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

Abstract

The Euler−Poinsot rigid body motion is a standard mechanical system and it is a 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 works on surfaces of revolution. Another related 2D-metric on S 2 associated to the dynamics of spin particles with Ising coupling is analysed using both geometric techniques and numerical simulations.. * Work supported by ANR Geometric Control Methods (project No. NT09 504490).

Training Schrödinger’s cat: quantum optimal control

“…Significant progress has been made in understanding of how to optimally control coupled spin systems with more than two spins [65,71,86,87,[313][314][315][356][357][358][359][360][361][362][363][364][365][366][367][368][369][370][371][372][373][374]. Recent advances include robust broadband and band-selective pulses in NMR and ESR.…”

Tracking of Lines in Spherical Images via Sub-Riemannian Geodesics in $${\text {SO(3)}}$$ SO(3)

“…In (39), the horizontal part follows from (36) and (31). It is known that the Hamiltonian system (39) is Liouville integrable [27,25], and it was explicitly integrated in [19,22]. In the next subsections, we classify the possible solutions by values of the parameter ξ, and we adapt the explicit solution to our coordinate chart (x, y, θ) ∈ M, where we follow the analogy to the closely related problem in SE (2).…”

“…In the next subsections, we classify the possible solutions by values of the parameter ξ, and we adapt the explicit solution to our coordinate chart (x, y, θ) ∈ M, where we follow the analogy to the closely related problem in SE (2). This allows us to obtain simpler formulas than in [19,22].…”

“…Here we describe the domains of parameter ξ > 0, which correspond to different dynamics of the Hamiltonian system (39). It is well known (see, e.g., [22]) that this system has two integrals of motion that depend only on momentum components h i (see illustration in Fig. 7)…”

“…The cut locus in a special case of the bi-invariant metric (for ξ = 1) was obtained in [15]. In [22], conjugate locus in the Riemannian case was constructed, and by considering a SR metric as a limiting case of the Riemannian metric, the corresponding formulas for the conjugate locus in SR case could be obtained.…”