Differential Geometric Methods Research Articles

Intrinsic Geometric Structure of Subcartesian Spaces

Every subset S of a Cartesian space Rd, endowed with differential structure C∞(S) generated by restrictions to S of functions in C∞(Rd), has a canonical partition M(S) by manifolds, which are orbits of the family X(S) of all derivations of C∞(S) that generate local one-parameter groups of local diffeomorphisms of S. This partition satisfies the frontier condition, Whitney’s conditions A and B. If M(S) is locally finite, then it satisfies all definitions of stratification of S. This result extends to Hausdorff locally Euclidean differential spaces. The partition M(S) of a subcartesian space S by smooth manifolds provides a measure for the applicability of differential geometric methods to the study of the geometry of S. If all manifolds in M(S) are single points, we cannot expect differential geometry to be an effective tool in the study of S. On the other extreme, if M(S) contains only one manifold M, then the subcartesian space S is a manifold, S=M, and it is a natural domain for differential geometric techniques.

Komplexe Analysis – Differential and Algebraic methods in Kähler spaces

Our workshop focused on recent results in our main field (complex geometry) and its connection with other branches of mathematics. The main theme of an important proportion of the talks was Hodge theory, combined with differential-geometric methods in the study of singular spaces. One special lecture was a very comprehensive introduction in Scholze-Clausen’s theory of condensed mathematics.

The non-linear perturbation of a black hole by gravitational waves. II. Quasinormal modes and the compactification problem

Recently, Friedrich’s generalized conformal field equations (GCFEs) have been implemented numerically and global quantities such as the Bondi energy and the Bondi–Sachs mass loss have been successfully calculated directly on null-infinity. Although being an attractive option for studying global quantities by way of local differential geometrical methods, how viable are the GCFE for study of quantities arising in the physical space-time? In particular, how long can the evolution track phenomena that need a constant proper physical timestep to be accurately resolved? We address this question by studying the curvature oscillations induced on the Schwarzschild space-time by a non-linear gravitational perturbation. For small enough amplitudes, these are the well approximated by the linear quasinormal modes, where each mode rings at a frequency determined solely by the Schwarzschild mass. We find that the GCFE can indeed resolve these oscillations, which quickly approach the linear regime, but only for a short time before the compactification becomes ‘too fast’ to handle numerically.

Reference shape of bodies with finite incompatible deformations

The work develops differential-geometric methods for modeling of finite incompatible deformations of hyperelastic solids. Deformation incompatibility can be caused, for example, by inhomogeneous temperature fields and distributed defects. As a result, residual stresses and distortion of geometric shape of the body occur. These factors determine the critical parameters of modern high-precision technologies, in particular, in additive manufacturing technologies. In this regard, the development of methods for their quantitative description is an urgent problem of modern solid mechanics.
 The application of methods of differential geometry is based on the representation of a body as a smooth manifold equipped with a metric and a non-Euclidean connection. This approach allows one to interpret the body as a global stress-free shape and to formulate the physical response and material balance equations with respect to this shape. Within the framework of the geometric method, deformations are characterized by embeddings of non-Euclidean shape into physical space, which is still considered to be Euclidean. Measures of incompatibility are identified with the invariants of the affine connection, namely, torsion, curvature, and nonmetricity, and the connection itself is determined by the type of physical process.

Application of Differential-Geometric Methods of Control Theory in the Theory of Partial Differential Equations. II

Application of Differential-Geometric Methods of Control Theory in the Theory of Partial Differential Equations. II

