THE SERRET-ANDOYER RIEMANNIAN METRIC AND EULER-POINSOT RIGID BODY MOTION

The Euler-Poinsot rigid body motion is a standard mechanical system and is the model for left-invariant Riemannian metrics on SO(3). In this article, using the Serret-Andoyer variables we parameterize the solutions and compute the Jacobi fields in relation with the conjugate locus evaluation. Moreover the metric can be restricted to a 2D surface and the conjugate points of this metric are evaluated using recent work [4] on surfaces of revolution.

where M = (M 1 , M 2 , M 3 ) is the angular momentum of the body measured in a specific moving frame. The motion is Liouville integrable and every trajectory is confined by the first integrals to a two-dimensional torus on which the motion is pseudo-periodic and characterized by two frequencies described using the Euler-Poinsot representation. Moreover the integral curves can be computed using elliptic integrals of the first and third kind, see [11].
A set of symplectic variables was introduced by Serret-Andoyer (see the survey [7]) to reduce the Hamiltonian to the form H (g, k, l, G, K, L) = 1 2 where G, K, L are the dual variables associated to g, k, l. If I 1 = I 2 the Hamiltonian depends only on momenta, but in general a further transformation is required to integrate the motion using Hamilton-Jacobi method. Finally such a transformation can be easily related to the standard action-angle variables to represent the rigid body motion [10] (they are crucial in perturbation analysis). The aim of this article is to study the Euler-Poinsot motion from the geometric optimal control point of view. Indeed such a system is related to the attitude control problem of a satellite [3] (assuming a direct control of the angular velocity by impulse torques) and the limit case I 1 → +∞ defines a left-invariant SR-metric on SO(3) [15] and is associated to the dynamics of spin systems [16]. In this framework the important and difficult problems are to determine the conjugate and cut loci: fixing R 0 ∈ SO(3), the conjugate locus C(R 0 ) is formed by the extremities of extremal curves where optimality is lost for the C 1 -neighboring curves while the cut locus C cut (R 0 ) is formed by extremities of extremals where the optimality is lost globally. Besides the interest of such computations in optimal control, the analysis of conjugate loci in Riemannian manifold has a long history in geometry, which goes back to Jacobi's study of the conjugate locus on ellipsoids, see [8] for recent advances. The conjugate locus for the rigid body in the case of two equal moment of inertia was descrided very recently in [2].
The main contribution of this article is to describe the conjugate locus of the Serret-Andoyer metric g a = dx 2 z(y) + associated to the restriction of H to the (g, l, G, L)-space and the principal moments of inertia being oriented according to I 1 > I 2 > I 3 . The analysis relies on recent studies of metrics on surfaces of revolution [13] and refined developments on twospheres of revolution [4], [14]. The key tool is to reduce the metric to the polar form g = dϕ 2 + m(ϕ)dθ 2 which is used to evaluate the Jacobi fields and the conjugate locus. The interest of interpreting the Serret-Andoyer metric in a geometric framework of surfaces of revolution is to show the analogy with the conjugate locus computations on oblate ellipsoids of revolution. It is a first step toward the computations of the conjugate locus for left-invariant metrics on SO(3) through the Jacobi fields parameterized using the polar normal form.
2. Riemannian metrics on surfaces of revolution. The objective of this section is to introduce the concepts and to recall the properties of the metrics on THE SERRET-ANDOYER METRIC AND EULER-POINSOT RIGID BODY MOTION   3 surfaces of revolution [13], presenting also the recent developments to compute the conjugate locus on two-spheres of revolution [4], [14].
2.1. Generalities of Riemannian metrics on surfaces of revolution. Taking a chart (U, q) the metric can be written in polar coordinates as One use Hamiltonian formalism on T * U, ∂ ∂p is the vertical space, ∂ ∂q is the horizontal space and α = pdq is the (horizontal) Liouville form. The associated Hamiltonian is and we denote exp t #-H the one-parameter group. Parameterizing by arc-length amounts to fix the level set to H = 1/2. Extremal solutions of #-H are denoted γ : t → (q(t, q 0 , p 0 ), p(t, q 0 , p 0 )) and fixing q 0 it defines the exponential mapping exp q0 : (t, p 0 ) → q(t, q 0 , p 0 ) = Π(exp t #-H(q 0 , p 0 )) where Π : (q, p) → q is the standard projection. Extremals are solutions of the equations , dp θ dt = 0. Assumptions. In the sequel we shall assume the following • (A1) ϕ = 0 is a parallel solution and the corrresponding parallel is called the equator. • (A2) The metric is reflectionally symmetric with respect to the equator: m(−ϕ) = m(ϕ).
Parameterizing by arc-length, one gets dϕ dt BERNARD BONNARD, OLIVIER COTS AND NATALIYA SHCHERBAKOVA By symmetry, one can assume θ(0) = 0, p θ ∈ [0, m(ϕ(0)] and dϕ dt (0) ≥ 0. Moreover one restricts our analysis to extremals such that ϕ(0) = 0. Arc-length parameterized extremal γ is defined by t → (ϕ(t, p θ ), θ(t, p θ )) and dϕ dt > 0 corresponds to an increasing branch and dϕ dt < 0 a decreasing branch. One has and for an increasing branch one can parameterize θ by ϕ and we get and we denote J(t) a Jacobi field, that is a non trivial solution of Jacobi equation. The time t c is said to be conjugate to t = 0 if there exists a Jacobi field J(t) = (δq(t), δp(t)) such that δq(0) = δq(t c ) = 0.
If γ is parameterized by arc-length two Jacobi fields are crucial in our analysis.
Then in this interval there exists no conjugate times.
Let I = (0, m(0)) be a positive interval such that for any p θ ∈ I, the derivative of the increasing branch starting at the equator vanishes at time T /4 and ϕ + = ϕ(T /4). Then the trajectory t → ϕ(t, p θ ) is periodic with period T given by and the first return to the equator is at time T /2 and the variation of θ is given by Definition 2.4. The mapping p θ ∈ I → T (p θ ) is called the period mapping and R : p θ → ∆θ is called the first return mapping.
Definition 2.5. The extremal flow is called tame on I if the first return mapping R is such that R < 0.
Proposition 2. For extremal curves with p θ ∈ I, in the tame case there exists no conjugate times on (0, T /2).
Proof. As in [4] if R < 0, the extremal curves initiating from the equator with p θ I are not intersecting before returning to the equator. As conjugate points are limits of intersecting extremal curves, conjugate points are not allowed before returning to the equator.
Assumptions. In the tame case we assume the following (A3) At the equator the Gauss curvature is positive and maximum.
Lemma 2.6. Under assumption (A3), the first conjugate point along the equator is at time π/ G(0) and realizes the minimum distance to the cut locus C cut (0). It is a cusp point of the conjugate locus.
Parameterization of the conjugate locus under assumptions (A1-2-3) for p θ ∈ I and ϕ(0) = 0. The conjugate locus will be computed by continuation, starting from the cusp point at the equator. Let p θ ∈ I and t ∈ (T /2, T /2 + T /4). One has the formulae According to proposition 7.2.1 [13] conjugate times are given by the relation Hence we deduce the following.
Lemma 2.7. For p θ ∈ I and conjugate times between (T /2, (1). One notes p θ → ϕ 1c (p θ ) the solution of the equation initiating from the equator. Differentiating one has

