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Variational Methods on Riemannian Manifolds: Theory and Applications
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Variational Methods on Riemannian Manifolds: Theory and Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Engineering Mathematics".

Deadline for manuscript submissions: closed (31 January 2024) | Viewed by 6331

Special Issue Editors

Dr. Margarida Camarinha

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Guest Editor
CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal
Interests: Riemannian geometry; Riemannian Lie groups; geometric control; geometric mechanics
Dr. Leonardo Colombo

E-Mail Website
Guest Editor
Institute of Mathematical Sciences (ICMAT), Calle Nicolás Cabrera 13-15, 28049 Madrid, Spain
Interests: control theory; geometric mechanics; hybrid and multi-agent systems; discrete mechanics; nonlinear and geometric control; variational problems on Riemannian manifolds

Special Issue Information

Dear Colleagues,

From the point of view of applications, Riemannian geometry interlaces with many other branches of mathematics, namely, calculus of variations, geometric control and geometric mechanics, which have had a great impact on other sciences.

Many problems of interest in engineering and physics are formulated on Riemannian manifolds, and mainly on Lie groups and homogeneous spaces. Applications such as trajectory generation for space and mobile robotics, navigation of autonomous vehicles and interpolation for 3D animation in computer graphics have been a motivation for developing variational methods and control techniques in the Riemannian setting.

The scope of this Special Issue includes research or review papers on recent developments of variational methods, geometric control and mechanics on Riemannian manifolds. It also aims to provide a wide range of applications of Riemannian geometry to physics, engineering and other sciences.

Dr. Margarida Camarinha
Dr. Leonardo Colombo
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Riemannian manifold
  • Lie group
  • symmetries
  • homogeneous space
  • calculus of variations
  • geometric control
  • mechanics in Riemannian manifolds
  • Hamiltonian and Lagrangian systems

Published Papers (6 papers)

