = cosh ψ + j sinh ψ (3.3) where j = iK. Since j2 = 1 we see that H1 is none other than the unit hyperbola in the algebra of split-complex− numbers; the analogue of the unit circle in the complex numbers. 1 3 Finally, we remark that H is a geodesic in H and that the action of SL2C can send any geodesic to H1.

∗ 3 ∗ 3 Proposition 3.1. Let Jb denote the momentum map for the action of SL2C on T H T H . × The critical points of Jb are those points (q1, p1, q2, p2) where p1 and p2 are each tangent to the ∗ 2 geodesic through q1 and q2. The corresponding momentum Ctot in sl2C has Ctot a non- negative real number. Conversely, C 2 is real if and only if p and p are each|| tangent|| to a || tot|| 1 2 Lobachevsky plane containing q1 and q2.

7 (a) Elliptic (b) Hyperbolic

Figure 1: Orbits of one-parameter subgroups on the Poincar´eball.

Proof. A point is a critical point of the momentum map if and only if the action at this point 1 is not locally free. We may suppose q1 and q2 belong to H . The isotropy subgroup fixing H1 is the group of rotations SO(2) SO(1, 3) in the xy-plane. Therefore, the action fails ⊂ to be locally free when p1 and p2 have no x or y component. A calculation in terms of the split-complex numbers reveals that L 2 is real and non-negative. || tot|| We may use the SL2C-action to place the first particle at the centre of the Poincaréball 1 2 where q1 = I. We can then rotate the ball so that q2 belongs to H and p2 to H . Since p1, q2 and p are all Hermitian, the imaginary part of C 2 is 2 || tot|| det(L + L ) det(L† + L†) = 2 p , [p , q−1] . 1 2 − 1 2 h 1 2 2 i 3 Observe that p1 is a traceless, Hermitian matrix since it is tangent to H at q1 = I, and that −1 [p2, q2 ] is a real, skew-symmetric matrix as p2 and q2 are each real and symmetric. It follows 2 that Ctot is real if and only if, either p2 commutes with q2, or if p1 is real. If p1 is real || || 2 then everything belongs to H . If q2 commutes with p2, then p2 must also be a split-complex 1 number, and hence, p2 is tangent to H at q1. We can therefore rotate the ball about the 1 2 H -axis to send p1 into H .

3.3 The right-reduced space

Instead of reducing by the full SL2C SL2C-symmetry we shall instead obtain an intermediate × reduced space by reducing by the action of right-multiplication by SL2C. The orbit quotient is given by sending (q1, p1, q2, p2) to (L1,L2, qR)

8 −1 where we have introduced the right-invariant quantity qR = q1q2 . One can directly perform this orbit quotient by using Corollary 3.8.5 in [8] or by applying the Semidirect Product Reduction ∗ ∗ 4 by Stages Theorem as in [1]. The reduced Poisson structure on sl2C sl2C C is the Poisson × × 4 structure on the dual of the complexiﬁed special-Euclidean Lie algebra se4C = so4C n C . The Hamiltonian H descends to 2 2 L1 L1 H(L1,L2, qR) = || || || || + V (z) (3.4) − 2m1 − 2m1 where z = q1, q2 = qR, 1 . We can use the explicit expression for the Lie-Poisson equations for a semidirecth producti h [7]i to obtain the equations of motion

L˙ 1 = +f(z) Im(qR),

L˙ 2 = f(z) Im(qR), (3.5) − L1 L2 q˙R = qR + qR . −m1 m2 In this set of equations f(z) = dV/dz and Im(q) = q q, 1 is the ‘imaginary part’ of a biquaternion q. − h i

3.4 Hyperbolic relative equilibria

∗ The action of left-multiplication by SL2C descends to the right-reduced space in se4C as g (L ,L , q ) = gL g−1, gL g−1, gq g−1 . (3.6) · 1 2 R 1 2 R ∗ 3 ∗ 3 A RE of the system on T CS T CS is a solution contained to an orbit of SL2C SL2C. It must therefore descend to a RE× on the right-reduced space with respect to the group× action above. Combined with Proposition 1.2 we see that RE of the hyperbolic 2-body problem can ∗ ∗ 3 ∗ 3 be found by classifying those RE in se4C which lift to solutions on T H T H . Without any loss of generality, we may suppose the RE are orbits of× one-parameter subgroups generated by S or N given in (3.1). We separate this subsection into two parts to handle both cases separately.