Analysis of equation
One can easily check that ∂f ∂p θ > 0 and ∂ 2 f Proposition 3. If ∆θ > 0 on I, then ∂ϕ1c ∂p θ = 0 and the curve p θ → (ϕ 1c (p θ ), θ 1c (p θ )) is a curve defined for p θ ∈ I and with no loop in the plane (ϕ, θ). In particular it is without cusp point.
Proof. By definition, the conjugate locus for p θ ∈ I is the envelop of the extremal curves initiating from the equator. One extremity is the cusp point at the equator and the curve can be continuated since it is geometrically clear that the conjugate point is located on (T /2, T /2 + T /4), that is before the second zero of dϕ dt . To simplify the computations one use [4].
Lemma 2.8. We have: Computations on the ellipsoid of revolution: the oblate case. The ellipsoid of revolution with respect to the z-axis is generated by the curve: y = sin ϕ, z = ε cos ϕ where 0 < ε < 1 corresponds to the oblate case while ε > 1 is the prolate case. The restriction of the Euclidian metric is where F 1 = cos 2 ϕ + ε 2 sin 2 ϕ, F 2 = sin 2 ϕ. The case ε = 1 is the round sphere and we shall restrict to the oblate case. In this case the Gauss curvature is positive and increasing from the north pole to the equator.
Hence the formulae for the period mapping is We introduce: Hence we have: where Z = cn u. Using [11] one gets: = 4αE(m).

BERNARD BONNARD, OLIVIER COTS AND NATALIYA SHCHERBAKOVA
A straightforward computation gives us Hence the conjugate locus can be continued from the horizontal cusp point at the equator to the vertical cusp point at the meridian θ = π , when p θ → 0. (Observe lim p θ →0 R(p θ ) = π ). By symmetry one gets easily the standard astroidal shape of the conjugate locus.
Observe the difference with the prolate case.
In this case the curvature is minimum at the equator, the first return mapping is increasing and the miminum is at p θ → 0 and is equal to π. In this case the conjugate locus is constructed by continuation from a vertical cusp at θ = π to an horizontal cusp at the equator.
In both cases the cut loci can be determined from the conjugate locus and the symmetries of the extremal flow: a segment of the equator in the oblate case versus a segment of the meridian in the prolate case.

