29 research outputs found
Superintegrability of Sub-Riemannian Problems on Unimodular 3D Lie Groups
Left-invariant sub-Riemannian problems on unimodular 3D Lie groups are
considered. For the Hamiltonian system of Pontryagin maximum principle for
sub-Riemannian geodesics, the Liouville integrability and superintegrability
are proved
Existence of planar curves minimizing length and curvature
In this paper we consider the problem of reconstructing a curve that is
partially hidden or corrupted by minimizing the functional , depending both on length and curvature . We fix
starting and ending points as well as initial and final directions.
For this functional we discuss the problem of existence of minimizers on
various functional spaces. We find non-existence of minimizers in cases in
which initial and final directions are considered with orientation. In this
case, minimizing sequences of trajectories can converge to curves with angles.
We instead prove existence of minimizers for the "time-reparameterized"
functional \int \| \dot\gamma(t) \|\sqrt{1+K_\ga^2} dt for all boundary
conditions if initial and final directions are considered regardless to
orientation. In this case, minimizers can present cusps (at most two) but not
angles
Minimization of length and curvature on planar curves
In this paper we consider the problem of reconstructing a curve that is partially hidden or corrupted by minimizing the functional ∫ √1+K 2 ds, depending both on length and curvature K. We fix starting and ending points as well as initial and final directions. For this functional, we find non-existence of minimizers on various functional spaces in which the problem is naturally formulated. In this case, minimizing sequences of trajectories can converge to curves with angles. We instead prove existence of minimizers for the "time-reparameterized" functional ∫γ(t)√1+Kγ2 dt for all boundary conditions if initial and final directions are considered regardless to orientation. ©2009 IEEE
Hopf fibration: geodesics and distances
Here we study geodesics connecting two given points on odd-dimensional
spheres respecting the Hopf fibration. This geodesic boundary value problem is
completely solved in the case of 3-dimensional sphere and some partial results
are obtained in the general case. The Carnot-Carath\'eodory distance is
calculated. We also present some motivations related to quantum mechanics.Comment: 22 pages, 6 figure
Sub-Riemannian structures on 3D Lie groups
We give the complete classification of left-invariant sub-Riemannian
structures on three dimensional Lie groups in terms of the basic differential
invariants. This classifications recovers other known classification results in
the literature, in particular the one obtained in [Falbel-Gorodski, 1996] in
terms of curvature invariants of a canonical connection. Moreover, we
explicitly find a sub-Riemannian isometry between the nonisomorphic Lie groups
and , where denotes the
group of orientation preserving affine maps on the real line
On 2-step, corank 2 nilpotent sub-Riemannian metrics
In this paper we study the nilpotent 2-step, corank 2 sub-Riemannian metrics
that are nilpotent approximations of general sub-Riemannian metrics. We exhibit
optimal syntheses for these problems. It turns out that in general the cut time
is not equal to the first conjugate time but has a simple explicit expression.
As a byproduct of this study we get some smoothness properties of the spherical
Hausdorff measure in the case of a generic 6 dimensional, 2-step corank 2
sub-Riemannian metric
Maxwell strata in sub-Riemannian problem on the group of motions of a plane
The left-invariant sub-Riemannian problem on the group of motions of a plane
is considered. Sub-Riemannian geodesics are parametrized by Jacobi's functions.
Discrete symmetries of the problem generated by reflections of pendulum are
described. The corresponding Maxwell points are characterized, on this basis an
upper bound on the cut time is obtained