The Lorentzian Problem on the Group $SU(2)$
We consider a left-invariant Lorentzian problem on SU(2). The following questions are addressed: complete controllability, existence of length maximizers, extremals, curvature.
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arXiv subjects
Yu. L. Sachkov
Publications and source records attributed to Yu. L. Sachkov.
Sectional Curvature and Structural Functions of a Two-Dimensional or Three-Dimensional Lorentzian Manifold
We derive a formula for the sectional curvature on an arbitrary two-dimensional or three-dimensional smooth manifold equipped with a Lorentzian metric.
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Sectional curvature and structure functions of a two-dimensional Lorentzian manifold
In this note we obtain a formula for the sectional curvature on an arbitrary two-dimensional smooth manifold $M$ equipped with a Lorentzian metric $g$.
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Lorentzian problems on the Heisenberg group
Three left-invariant Lorentzian problems on the Heisenberg group are considered. The Pontryagin maximum principle was applied to both problems and a parameterization of abnormal and normal extremal trajectories was obtained. Reachability sets and the existence of optimal trajectories are investigated.
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Lorentzian anti-de Sitter plane
In this paper, we study a two-dimensional Lorentzian problem on the anti-de Sitter plane. Using methods of geometric control theory and differential geometry, it was possible to construct an orthonormal frame, calculate extremal trajectories, describe the reachable set, construct an optimal synthesis, and describe the Lorentzian distance.
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Flat sub-Lorentzian structures on Martinet distribution
Two flat sub-Lorentzian problems on the Martinet distribution are studied. For the first one, the attainable set has a nontrivial intersection with the Martinet plane, but for the second one it does not. Attainable sets, optimal trajectories, sub-Lorentzian distances and spheres are described.
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Lorentzian distance on the Lobachevsky plane
Left-invariant Lorentzian structures on the 2D solvable non-Abelian Lie group are studied. Sectional curvature, attainable sets, Lorentzian length maximizers, distance, spheres, and infinitesimal isometries are described.
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Sub-Lorentzian distance and spheres on the Heisenberg group
The left-invariant sub-Lorentzian problem on the Heisenberg group is considered. An optimal synthesis is constructed, the sub-Lorentzian distance and spheres are described.
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Extremals for a series of sub-Finsler problems with 2-dimensional control via convex trigonometry
We consider a series of optimal control problems with 2-dimensional control lying in an arbitrary convex compact set $Ω$. The considered problems are well studied for the case when $Ω$ is a unit disc, but barely studied for arbitrary $Ω$. We derive extremals to these problems in general case by using machinery of convex trigonometry, which allows us to do this identically and independently on the shape of $Ω$. The paper describes geodesics in (i) the Finsler problem on the Lobachevsky hyperbolic plane; (ii) left-invariant sub-Finsler problems on all unimodular 3D Lie groups (SU(2), SL(2), SE(2), SH(2)); (iii) the problem of rolling ball on a plane with distance function given by $Ω$; (iv) a series of "yacht problems" generalizing Euler's elastic problem, Markov-Dubins problem, Reeds-Shepp problem and a new sub-Riemannian problem on SE(2); and (v) the plane dynamic motion problem.
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Maxwell Strata and Cut Locus in Sub-Riemannian Problem on Engel group
We consider the nilpotent left-invariant sub-Riemannian structure on the Engel group. This structure gives a fundamental local approximation of a generic rank 2 sub-Riemannian structure on a 4-manifold near a generic point (in particular, of the kinematic models of a car with a trailer). On the other hand, this is the simplest sub-Riemannian structure of step three. We describe the global structure of the cut locus (the set of points where geodesics lose their global optimality), the Maxwell set (the set of points that admit more than one minimizer), and the intersection of the cut locus with the caustic (the set of conjugate points along all geodesics). The group of symmetries of the cut locus is described: it is generated by a one-parameter group of dilations $\mathbb{R}_+$ and a discrete group of reflections $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$. The cut locus admits a stratification with 6 three-dimensional strata, 12 two-dimensional strata, and 2 one-dimensional strata. Three-dimen-sional strata of the cut locus are Maxwell strata of multiplicity 2 (for each point there are 2 minimizers). Two-dimensional strata of the cut locus consist of conjugate points. Finally, one-dimensional strata are Maxwell strata of infinite multiplicity, they consist of conjugate points as well. Projections of sub-Riemannian geodesics to the 2-dimensional plane of the distribution are Euler elasticae. For each point of the cut locus, we describe the Euler elasticae corresponding to minimizers coming to this point. Finally, we describe the structure of the optimal synthesis, i.e., the set of minimizers for each terminal point in the Engel group.