Remark 2.
In the oblate case, one can similarly evaluate the different conjugate loci corresponding to i th -conjugate points, i = 1, 2, . . . . For instance for the second locus, we replace in lemma 2.7, θ by the formulae for t ∈ [T, T + T /4], ϕ > 0 and the "four cusps Jacobi conjecture of the conjugate loci" can be checked as an exercice left to the reader.

Left-invariant metrics on SO(3) and the Serret-Andoyer formalism in
Euler-Poinsot rigid body motion.

Left-invariant metrics on SO(3).
We recall the geometric framework using [9], [1]. We note (e 1 , e 2 , e 3 ) the fixed frame and (E 1 (t), E 2 (t), E 3 (t)) the moving frame attached to the body and formed by principal axis. The position of the body is represented by the matrix R(t) = (E 1 (t), E 2 (t), E 3 (t)) and is solution of dR dt = i=1,3 u i RA i . The motion is obtained by minimizing T 0 Ldt, with L = i=1,3 I i u 2 i where the principal moment of inertia in the distinct case are oriented using I 1 > I 2 > I 3 . The rigid body dynamics can be derived using Pontryagin maximum principle and appropriate coordinates. Le ξ be an element of T * R (SO(3)) and denotes H i = ξ(RA i ), i = 1, 2, 3 the symplectic lifts of the vector fields RA i . The pseudo-Hamiltonian takes the form and the maximization condition of the maximum principle implies ∂H ∂u = 0. Hence u i = H i /I i and plugging such u i into H we get the true Hamiltonian H e = 1 2 Note that the vector H=(H 1 , H 2 , H 3 ) represents the angular momentum of the body measured in the moving frame and related by H i = I i Ω i to the angular velocity Ω = (Ω 1 Ω 2 , Ω 3 ).
The Euler equation describing the evolution of the angular velocity is and {, } denotes the Poisson bracket. The Lie bracket of two matrices is computed using the convention [A, B] = AB − BA and we have the relation and Euler equation is The following proposition is standard.

Proposition 4. The Euler equation is integrable by quadratures using the two first integrals: the Hamiltonian H e and the Casimir
The solutions of Euler equations are called polhodes in classical mechanics. The limit case with two or three equal principal moments of inertia can be treated similarly. Also, as pointed in [9], the above calculations hold for every left-invariant Hamiltonian H = f (H 1 , H 2 , H 3 ) on SO(3). In particular the SR-case is derived next as a limit of the Riemannian case.

3.2.
The sub-Riemannian case. Setting u 1 = εv 1 , with ε → 0, one gets a control system with two inputs only. Since u i = H i /I i this is equivalent to I 1 → +∞. Then with the corresponding Euler-Lagrange equation. This is the model for left-invariant SR-metrics on SO(3) depending upon one parameter k 2 = I 2 /I 3 .
Once the Euler equation is integrated the next step is to parameterize the solution. It relies on the following general property [9]. Note that he additional first integrals are simply deduced from Noether theorem in Hamiltonian form.

Explicit integration and Euler angles.
Euler angles are introduced on SO(3) to complete the computations. They are denoted φ 1 , φ 2 , φ 3 and defined using the following convention: As it is shown in [9], the angles φ 2 and φ 3 can be founded using the relations while φ 1 is computed by integrating the equation The following proposition is useful for the computations.