Tracking the state of pacemakers modeled by the van der Pol equation

It is known that in many applications of physics, biology and other sciences as an approximate dynamic model of complex nonlinear oscillatory processes a model of one or more interconnected van der Pol oscillators or some of its modifications is used \cite{ShD01}. For this reason, nonlinear oscillators are studied as a method of modeling, analysis or even control in various fields, such as electronics \cite{ShD02}, control, robotics \cite{ShD03, ShD04}, biomedical research \cite{ShD05}, geology \cite{ShD06} and others. Naturally, in such modeling there are problems in determining the state and parameters of the models based on the results of measuring the output signals in real time. One of these problems, namely: the problem of determining the state of pacemaker models, which are obtained as certain modifications of the van der Pol oscillator equation, is considered in this paper. In the scientific literature, publications on the modeling of cardiovascular activity using oscillatory systems are widely represented. In recent years, among them there are works that are related to the solution of inverse problems for such models. In particular, in \cite{ShD07} using differential-geometric methods of control theory, a general scheme for constructing asymptotically accurate estimates of the state of a two-dimensional dynamical system is proposed. The obtained results are used to effectively solve the problem of observing two models of pacemakers. In our case in this observation problem we used the method of invariant relations \cite{ShD08}, which was developed in analytical mechanics to find partial solutions (dependencies between variables) in the problems of dynamics of a rigid body with a fixed point. Modification of this method to the problems of control theory, observation, identification allowed to synthesize between known and unknown values of the original system additional connections that arise during the motion of its extended model \cite{ShD09, ShD10, ShD11}. The corresponding technique is to expand the original system by introducing additional controlled differential equations and immersing the original system in a system of greater dimension, which due to its sufficiently free structure is more suitable for constructing an observer or identifier. Controls in an extended system are used to synthesize on its trajectories pre-proposed relations that define the unknown components of the mathematical model (phase vector, parameters) as functions of known quantities. The obtained theoretical results are illustrated by numerical simulations of the corresponding nonlinear observers in Section 5.

Jacobi stability analysis and impulsive control of a 5D self-exciting homopolar disc dynamo

<p style='text-indent:20px;'>In this paper, we make a thorough inquiry about the Jacobi stability of 5D self-exciting homopolar disc dynamo system on the basis of differential geometric methods namely Kosambi-Cartan-Chern theory. The Jacobi stability of the equilibria under specific parameter values are discussed through the characteristic value of the matrix of second KCC invariants. Periodic orbit is proved to be Jacobi unstable. Then we make use of the deviation vector to analyze the trajectories behaviors in the neighborhood of the equilibria. Instability exponent is applicable for predicting the onset of chaos quantitatively. In addition, we also consider impulsive control problem and suppress hidden attractor effectively in the 5D self-exciting homopolar disc dynamo.</p>

Application of Differential-Geometric Methods of Control Theory to the Theory of Partial Differential Equations. I

Application of Differential-Geometric Methods of Control Theory to the Theory of Partial Differential Equations. I

Generalized vanishing theorems for Yukawa couplings in heterotic compactifications

Heterotic compactifications on Calabi-Yau threefolds frequently exhibit textures of vanishing Yukawa couplings in their low energy description. The vanishing of these couplings is often not enforced by any obvious symmetry and appears to be topological in nature. Recent results used differential geometric methods to explain the origin of some of this structure [1, 2]. A vanishing theorem was given which showed that the effect could be attributed, in part, to the embedding of the Calabi-Yau manifolds of interest inside higher dimensional ambient spaces, if the gauge bundles involved descended from vector bundles on those larger manifolds. In this paper, we utilize an algebro-geometric approach to provide an alternative derivation of some of these results, and are thus able to generalize them to a much wider arena than has been considered before. For example, we consider cases where the vector bundles of interest do not descend from bundles on the ambient space. In such a manner we are able to highlight the ubiquity with which textures of vanishing Yukawa couplings can be expected to arise in heterotic compactifications, with multiple different constraints arising from a plethora of different geometric features associated to the gauge bundle.

Jacobi Stability Analysis and the Onset of Chaos in a Two-Degree-of-Freedom Mechanical System

In this paper, the Jacobi stability of a two-degree-of-freedom mechanical system is studied by the innovative application of KCC-theory, namely differential geometric methods. We discuss the Jacobi stability of two equilibria and a periodic orbit by constructing geometric invariants. Both the regions of Jacobi stability and Lyapunov stability are presented to show the difference. We draw the phase portraits of the deviation vector near two equilibria under specific parameter values and initial conditions, and point out the sensitivity of deviation vector to initial conditions. In addition, the corresponding instability exponent and curvature are applicable for predicting the onset of chaos, which help us to detect chaotic behaviors quantitatively.

