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Search Results (5)

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Keywords = Legendrian duality

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16 pages, 419 KiB  
Article
Singularities for Focal Sets of Timelike Sabban Curves in de Sitter 3-Space
by Yongqiao Wang, Lin Yang, Yuxin Liu and Yuan Chang
Symmetry 2022, 14(12), 2471; https://doi.org/10.3390/sym14122471 - 22 Nov 2022
Cited by 4 | Viewed by 1437
Abstract
In the theory of cosmology, de Sitter space is the symmetrical model of accelerated expansions of the universe. It is derived from the solution of the Einstein field equation, which has a positive cosmological constant. In this paper, we define the evolutes and [...] Read more.
In the theory of cosmology, de Sitter space is the symmetrical model of accelerated expansions of the universe. It is derived from the solution of the Einstein field equation, which has a positive cosmological constant. In this paper, we define the evolutes and focal surfaces of timelike Sabban curves in de Sitter space. We find that de Sitter focal surfaces can be regarded as caustics and de Sitter evolutes corresponding to the locus of the polar vectors of osculating de Sitter subspaces. By using singularity theory, we classify the singularities of the de Sitter focal surfaces and de Sitter evolutes and show that there is a close relationship between a new geometric invariant and the types of singularities. Moreover, the Legendrian dual relationships between the hyperbolic tangent indicatrix of timelike Sabban curves and the focal surfaces are given. Finally, we provide an example to illustrate our main results. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology)
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21 pages, 546 KiB  
Article
Singularities of Osculating Developable Surfaces of Timelike Surfaces along Curves
by Yongqiao Wang, Lin Yang, Pengcheng Li and Yuan Chang
Symmetry 2022, 14(11), 2251; https://doi.org/10.3390/sym14112251 - 26 Oct 2022
Cited by 3 | Viewed by 1727
Abstract
In this paper, we focus on a developable surface tangent to a timelike surface along a curve in Minkowski 3-space, which is called the osculating developable surface of the timelike surface along the curve. The ruling of the osculating developable surface is parallel [...] Read more.
In this paper, we focus on a developable surface tangent to a timelike surface along a curve in Minkowski 3-space, which is called the osculating developable surface of the timelike surface along the curve. The ruling of the osculating developable surface is parallel to the osculating Darboux vector field. The main goal of this paper is to classify the singularities of the osculating developable surface. To this end, two new invariants of curves are defined to characterize these singularities. Meanwhile, we also research the singular properties of osculating developable surfaces near their lightlike points. Moreover, we give a relation between osculating Darboux vector fields and normal vector fields of timelike surfaces along curves from the viewpoint of Legendrian dualities. Finally, some examples with symmetrical structures are presented to illustrate the main results. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology)
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14 pages, 312 KiB  
Article
Extended Legendrian Dualities Theorem in Singularity Theory
by Haiming Liu and Jiajing Miao
Symmetry 2022, 14(5), 982; https://doi.org/10.3390/sym14050982 - 11 May 2022
Cited by 1 | Viewed by 1644
Abstract
In this paper, we find some new information on Legendrian dualities and extend them to the case of Legendrian dualities for continuous families of pseudo-spheres in general semi-Euclidean space. In particular, we construct all contact diffeomorphic mappings between the contact manifolds and display [...] Read more.
In this paper, we find some new information on Legendrian dualities and extend them to the case of Legendrian dualities for continuous families of pseudo-spheres in general semi-Euclidean space. In particular, we construct all contact diffeomorphic mappings between the contact manifolds and display them in a table that contains all information about Legendrian dualities. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology)
8 pages, 224 KiB  
Article
Necessary Optimality Conditions in Isoperimetric Constrained Optimal Control Problems
by Silviu-Aurelian Urziceanu
Symmetry 2019, 11(11), 1380; https://doi.org/10.3390/sym11111380 - 7 Nov 2019
Cited by 4 | Viewed by 2179
Abstract
In this paper, we focus on a new class of optimal control problems governed by a simple integral cost functional and isoperimetric-type constraints (constant level sets of some simple integral functionals). By using the notions of a variational differential system and adjoint equation, [...] Read more.
In this paper, we focus on a new class of optimal control problems governed by a simple integral cost functional and isoperimetric-type constraints (constant level sets of some simple integral functionals). By using the notions of a variational differential system and adjoint equation, necessary optimality conditions are established for a feasible solution in the considered optimization problem. More precisely, under simplified hypotheses and using a modified Legendrian duality, we establish a maximum principle for the considered optimization problem. Full article
8 pages, 223 KiB  
Article
Noether-Type First Integrals Associated with Autonomous Second-Order Lagrangians
by Savin Treanţă
Symmetry 2019, 11(9), 1088; https://doi.org/10.3390/sym11091088 - 31 Aug 2019
Cited by 7 | Viewed by 1953
Abstract
In this paper, the analysis is centered on Noether-type first integrals in Lagrange-Hamilton dynamics based on autonomous second-order Lagrangians. More precisely, by using the classical Noether’s theorem and a non-standard Legendrian duality, the single-time and multi-time versions of Noether’s result are investigated for [...] Read more.
In this paper, the analysis is centered on Noether-type first integrals in Lagrange-Hamilton dynamics based on autonomous second-order Lagrangians. More precisely, by using the classical Noether’s theorem and a non-standard Legendrian duality, the single-time and multi-time versions of Noether’s result are investigated for autonomous second-order Lagrangians. A correspondence is established between the invariances under flows and the first integrals for autonomous second-order Lagrangians. In this way, our results extend, unify and improve several existing theorems in the current literature. Full article
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