Coordinate-free Approach Research Articles

On the internal approach to differential equations 2. The controllability structure

AbstractThe article concerns the geometrical theory of general systems Ω of partial differential equations in theabsolute sense, i.e., without any additional structure and subject to arbitrary change of variables in the widest possible meaning. The main result describes the uniquely determined composition series Ω0⊂ Ω1⊂ … ⊂ Ω where Ωkis the maximal system of differential equations “induced” by Ω such that the solution of Ωkdepends on arbitrary functions ofkindependent variables (on constants ifk= 0). This is a well-known result only for the particular case of underdetermined systems of ordinary differential equations. Then Ω = Ω1and we have the composition series Ω0⊂ Ω1= Ω where Ω0involves all first integrals of Ω, therefore Ω0is trivial (absent) in the controllable case. The general composition series Ω0⊂ Ω1⊂ … ⊂ Ω may be regarded as a “multidimensional” controllability structure for the partial differential equations.Though the result is conceptually clear, it cannot be included into the common jet theory framework of differential equations. Quite other and genuinely coordinate-free approach is introduced.

A Convex Characterization of Multidimensional Linear Systems Subject to SQI Constraints

This technical note proposes a convex characterization of the set of all stable closed-loop linear systems that are obtained from a given plant, which may be multidimensional, by interconnecting it in feedback with controllers that satisfy a certain pre-selected constraint. We take an approach that is particularly useful when a doubly-coprime factorization of the plant is difficult to obtain, and a stable stabilizing controller may not exist (the plant is not strongly-stabilizable) or when one may be difficult to find; most related work requires one of these. We adopt the so-called coordinate-free approach, which, unlike Youla's parametrization, does not rely on a doubly-coprime factorization of the plant. We show that if constraints which satisfy a condition called strong quadratic invariance (SQI) are imposed on the controllers then the set of all stable closed-loop multidimensional linear systems has a convex representation, and norm-optimal control problems can be cast in convex form. Although the SQI condition is in general slightly stronger than quadratic invariance (QI), which was developed in related work, they are equivalent for common classes of problems arising in decentralized control.

Kwadratowa estymacja komponentów wariancyjnych w modelach liniowych

In the paper the theory of quadratic estimation of variance components in linear models and its applications are presented. In the presentation of the theory the coordinate-free approach is used. The applications concern estimation of variance components in general linear regression model and its special cases. The problem of admissibility of quadratic estimates in mixed linear models with two variance components is considered separately.

Reaction Coordinate-Free Approach to Recovering Kinetics from Potential-Scaled Simulations: Application of Kramers' Rate Theory.

Enhanced sampling techniques are used to increase the frequency of "rare events" during computer simulations of complex molecules. Although methods exist that allow accurate thermodynamics to be recovered from enhanced simulations, recovering kinetics proves to be more challenging. Here we present an extrapolation approach that allows reliable kinetics to be recovered from potential-scaled MD simulations. The approach, based on Kramers' rate theory, is simple and computationally efficient, and allows kinetics to be recovered without defining reaction coordinates. To test our approach, we use it to determine the kinetics of barrier crossing between two metastable states on the 2D-Müller potential and the C7eq to αR transition in alanine dipeptide. The mean first passage time estimates obtained are in excellent agreement with reference values obtained from direct simulations on the unscaled potentials performed over times that are orders of magnitude longer.

On the use of coordinate-free matrix calculus

For a standard tool in econometrics, matrix calculus, an approach is illustrated in this note that is unusual in that context, a coordinate-free approach. It can help to eliminate the persistent use of non-standard conventions. The Kronecker product and its use can be better understood. The complications and pitfalls of defining twice differentiability by partial derivatives are avoided. Its use is demonstrated, for example by giving a coordinate-free determination of the derivative of the determinant.

Attitude synchronization of satellites with internal actuation

We present control synthesis for attitude synchronization of a cluster of satellites, that are internally actuated, using a coordinate-free approach. The control law is designed using the notion of potential energy shaping that rests on the use of modified trace functions. The distinguishing aspect of the internal actuation model is that the drift vector field in the system dynamics explicitly depends on the conserved total angular momentum. The control law presented is applicable to a general undirected graph of communication between the satellites, with one satellite of the cluster communicating with the leader. Further, the control law is applicable with just two vector observations in each satellite, and hence making redundant, complete attitude information.

Higher order corrections to adiabatic invariants of generalized slow-fast Hamiltonian systems

We present a coordinate-free approach for constructing approximate first integrals of generalized slow-fast Hamiltonian systems, based on the global averaging method on parameter-dependent phase spaces with \documentclass[12pt]{minimal}\begin{document}$\mathbb {S}^1$\end{document}S1-symmetry. Explicit global formulas for approximate second-order first integrals are derived. As examples, we analyze the case quadratic in the fast variables (in particular, the elastic pendulum), and the charged particle in a slowly-varying magnetic field.