Stabilization of Equilibrium for Underactuated Mechanical Systems Without Potential Energy

This paper investigates the equilibrium stabilization problem for a class of un-deractuated mechanical systems which do not possess potential energy. The dynamics of the system is established under the framework of Rimannian geometry, and differential geometric methods are employed in the design of stabilization controller. The main novelty of this paper is that we stabilize the equilibrium by constructing an artificial potential for the closed-loop system, which is related to the designed configuration feedback. Once the artificial potential satisfy certain requirements with respect to the equilibrium, the stability of the system can be guaranteed. Furthermore, by incorporating dissipative feedback into the control strategy, we successfully obtain the exponential stability of the equilibrium.

Geometric Robot Dynamic Identification: A Convex Programming Approach

Recent work has shed light on the often unreliable performance of constrained least-squares estimation methods for robot mass-inertial parameter identification, particularly for high degree-of-freedom systems subject to noisy and incomplete measurements. Instead, differential geometric identification methods have proven to be significantly more accurate and robust. These methods account for the fact that the mass-inertial parameters reside in a curved Riemannian space, and allow perturbations in the mass-inertial properties to be measured in a coordinate-invariant manner. Yet, a continued drawback of existing geometric methods is that the corresponding optimization problems are inherently nonconvex, have numerous local minima, and are computationally highly intensive to solve. In this paper, we propose a convex formulation under the same coordinate-invariant Riemannian geometric framework that directly addresses these and other deficiencies of the geometric approach. Our convex formulation leads to a globally optimal solution, reduced computations, faster and more reliable convergence, and easy inclusion of additional convex constraints. The main idea behind our approach is an entropic divergence measure that allows for the convex regularization of the inertial parameter identification problem. Extensive experiments with the 3-DoF MIT Cheetah leg, the 7-DoF AMBIDEX tendon-driven arm, and a 16-link articulated human model show markedly improved robustness and generalizability vis-a-vis existing vector space methods while ensuring fast, guaranteed convergence to the global solution.

Invariants of algebraic group actions from differential point of view

In this paper we suggest an approach to study actions of semisimple (or reductive) algebraic groups in their algebraic complex representations. We use differential-geometric methods instead of classical algebraic constructions. Namely, according to Borel–Weil–Bott theorem, every algebraic representation of semisimple algebraic group is isomorphic to the action of this group on the module of holomorphic sections of some reductive bundle over homogeneous space. Using this, we give a complete description of the field of differential invariants for this action and obtain a criterion, which separates regular orbits.

The Bountiful Intersection of Differential Geometry, Mechanics, and Control Theory

The areas of mechanics and control theory have a rich and productive history of interaction with the broad mathematical subject of differential geometry. This article provides an overview of these sorts of interplay in the areas of Riemannian and affine differential geometry and the geometry of vector distributions. It emphasizes areas where differential geometric methods have played a crucial role in solving problems whose solutions are difficult to achieve without access to these methods. It also emphasizes a concise and elegant presentation of the approach, rather than a detailed and concrete presentation. The results overviewed, while forming a coherent and elegant body of work, are limited in scope. The review closes with a discussion of why the approach is limited and a brief consideration of issues that must be resolved before the results of the type presented here can be extended.

Geometric Methods in Analysis and Control of Implicit Differential Systems

In this article, we discuss the application of differential geometric methods in analyzing the structure and designing control laws for implicit differential systems or differential algebraic equation (DAE) systems. While there have been several efforts toward numerical and quantitative analysis of DAE problems, the theoretical contributions especially in the case of nonlinear systems are scarce. We discuss two popular techniques from differential geometric control theory and bring out their merits in addressing implicit differential systems. In the first section, we review the theory of noninteracting control via input–output decoupling and its application in analyzing the intrinsic structure of DAE control problems. In particular, we focus on addressing the problem of well-posedness of DAE systems as well as feedback control design through a regularization process which allows one to solve the DAE by expressing the constraint variable as a dynamically dependent endogenous function of the states, inputs, and their derivatives. Further, extensions of these techniques to stochastic differential algebraic equations have been presented. In the second section, we review the theory of differential flatness and its applicability to feedback control design for a class of DAE systems. Here, the DAE system is expressed as a Cartan field on a manifold of jets of infinite order, and necessary and sufficient conditions for its equivalence to a linear, controllable system have been derived in order to design globally stabilizing nonlinear feedback laws. Examples from constrained mechanics have been presented in order to demonstrate the practical applicability of these methods.