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Research

12 pages, 289 KiB  
Article
A Coordinate-Free Variational Approach to Fourth-Order Dynamical Systems on Manifolds: A System and Control Theoretic Viewpoint
by Simone Fiori
Mathematics 2024, 12(3), 428; https://0-doi-org.brum.beds.ac.uk/10.3390/math12030428 - 29 Jan 2024
Viewed by 569
Abstract
The present paper describes, in a theoretical fashion, a variational approach to formulate fourth-order dynamical systems on differentiable manifolds on the basis of the Hamilton–d’Alembert principle of analytic mechanics. The discussed approach relies on the introduction of a Lagrangian function that depends on [...] Read more.
The present paper describes, in a theoretical fashion, a variational approach to formulate fourth-order dynamical systems on differentiable manifolds on the basis of the Hamilton–d’Alembert principle of analytic mechanics. The discussed approach relies on the introduction of a Lagrangian function that depends on the kinetic energy and the covariant acceleration energy, as well as a potential energy function that accounts for conservative forces. In addition, the present paper introduces the notion of Rayleigh differential form to account for non-conservative forces. The corresponding fourth-order equation of motion is derived, and an interpretation of the obtained terms is provided from a system and control theoretic viewpoint. A specific form of the Rayleigh differential form is introduced, which yields non-conservative forcing terms assimilable to linear friction and jerk-type friction. The general theoretical discussion is complemented by a brief excursus about the numerical simulation of the introduced differential model. Full article
(This article belongs to the Special Issue Variational Methods on Riemannian Manifolds: Theory and Applications)
31 pages, 1755 KiB  
Article
Time-Optimal Problem in the Roto-Translation Group with Admissible Control in a Circular Sector
by Alexey Mashtakov and Yuri Sachkov
Mathematics 2023, 11(18), 3931; https://0-doi-org.brum.beds.ac.uk/10.3390/math11183931 - 15 Sep 2023
Viewed by 535
Abstract
We study a time-optimal problem in the roto-translation group with admissible control in a circular sector. The problem reveals the trajectories of a car model that can move forward on a plane and turn with a given minimum turning radius. Our work generalizes [...] Read more.
We study a time-optimal problem in the roto-translation group with admissible control in a circular sector. The problem reveals the trajectories of a car model that can move forward on a plane and turn with a given minimum turning radius. Our work generalizes the sub-Riemannian problem by adding a restriction on the velocity vector to lie in a circular sector. The sub-Riemannian problem is given by a special case when the sector is the full disc. The trajectories of the system are applicable in image processing to detect salient lines. We study the local and global controllability of the system and the existence of a solution for given arbitrary boundary conditions. In a general case of the sector opening angle, the system is globally but not small-time locally controllable. We show that when the angle is obtuse, a solution exists for any boundary conditions, and when the angle is reflex, a solution does not exist for some boundary conditions. We apply the Pontryagin maximum principle and derive a Hamiltonian system for extremals. Analyzing a phase portrait of the Hamiltonian system, we introduce the rectified coordinates and obtain an explicit expression for the extremals in Jacobi elliptic functions. We show that abnormal extremals are of circular type, and they correspond to motions of a car along circular arcs of minimal possible radius. The normal extremals in a general case are given by concatenation of segments of sub-Riemannian geodesics in SE2 and arcs of circular extremals. We show that, in a general case, the vertical (momentum) part of the extremals is periodic. We partially study the optimality of the extremals and provide estimates for the cut time in terms of the period of the vertical part. Full article
(This article belongs to the Special Issue Variational Methods on Riemannian Manifolds: Theory and Applications)
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19 pages, 337 KiB  
Article
An Intrinsic Version of the k-Harmonic Equation
by Lígia Abrunheiro and Margarida Camarinha
Mathematics 2023, 11(17), 3628; https://0-doi-org.brum.beds.ac.uk/10.3390/math11173628 - 22 Aug 2023
Viewed by 534
Abstract
The notion of k-harmonic curves is associated with the kth-order variational problem defined by the k-energy functional. The present paper gives a geometric formulation of this higher-order variational problem on a Riemannian manifold M and describes a generalized Legendre transformation [...] Read more.
The notion of k-harmonic curves is associated with the kth-order variational problem defined by the k-energy functional. The present paper gives a geometric formulation of this higher-order variational problem on a Riemannian manifold M and describes a generalized Legendre transformation defined from the kth-order tangent bundle TkM to the cotangent bundle T*Tk−1M. The intrinsic version of the Euler–Lagrange equation and the corresponding Hamiltonian equation obtained via the Legendre transformation are achieved. Geodesic and cubic polynomial interpolation is covered by this study, being explored here as harmonic and biharmonic curves. The relationship of the variational problem with the optimal control problem is also presented for the case of biharmonic curves. Full article
(This article belongs to the Special Issue Variational Methods on Riemannian Manifolds: Theory and Applications)
23 pages, 524 KiB  
Article
Endpoint Geodesic Formulas on Graßmannians Applied to Interpolation Problems
by Knut Hüper and Fátima Silva Leite
Mathematics 2023, 11(16), 3545; https://0-doi-org.brum.beds.ac.uk/10.3390/math11163545 - 16 Aug 2023
Viewed by 1093
Abstract
Simple closed formulas for endpoint geodesics on Graßmann manifolds are presented. In addition to realizing the shortest distance between two points, geodesics are also essential tools to generate more sophisticated curves that solve higher order interpolation problems on manifolds. This will be illustrated [...] Read more.
Simple closed formulas for endpoint geodesics on Graßmann manifolds are presented. In addition to realizing the shortest distance between two points, geodesics are also essential tools to generate more sophisticated curves that solve higher order interpolation problems on manifolds. This will be illustrated with the geometric de Casteljau construction offering an excellent alternative to the variational approach which gives rise to Riemannian polynomials and splines. Full article
(This article belongs to the Special Issue Variational Methods on Riemannian Manifolds: Theory and Applications)
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24 pages, 381 KiB  
Article
Rolling Geodesics, Mechanical Systems and Elastic Curves
by Velimir Jurdjevic
Mathematics 2022, 10(24), 4827; https://0-doi-org.brum.beds.ac.uk/10.3390/math10244827 - 19 Dec 2022
Cited by 3 | Viewed by 1019
Abstract
This paper defines a large class of differentiable manifolds that house two distinct optimal problems called affine-quadratic and rolling problem. We show remarkable connections between these two problems manifested by the associated Hamiltonians obtained by the Maximum Principle of optimal control. We also [...] Read more.
This paper defines a large class of differentiable manifolds that house two distinct optimal problems called affine-quadratic and rolling problem. We show remarkable connections between these two problems manifested by the associated Hamiltonians obtained by the Maximum Principle of optimal control. We also show that each of these Hamiltonians is completely intergrable, in the sense of Liouville. Finally we demonstrate the significance of these results for the theory of mechanical systems. Full article
(This article belongs to the Special Issue Variational Methods on Riemannian Manifolds: Theory and Applications)
56 pages, 563 KiB  
Article
Riemannian Calculus of Variations Using Strongly Typed Tensor Calculus
by Victor Dods
Mathematics 2022, 10(18), 3231; https://0-doi-org.brum.beds.ac.uk/10.3390/math10183231 - 06 Sep 2022
Viewed by 1469
Abstract
In this paper, the notion of strongly typed language will be borrowed from the field of computer programming to introduce a calculational framework for linear algebra and tensor calculus for the purpose of detecting errors resulting from inherent misuse of objects and for [...] Read more.
In this paper, the notion of strongly typed language will be borrowed from the field of computer programming to introduce a calculational framework for linear algebra and tensor calculus for the purpose of detecting errors resulting from inherent misuse of objects and for finding natural formulations of various objects. A tensor bundle formalism, crucially relying on the notion of pullback bundle, will be used to create a rich type system with which to distinguish objects. The type system and relevant notation is designed to “telescope” to accommodate a level of detail appropriate to a set of calculations. Various techniques using this formalism will be developed and demonstrated with the goal of providing a relatively complete and uniform method of coordinate-free computation. The calculus of variations pertaining to maps between Riemannian manifolds will be formulated using the strongly typed tensor formalism and associated techniques. Energy functionals defined in terms of first-order Lagrangians are the focus of the second half of this paper, in which the first variation, the Euler–Lagrange equations, and the second variation of such functionals will be derived. Full article
(This article belongs to the Special Issue Variational Methods on Riemannian Manifolds: Theory and Applications)
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