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Symmetric Riemannian problem on the group of proper isometries of hyperbolic plane
We consider the Lie group PSL(2) (the group of orientation preserving isometries of the hyperbolic plane) and a left-invariant Riemannian metric on this group with two equal eigenvalues that correspond to space-like eigenvectors (with respect to the Killing form). For such metrics we find a parametrization of geodesics, the conjugate time, the cut time and the cut locus. The injectivity radius is computed. We show that the cut time and the cut locus in such Riemannian problem converge to the cut time and the cut locus in the corresponding sub-Riemannian problem as the third eigenvalue of the metric tends to infinity. Similar results are also obtained for SL(2).
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Cut locus of a left invariant Riemannian metric on SO(3) in the axisymmetric case
We consider a left invariant Riemannian metric on SO(3) with two equal eigenvalues. We find the cut locus and the equation for the cut time. We find the diameter of such metric and describe the set of all most distant points from the identity. Also we prove that the cut locus and the cut time converge to the cut locus and the cut time in the sub-Riemannian problem on SO(3) as one of the metric eigenvalues tends to infinity.
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Sub-Riemannian and almost-Riemannian geodesics on SO(3) and $S^2$
In this paper we study geodesics of left-invariant sub-Riemannian metrics on SO(3) and almost-Riemannian metrics on $S^2$. These structures are connected with each other, and it is possible to use information about one of them to obtain results about another one. We give an explicit parameterization of sub-Riemannian geodesics on SO(3) and use it to get a parameterization of almost-Riemannian geodesics on $S^2$. We use symmetries of the exponential map to obtain some necessary optimality conditions. We present some upper bounds on the cut time in both cases and describe periodic geodesics on SO(3).
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Cut time in sub-Riemannian problem on Engel group
The left-invariant sub-Riemannian problem on the Engel group is considered. The problem gives the nilpotent approximation to generic nonholonomic systems in four-dimensional space with two-dimensional control, for instance to a system which describes motion of mobile robot with a trailer. The global optimality of extremal trajectories is studied via geometric control theory. The global diffeomorphic structure of the exponential mapping is described. As a consequence, the cut time is proved to be equal to the first Maxwell time corresponding to discrete symmetries of the exponential mapping.
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Conjugate points in nilpotent sub-Riemannian problem on the Engel group
The left-invariant sub-Riemannian problem on the Engel group is considered. This problem is very important as nilpotent approximation of nonholonomic systems in four-dimensional space with two-dimensional control, for instance of a system which describes movement of mobile trailer robot. We study the local optimality of extremal trajectories and estimate conjugate time in this article.
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Cut time and optimal synthesis in sub-Riemannian problem on the group of motions of a plane
A solution to the left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is obtained. Local and global optimality of extremal trajectories is characterized. Lower and upper bounds on the first conjugate time are proved. The cut time is shown to be equal to the first Maxwell time corresponding to the group of discrete symmetries of the exponential mapping. An explicit global description of the cut locus is obtained. Optimal synthesis is described.
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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.
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Conjugate points in Euler's elastic problem
For the classical Euler's elastic problem, conjugate points are described. Inflectional elasticae admit the first conjugate point between the first and the third inflection points. All the rest elasticae do not have conjugate points.
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