The Serret-Andoyer variables and the associated metric. Euler-Poinsot
Hamiltonian can be computed in the symplectic Andoyer variables, see [7]. The moment of inertia are oriented according to I 1 > I 2 > I 3 and the Andoyer variables are denoted by (g, k, l, G, K, L) where G, K, L denote the canonical impulses associated to (g, k, l) . They are defined by Hence G = |H| and the Hamiltonian takes the form H e (g, k, l, G, K, L) = 1 2 The complete relations between Andoyer variables and Euler angles are: In the sequel we use the following notations for the Andoyer representation where p x = G = |H|, y = l = arctan H 1 H 2 , z = 2 A sin 2 y + B cos 2 y , , and the Hamiltonian takes the form H e (x, y, w, p x , p y , p w ) = 1 2 Thus H e is associated to the so-called Riemannian Serret-Andoyer metric Remark 4. This metric is not related to the original metric on SO(3) since the symplectic transformation mixed state variables and impulses. In particular, conjugate points of both metrics are not in correspondance. Nevertheless we have the following. Proof. This is clear since Andoyer variables are symplectic coordinates.
3.5. The pendulum representation. The dynamics of rigid body is well understood using Euler-Poinsot interpretation. The polhodes describes the evolution of the angular velocity in the moving frame and are contained in algebraic curves intersecting the ellipsoids AH 2 1 + BH 2 2 + CH 2 3 = c 1 with the spheres of constant angular momentum H 2 1 + H 2 2 + H 2 3 = c 2 . They form periodic curves except pair of separatrices contained in the planes H 3 = ± B−A C−B H 1 . Besides those remarkable properties they can be parameterized using Jacobi or Weierstrass elliptic functions, see [11]. In particular the Euler equation can be used to define the cn − dn Jacobi elliptic functions. To complete the integration one uses Euler-Poinsot interpretation. According to this representation the end-point of the angular velocity in a fixed frame describes a curve in a plane orthogonal to the angular momentum. Such a curve is called an herpolhode in mechanics. Its description by an angle and its parameterization using the elliptic integral of the third kind Π is given in [11]. Altogether this gives a pseudo-periodic motion described by two frequencies.
Serret-Andoyer representation leads to a different geometric interpretation that we describe next.
Using H e (x, y, w, p x , p y , p w ) = 1 2 z 2 p 2 x + C − z 2 p 2 y one gets the equations dx dt = p x A sin 2 y + B cos 2 y , dp x dt = 0, completed by the trivial equations dw dt = dp w dt = 0.
The reduced system (8) describes the evolution of the extremals of the Andoyer metric. One has dy dt = p y (C − z 2 ) and the isoenergetic curves H e = h gives us where h > 0 since H e > 0. By homogeneity one can set h = 1 2 which amounts to parameterize the extremals by arc-length. The Hamiltonian function is π-periodic with respect to the y-variable. It verifies the following relations: H e (y, p y ) = H e (y, −p y ), H e (y, p y ) = H e (−y, p y ).
The equilibrium points are p y = 0, y = kπ 2 . In the neighborhood of the point y = p y = 0 the eigenvalues of the linearized system are solutions of λ 2 = (B − A)(C − B)p 2 x and they are real since A < B < C and in the neighborhood of y = π 2 , p y = 0 they are purely imaginary. In consequence in order to parameterize all phase trajectories in the plane (y, p y ) it is sufficient to integrate with y(0) = π 2 . To resume we have.
Proposition 7. The Euler-Poinsot motion projects in the (y, p y ) plane into a pendulum motion which can be ineterpreted on the cylinder y ∈ [0, π], with a stable equilibrium at y = π 2 and an an unstable at y = 0. They exist two types of periodic trajectories: oscillating trajectories homotopic to zero and rotating trajectories. The non periodic trajectories are separatrices joining 0 to π and represent separating polhodes. The trajectories are reflectionally symmetric with respect to the two axes: y = 0 and p y = 0.

4.2.
Geometric analysis. The Hamiltonian associated to the metric is H = 1 2 (p 2 ϕ + p 2 θ m(ϕ) ) and parameterizing using H = 1 2 one gets the mechanical system ( dϕ dt ) 2 + V (ϕ, p θ ) = 1 where the potential is given by The potential is symmetric with respect to the equator ϕ = 0: V (ϕ) = V (−ϕ) and is periodic of period 2K(k) α . It has a minimum at ϕ = 0 given by p 2 θ A and a maximum at ϕ = K α given by p 2 θ B and is monotonic on [0, K α ]. To compute the extremals starting from the equator ϕ = 0 we proceed as follows. One can restrict to p θ ≥ 0 and one must have p θ ∈ [0, √ I 1 ]. We have two types of solutions associated to the pendulum representation : • Rotating trajectories: for physical solution one has p θ ∈ ( √ I 3 , √ I 2 ) but to get a complete metric they are extended to p θ ∈ (0, √ I 2 ). • Oscillating trajectories: Except the separatrices all the trajectories are periodic and due to the symmetry with respect to the equator ϕ = 0 it is sufficient to parameterize the following: • For oscillating trajectories denote ϕ + the intersection of the potential with the level set 1. We parameterize the branch between [0, ϕ + ]. • For rotating trajectories, one may assume ϕ ∈ [0, K α ].
To integrate, we use a specific homographic integration because the roots are explicit. Besides the geometric interest it is related to the uniformization of the computations of the conjugate locus in relation with [5]. We proceed as follows. 4.3.1. Parameterization of oscillating trajectories. We have for p θ ∈ ( √ I 2 , √ I 1 ) four distinct roots for the polynomial at the right-hand member of (10): 0 < Y + < 1 < 1 k 2 and for the non fixed roots we use the notation Y 1 = Y + and Y 2 = 1 k 2 and the motion can be restricted to the interval [0, Y 1 ]. The equation (10) takes the form dY dt and we set and η is monotone increasing from 0 to 1. We have and after simplication the equation (11) takes the normal form dη dt with Denoting one gets the parameterization η(t) = cn(M t + ψ 0 , m).
Therefore we have To integrate the θ-variable one uses the following relations [11], p. 68.