3.4.1 Semisimple RE By diﬀerentiating the action of the one-parameter subgroup g(t) = exp(St) in (3.6) and setting this to equal the equations of motion in (3.5) we obtain

[S,L1] = +f Im(qR), (3.7) [S,L ] = f Im(q ), (3.8) 2 − R L1 L2 [S, qR] = qR + qR . (3.9) −m1 m2 The ﬁrst two equations imply Im(q ) is orthogonal to S = iηI, where we are now using the R − biquaternion notation from (2.1). Therefore, we may suppose qR is a biquaternion of the form u1 + zK. For hyperbolic RE we must have qR, 1 = q1, q2 = cosh ψ, and so it follows that qR is the split-complex number eψj from (3.3).h Equationsi h (3.7)i and (3.8) are now satisﬁed for

L = x I + yJ and L = x I yJ, (3.10) 1 1 2 2 − 9 where f sinh ψ y = , (3.11) 2η and x1 and x2 are complex numbers yet to be determined. Expanding Equation (3.9) using the multiplication of biquaternions results in a linear system of equations in x1 and x2. For ψ = 0 the solutions are uniquely given by 6

m x = iy coth 2ψ + 1 csch 2ψ + im η, 1 m 1 2 (3.12) m2 x2 = iy coth 2ψ + csch 2ψ + im2η. m1

3.4.2 Non-existence of parabolic RE For a parabolic RE we replace S with N in Equations (3.7)–(3.9). By considering the action of the isotropy subgroup of N in SL2C we may suppose that qR is of the form cosh ψ + i sinh ψI. In this case, Equations (3.7) and (3.8) are satisﬁed for Ls = ysJ + zsK where

y iz = y + iz = f sinh ψ. 1 − 1 − 2 2 Expanding Equation 3.9 reveals that we require

ψ −ψ 0 = f sinh ψ(m1e + m2e ).

For a strictly attractive potential f is never zero, and so the equation above cannot hold for ψ = 0, and hence, no such parabolic RE exist. 6 3.5 Reconstruction and classiﬁcation

ψj According to the matrix representation of biquaternions in (2.2) qR = e is Hermitian. For a −1 hyperbolic RE q1 and q2 are also Hermitian, therefore, since qR = q1q2 we see that this implies q1 and q2 each commute with qR. This is only possible if they too belong to the split-complex χ1j −χ2j numbers. Therefore, we must have q1 = e and q2 = e for χ1 and χ2 real numbers satisfying χ1 + χ2 = ψ. For each particle qs we may diﬀerentiate the SL2C-action in (2.6) for the orbit of g(t) = † exp(St) to ﬁnd the velocity vectorq ˙s = Sq + qS . The momentum ps is then given by msq˙s. Recall that this minus sign is a consequence of the negative kinetic energy term in (2.8).− We can then calculate L1 and L2 explicitly and compare this with the expression in (3.10). From this calculation we obtain

xs = ims(η + η cosh 2χs) and y = msη sinh 2χs. (3.13)

By setting these expressions to equal those in Equations (3.11) and (3.12) we obtain the full classiﬁcation of RE for the 2-body problem on H3.