Growing Solids and Thin-Walled Structures

The present paper provides a systematic treatment of modern differential-geometrical methods for modeling of incompatible finite deformations in solids and thin walled structures. The incompatibility of deformations may be caused by a variety of physical phenomena; among them are: growth, non-uniform thermal fields, shrinkage, etc. Incompatible deformations results in residual stresses and distortion of geometrical shape. These factors are associated with critical parameters in modern high-precision technologies, particularly, in additive manufacturing, and considered to be essential constituents in corresponding mathematical models. The methods in question are based on the representation of a body and physical space in terms of differentiable manifolds, namely material manifold and physical manifold. These manifolds are equipped with specific metrics and connections, non-Euclidian in general.

Hermitian and gauge-covariant Hamiltonians for a particle in a magnetic field on cylindrical and spherical surfaces

We construct the Hermitian Schrödinger Hamiltonian of spin-less particles and the gauge-covariant Pauli Hamiltonian of spin one-half particles in a magnetic field, which are confined to cylindrical and spherical surfaces. The approach does not require the use of involved differential-geometrical methods and is intuitive and physical, relying on the general requirements of Hermicity and gauge-covariance. The surfaces are embedded in the full three-dimensional space and confinement to the surfaces is achieved by strong radial potentials. We identify the Hermitian and gauge-covariant (in the presence of a magnetic field) physical radial momentum in each case and set it to zero upon confinement to the surfaces. The resulting surface Hamiltonians are seen to be automatically Hermitian and gauge-covariant. The well-known geometrical kinetic energy also emerges naturally.

Local Decompositions of Second Order Infinite‐Dimensional Systems

AbstractThis article deals with the local system decomposition of infinite‐dimensional systems, which are described by second‐order nonlinear partial differential equations. It will be shown that the existence of a certain codistribution allows a local triangular decomposition of the generalized system vector field and the boundary conditions. Based on this triangular decomposition, non‐accessibility follows by a simple structural analysis. Throughout the article differential geometric methods are applied, highlighting the geometric picture behind the system description. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Stochastic Global Optimization Algorithm for the Two-Frame Sensor Calibration Problem

In the two-frame sensor calibration problem, the objective is to find rigid-body homogeneous transformation matrices $X,Y$ that best fit a set of equalities of the form $A_i X = Y B_i$ , $i=1, \ldots, N$ , where the $\{(A_i, B_i)\}$ are pairs of homogeneous transformations obtained from sensor measurements. The measurements are often subject to varying levels of noise and the resulting optimization can have numerous local minima that exhibit high sensitivity in the choice of optimization parameters. As a first contribution, we present a fast and numerically robust local optimization algorithm for the two-frame sensor calibration objective function. Using coordinate-invariant differential geometric methods that take into account the matrix Lie group structure of the rigid-body transformations, our local descent method makes use of analytic gradients and Hessians, and a strictly descending fast step-size estimate to achieve significant performance improvements. As a second contribution, we present a two-phase stochastic geometric optimization algorithm for finding a stochastic global minimizer based on our earlier local optimizer. Numerical studies demonstrate the considerably enhanced robustness and efficiency of our algorithm over existing unit quaternion-based methods.

Local Decomposition and Accessibility of Nonlinear Infinite-Dimensional Systems

This article deals with the local system decomposition of infinite-dimensional systems, which are described by second-order nonlinear partial differential equations. We show that if there exists a certain codistribution which is invariant under the generalized system vector field, a local triangular decomposition can be obtained. Furthermore, we draw connections to a different approach which is based on transformation groups. Throughout the article we apply differential geometric methods, highlighting the geometric picture behind the system description. The article is closed with a nonlinear example.