The conjugate locus for the Euler top I. The axisymmetric case

Abstract

The flow of the Euler top is a geodesic flow on SO(3) with a left invariant metric. We determine the conjugate locus for this geodesic flow in the case that the metric has an S 1 invariance, which is the case when two of the three moments of inertia are equal. Depending on the ratios of these moments, the conjugate locus is either a segment or circle (if the body is oblate) or a non-injective mapping of an astroid of revolution (if the body is prolate).

Conjugate points for systems of second-order ordinary differential equations

“…This coincides with results in the literature (see e.g. [1,22]). The conjugate time (for a general geodesic, not necessarily a relative equilibrium) is given in e.g.…”

Cut locus of a left invariant Riemannian metric on SO(3) in the axisymmetric case

“…Below we consider only the Lagrange case I 1 = I 2 (the case I 1 = I 2 = I 3 is called the Euler case). In the Lagrange case the Hamiltonian system (1) is integrated in elementary functions [2]:…”

“…Parametrization of left invariant Riemannian geodesics on SO 3 is a classical L. Euler's result [1]. L. Bates and F. Fassò [2] found an equation for the conjugate time (in the Lagrange case) and the conjugate locus depending on the ratio of eigenvalues. Thus local optimality of geodesics was studied.…”

Riemannian metrics on 2D-manifolds related to the Euler−Poinsot rigid body motion

“…Very recently a detailed analysis was done for geodesic flows of Riemannian metrics on 2-spheres of revolution, wich are reflectionally symmetric with respect to the equator [6,25] with an application to the case of an ellipsoid of revolution. It was extended to a general ellipsoid [16], such computations were also done in the geometric control context [8,11] and in [3] for the axisymmetric ellipsoid of inertia.…”

“…In such a representation the Hamiltonian H, which does not depend explicitly on w and p w , describes a Riemannian metric in the (x, y)-variables. In particular, this reduction is crucial for the integration of the system and for the analysis of the Jacobi equation associated to the left-invariant metric on SO (3).…”