Theorem 3.2. Every semisimple RE for the 2-body problem on H3 is conjugate to a RE generated by the biquaternion S = iηI in sl2C M2(C) for a complex number η. These solutions are classiﬁed up to conjugacy− by the separation⊂ ψ > 0 between the particles and the

10 phase θ = arg η [0, π). For each (θ, ψ) we may suppose q = eχ1j and q = e−χ2j for χ and ∈ 1 2 1 χ2 positive real numbers uniquely determined by χ1 + χ2 = ψ and

m1 sinh 2χ1 = m2 sinh 2χ2.

The modulus of η satisﬁes f sinh ψ η 2 = | | 2ζ where ζ = m1 sinh 2χ1 = m2 sinh 2χ2. Here f = dV/dz where z = cosh ψ and V is the potential. For a strictly attractive potential there are no parabolic RE.

4 Stability

4.1 Some invariant theory We may further reduce the right-reduced space by the residual action of left-multiplication in (3.6) to obtain the full reduced space. To do this we shall consider the (holomorphic) Poisson ∗ reduction of se4C by the SL2C-action. By writing (L1,L2, qR) as (L1,L2, Im(qR), z) the action decomposes into three irreducible sl2C-components and one trivial component. The isomorphism sl2C ∼= so3C intertwines the SL2C-adjoint-representation with the standard vector representation of SO3C. We can therefore employ the First Fundamental Theorem of Invariant Theory to obtain generators of the SL2C-invariant ring, see for instance [5]. If we let (v1, v2, v3) denote (L1,L2, Im(qR)) then the generators of this ring are the pairwise products kij = vi, vj and the determinant h 2 i δ = v3, [v1, v2] . These satisfy the algebraic relations kij = kji and δ = det kij. Theh categoricali quotient for this action is the map

8 π :(L1,L2, qR) ( kij i≤j, δ, z) C . (4.1) 7−→ { } ∈ Unfortunately, this is not an orbit-map since there exist orbits which are not separated by the invariants. Geometric Invariant Theory arises to resolve this issue by restricting to a subset of so-called stable points upon which this map is a geometric quotient, or in other words, an orbit-map [10].

Lemma 4.1. If v1, v2 and v3 are each semisimple elements in sl2C, and not all colinear, then (v1, v2, v3) is stable with respect to the diagonal SL2C-action.

Proof. Consider the action of a one-parameter subgroup C× generated by a semisimple gener- ator S as in (3.1). The space sl2C decomposes as V0 V−1 V+1 where the action on Vw is w ⊕ ⊕ t v = t v. Note that V+1 and V−1 must therefore be isotropic subspaces with respect to the complex· bilinear form , . The Hilbert-Mumford criterion tells us that a point is stable if there h i does not exist any such one-parameter subgroup for which v1, v2, and v3 all belong to either V0 V−1 or V0 V+1. Suppose vi and vj belong to such a subspace. This subspace is degenerate with⊕ respect to⊕ , and so 4(k k k2 ) = det([v , v ]) must be zero. By semisimplicity of v h i ii jj − ij i j i and vj this implies [vi, vj] = 0, and therefore, vi and vj are colinear.

11 4.2 The full reduced space

∗ 3 ∗ 3 Let M denote the subset of stable points of T CS T CS with respect to the G = SL2C × × SL2C-action. The elements L1, L2, and Im(qR) are always semisimple on the hyperbolic real form. Moreover, they are only ever colinear when the SL2C-action fails to be locally free, and hence, by Proposition 3.1 we see that the real form M R is

(T ∗H3 T ∗H3) Crit J.b × \ The group G acts freely on M R, and therefore, thanks to R-compatibility, the intersection of a G-orbit with M R is an orbit of Gρ. This implies that M R , M descends to an injection of orbit spaces M R/Gρ , M/G. It follows that π restricted to →M R deﬁnes an orbit map for the →∗ 3 ∗ 3 action of SL2C on (T H T H ) Crit Jb. In fact, although we shall× not pursue\ this further, we remark that because the action of G is R-compatible, R descends to a real structure Re on the reduced space M/G. The real reduced space M R/Gρ belongs to the ﬁxed-point set of Re which we call a real-Poisson form. For our purposes, the most important consequence of this is that the symplectic leaves of M R/Gρ are real-symplectic forms of the holomorphic symplectic leaves in M/G [2, Theorem 2.2].

