Geodesics in the sub-Riemannian problem on the group $\mathrm{SO}(3)$

Бесчастный, Иван Юрьевич; Beschatnyi, Ivan Yurevich; Сачков, Юрий Леонидович; Sachkov, Yu. L.

doi:10.4213/sm8555

Математический сборник

2016

DOI: 10.4213/sm8555

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Geodesics in the sub-Riemannian problem on the group $\mathrm{SO}(3)$

Иван Юрьевич Бесчастный¹,

Ivan Yurevich Beschatnyi²,

Юрий Леонидович Сачков³

et al.

Abstract: Изучаются геодезические левоинвариантных субримановых структур на группе $\mathrm{SO}(3)$. Приводится полное описание периодических геодезических и их простейшие свойства, некоторые необходимые условия минимальности, оценки на время разреза и на диаметр метрики. Библиография: 32 названия.

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Cited by 4 publications

References 25 publications

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Closure of the Laplace-Beltrami Operator on 2D Almost-Riemannian Manifolds and Semi-Fredholm Properties of Differential Operators on Lie Manifolds

Beschastnyi

2023

Results Math

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The problem of determining the domain of the closure of the Laplace-Beltrami operator on a 2D almost-Riemannian manifold is considered. Using tools from theory of Lie groupoids natural domains of perturbations of the Laplace-Beltrami operator are found. The main novelty is that the presented method allows us to treat geometries with tangency points. This kind of singularity is difficult to treat since those points do not have a tubular neighbourhood compatible with the almost-Riemannian metric.

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Closure of the Laplace-Beltrami Operator on 2D Almost-Riemannian Manifolds and Semi-Fredholm Properties of Differential Operators on Lie Manifolds

Beschastnyi

2023

Results Math

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Quantum Confinement for the Curvature Laplacian −Δ + cK on 2D-Almost-Riemannian Manifolds

Beschastnyi¹,

Boscain

Pozzoli

2021

Potential Anal

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Two-dimension almost-Riemannian structures of step 2 are natural generalizations of the Grushin plane. They are generalized Riemannian structures for which the vectors of a local orthonormal frame can become parallel. Under the 2-step assumption the singular set , where the structure is not Riemannian, is a 1D embedded submanifold. While approaching the singular set, all Riemannian quantities diverge. A remarkable property of these structures is that the geodesics can cross the singular set without singularities, but the heat and the solution of the Schrödinger equation (with the Laplace-Beltrami operator Δ) cannot. This is due to the fact that (under a natural compactness hypothesis), the Laplace-Beltrami operator is essentially self-adjoint on a connected component of the manifold without the singular set. In the literature such phenomenon is called quantum confinement.In this paper we study the self-adjointness of the curvature Laplacian, namely −Δ + , for ∈ (0, 1/2) (here is the Gaussian curvature), which originates in coordinate-free quantization procedures (as for instance in path-integral or covariant Weyl quantization). We prove that there is no quantum confinement for this type of operators.

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Left-invariant optimal control problems on Lie groups that are integrable by elliptic functions

Сачков¹,

Sachkov

2023

Uspekhi Mat. Nauk

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Левоинвариантные задачи оптимального управления на группах Ли образуют важный класс задач с большой группой симметрий. Они интересны в теоретическом плане, так как часто допускают полное исследование и на этих модельных задачах можно изучить общие закономерности. В частности, задачи на нильпотентных группах Ли доставляют фундаментальную нильпотентную аппроксимацию общих задач. Левоинвариантные задачи также часто возникают в приложениях: в классической и квантовой механике, геометрии, робототехнике, моделях зрения и обработке изображений. Цель данной работы - дать обзор основных понятий, методов и результатов, относящихся к левоинвариантным задачам оптимального управления на группах Ли, интегрируемым в эллиптических функциях. Основное внимание уделено описанию экстремальных траекторий и их оптимальности, времени разреза и множества разреза, оптимального синтеза. Библиография: 162 названия.

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Geodesics in the sub-Riemannian problem on the group $\mathrm{SO}(3)$

Cited by 4 publications

References 25 publications

Closure of the Laplace-Beltrami Operator on 2D Almost-Riemannian Manifolds and Semi-Fredholm Properties of Differential Operators on Lie Manifolds

Closure of the Laplace-Beltrami Operator on 2D Almost-Riemannian Manifolds and Semi-Fredholm Properties of Differential Operators on Lie Manifolds

Quantum Confinement for the Curvature Laplacian −Δ + cK on 2D-Almost-Riemannian Manifolds

Left-invariant optimal control problems on Lie groups that are integrable by elliptic functions

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