4.3 Degenerate RE Thanks to the classiﬁcation in Theorem 3.2 we may parameterise the hyperbolic RE in the full reduced space by (θ, ψ). This parameterises a surface Fix H of ﬁxed-points in the full reduced space. The critical values of the energy-Casimir map

R ρ 2 µ: M /G R C;( kij i≤j, δ, z) (H, Ctot ) → × { } 7−→ || || are precisely the image of Fix H. We call this image the energy-Casimir diagram. For the gravitational potential in (2.10) this image is included in Figure 2. In practice, to obtain this image it is ﬁrst necessary to show that

−1 ψ ζ = m1m2Z e sinh 2ψ (4.2) where p 2ψ 2ψ Z = (m1 + m2e )(m2 + m1e ). (4.3) 2 One can then express χ1, χ2, and η in terms of (θ, ψ) and use (3.13) to carry out the computation. Observe that this image| is| pinched along a singular cusp. As the next proposition shows, this is indicative of degenerate RE. The proof is immediate.

Proposition 4.2. Let P be a Poisson manifold and suppose the symplectic leaf M through x is the regular level set of a Casimir function C. Consider the Hamiltonian flow generated by H and suppose Fix H is an immersed submanifold containing x. The subspace ker Dx(C Fix H ) TxM is fixed by the linearised flow of H at x. | ⊂

As a corollary we see that the cusp in Figure 2 must correspond to degenerate RE. Indeed, since this cusp consists of singular values of µ restricted to Fix H, they must also be singular values of the Casimir map restricted to Fix H.

12 1 −

0 0 2 Re Ctot || || Im C 2 || tot||

Figure 2: Energy-Casimir diagram for m1 = 1 and m2 = 2. The lines emanating from the focal point are curves of constant θ and those transversal to them are of constant ψ.

Γ > 0 ψcrit

Γ < 0

0 π θ

Figure 3: The curve Γ(θ, ψ) = 0 of degenerate RE.

13 Proposition 4.3. A RE with separation ψ and phase θ is degenerate if and only if 1 + tanh ψ Γ(θ, ψ) = (m + m )Z cos 2θ + (m + m cosh 2ψ)(m + m cosh 2ψ) (4.4) 1 2 2 cosh ψ 1 2 2 1 is equal to zero. These RE correspond to the singular points along the cusp of the energy-Casimir diagram.

Proof. Thanks to our formulation of the problem as the complexiﬁcation of the 2-body problem on the sphere, the fully reduced equations of motion in C8 coincide with those in Equation 2.10 of [2]. These may be linearised at a ﬁxed-point in a symplectic leaf. A RE is degenerate if and only if the constant term of the characteristic polynomial of this linearisation is zero. This polynomial is derived in Equation 3.16 of [2], however for a hyperbolic RE in Fix H we must remember to replace every instance of a trigonometric function with its hyperbolic counterpart in ψ. The constant term c0 is then given by 2 k11 k22 2 k11 m2 k22 m1 2 2 2 2 +2 coth θ csch θ 1 + + 1 + + (m1 + m2) coth θ csch θ . m1 − m2 m1 m1 m2 m2 The expressions in (3.13) allow us to write

k = 2 η 2m2(cos 2θ + cosh 2χ ), 11 − | | 1 1 which can then be rewritten using (4.2) and (4.3) as

2 m1 −ψ k11 = Ze cos 2θ + (m1 cosh 2ψ + m2) . −2 sinh3 ψ cosh ψ

A similar expression holds for k22. If we substitute this into the constant term above we ﬁnd −ψ 6 that c0 = 2e Γ(θ, ψ) csch ψ. In Figure 3 we illustrate the solutions to Γ(θ, ψ) = 0. As this deﬁnes a connected− curve in (θ, ψ)-space it must correspond exactly to the degenerate RE along the singular cusp in Figure 2.

4.4 The Energy-Casimir method If C = C 2 is real then by Proposition 3.1 the orbit-reduced space may be identiﬁed with || tot|| the space of SL2C-orbits which intersect

∗ 2 ∗ 2 2 (q1, p1, q2, p2) T H T H Ctot sl2R, Ctot = C . (4.5) ∈ × | ∈ || ||

Observe that q1, p1, q2, p2 spans the space of real, symmetric matrices in M2(C). From (2.6) we { } see that the orbits contained within this space are equivalently the orbits of the subgroup SL2R. 2 Therefore, thanks to the isomorphism PSL2R ∼= SO(1, 2) the reduced space for C = Ctot real is identical to the orbit-reduced space for the 2-body problem on H2 with Casimir||C. || The Hessian and linearisation of the reduced Hamiltonian for the problem on H2 is found in [3]. We can therefore apply a continuity argument to extend their results to H3, noting that the behaviour of the ﬁxed-points can only change when they cross a degenerate point whereupon an eigenvalue becomes zero. Theorem 4.4. The stability of a RE for the gravitational 2-body problem on H3 is determined by Γ(θ, ψ) as follows:

14 1. For Γ < 0 the Hessian of the Hamiltonian at the ﬁxed-point in reduced space is positive deﬁnite, and thus, the RE is Lyapunov stable.

2. For Γ > 0 the Hessian at the ﬁxed-point in reduced space has signature (+ + + ), and thus, the RE is linearly unstable. −

In particular, a RE is unstable whenever cos 2θ > 0, or whenever ψ > ψcrit.

References

[1] P. Arathoon. Singular reduction of the 2-body problem on the 3-sphere and the 4- dimensional spinning top. Regular and Chaotic Dynamics, 24(4):370–391, 2019.

[2] P. Arathoon and M. Fontaine. Real forms of holomorphic Hamiltonian systems. arXiv e-prints, 2020. arXiv:2009.10417.

[3] A. V. Borisov, L. C. Garc´ıa-Naranjo, I. S. Mamaev, and J. Montaldi. Reduction and relative equilibria for the two-body problem on spaces of constant curvature. Celestial Mechanics and Dynamical Astronomy, 130(6):43, 2018.

[4] F. Diacu, E. P´erez-Chavela, and J. G. Reyes. An intrinsic approach in the curved n-body problem: the negative curvature case. Journal of Diﬀerential Equations, 252(8):4529 – 4562, 2012.

[5] W. Fulton and J. Harris. Representation theory: a ﬁrst course, volume 129. Springer Science & Business Media, 2013.

[6] L. C. Garc´ıa-Naranjo, J. C. Marrero, E. Pérez-Chavela, and M. Rodr´ıguez-Olmos. Clas- sification and stability of relative equilibria for the two-body problem in the hyperbolic space of dimension 2. Journal of Differential Equations, 260(7):6375–6404, 2016.

[7] D. D. Holm, J. E. Marsden, and T. S. Ratiu. The Euler–Poincar´eequations and semidirect products with applications to continuum theories. Advances in Mathematics, 137(1):1–81, 1998.

[8] J. E. Marsden. Lectures on Mechanics. London Mathematical Society Lecture Note Series. Cambridge University Press, 1992.

[9] J. E. Marsden, G. Misiolek, J. P. Ortega, M. Perlmutter, and T. S. Ratiu. Hamiltonian reduction by stages. Springer, 2007.

[10] P. E. Newstead. Introduction to moduli problems and orbit spaces. Narosa Publishing House, New Delhi, 2012.

[11] L. O’Shea and R. Sjamaar. Moment maps and Riemannian symmetric pairs. Mathematis- che Annalen, 317(3):415–457, 2000.

P. Arathoon, University of Manchester [email protected]