Oblique circular torus, Villarceau circles, and four types of Bennett linkages

The oblique circular torus (OCT) and its main geometric properties are introduced. Intrinsic vector calculation is utilized to mathematically describe the OCT. The coordinate-free approach leads to the algebraic equation of an OCT in a privileged Cartesian reference frame. The OCT equation is used to confirm a theorem of Euclidean geometry. In a broad category of OCT, through any point five circles can be drawn on the surface, namely the parallel of latitude and four circular generatrices whose planes pass through the OCT center of symmetry. In the special case of a right circular torus, the Villarceau theorem is verified. Next, consider the four RRS open chains whose S spherical-joint centers move on the same OCT and their possible in-parallel assemblies in single-loop RRRS chains. From a category of the foregoing RRRS chains, a new derivation of the amazing Bennett 4R linkage is proposed. Two kinds of Bennett linkages are further verified and each kind contains two enantiomorphic or symmetric linkages. Four types of Bennett linkages associated with one OCT are established by uniquely specifying the link twist as an acute value. Two cases of special type, rectangular and equilateral configurations, are also confirmed.

On Cesàro-Volterra Method in Orthotropic Saint-Venant Beam

The linearly elastic and orthotropic Saint-Venant beam model, with a spatially constant Poisson tensor and fiberwise homogeneous elastic moduli, is investigated by a coordinate-free approach. A careful reasoning reveals that the elastic strain, fulfilling the whole set of differential conditions of integrability and a differential condition imposed by equilibrium, is defined on the whole ambient space in which the beam is immersed. At this stage the shape of the beam cross-section is inessential and Cesaro-Volterra formula provides the general integral of the differential conditions of kinematic compatibility. The cross-section geometrical shape comes into play only when differential and boundary equilibrium conditions are imposed to evaluate the warping displacement field. The treatment of an orthotropic Saint-Venant beam is applied to investigate about the locations of the shear and twist centres. It is shown that the position of the shear centre can be expressed in terms of the sole cross-section twist warping. The advantage with respect to treatments in the literature is that the solution of a single Neumann-like problem is required.

On the compatibility relation for the Föppl–von Kármán plate equations

By using a coordinate-free approach we propose a new derivation of the compatibility equation for the Föppl–von Kármán nonlinear plate theory.

Coordinate-Free Classic Geometries

This paper is devoted to a coordinate-free approach to several classic geometries such as hyperbolic (real, complex, quaternionic), elliptic (spherical, Fubini-Study), and lorentzian (de Sitter, anti de Sitter) ones. These geometries carry a certain simple structure that is in some sense stronger than the riemannian structure. Their basic geometrical objects have linear nature and provide natural compactifications of classic spaces. The usual riemannian concepts are easily derivable from the strong structure and thus gain their coordinate-free form. Many examples illustrate fruitful features of the approach. The framework introduced here has already been shown to be adequate for solving problems concerning particular classic spaces.

Histories and observables in covariant field theory

Motivated by DeWitt’s viewpoint of covariant field theory, we define a general notion of a non-local classical observable that applies to many physical Lagrangian systems (with bosonic and fermionic variables), by using methods that are now standard in algebraic geometry. We review the methods of local functional calculus, as they are presented by Beilinson and Drinfeld, and relate them to our construction. We partially explain the relation of these with Vinogradov’s secondary calculus. The methods present here are all necessary to understand mathematically properly, and with simple notions, the full renormalization of the standard model, based on functional integral methods. Our approach is close in spirit to non-perturbative methods since we work with actual functions on spaces of fields, and not only formal power series. This article can be seen as an introduction to well-grounded classical physical mathematics, and as a good starting point to study quantum physical mathematics, which make frequent use of non-local functionals, like for example in the computation of Wilson’s effective action. We finish by describing briefly a coordinate-free approach to the classical Batalin–Vilkovisky formalism for general gauge theories, in the language of homotopical geometry.

Generalization of ray tracing in a linear inhomogeneous anisotropic medium: a coordinate-free approach

The Hamiltonian of an optical medium is important in both the design and the description of optical devices in the geometrical optics limit. The results calculated in this article show in detail how ray tracing in anisotropic materials in arbitrary coordinate systems and curved spaces can be carried out. Writing Maxwell's equations in the most general form, we derive a coordinate-free form for the eikonal equation and hence the Hamiltonian of a general purpose medium. The expression works for both orthogonal and non-orthogonal coordinate systems, and we show how it can be simplified for biaxial and uniaxial media in orthogonal coordinate systems. In order to show the utility of the equations in a real case, we study both the isotropic and the uniaxially transmuted birefringent Eaton lens and derive the ray trajectories in spherical coordinates for each case.

Relativistic optics in moving media with spacetime algebra

Relativistic optics in moving media is an old topic in electromagnetic theory. The study of bianisotropic media – a very important class of metamaterials – stems from the research in that topic. However, a coordinate-free approach to moving media leads to rather cumbersome algebraic manipulations with tensors (or dyadics). In this article we present an alternative framework to deal with moving media: using spacetime algebra, the Clifford (geometric) algebra of Minkowski spacetime, we show that the inherent complexity of this subject can be substantially reduced through a transformation – the vacuum form reduction – that maps our problem into an equivalent spacetime with the same formal spacetime constitutive relation as real vacuum. This is made clear by showing that the two Maxwell equations (in spacetime) are reduced to a single Maxwell equation as in vacuum. We apply this technique to plane wave propagation in moving isotropic media which, from the laboratory perspective (that sees the movement), are actually nonreciprocal bianisotropic media. Our results are in full agreement with the conventional tensor and dyadic analyses.

On the existence of the Gauss-Markov estimators in linear mixed models

The purpose of this article is to present a simpler way to verify the existence of the Gauss-Markov estimators (GME) of the expected mean which are both the best linear unbiased estimators (BLUE) of the mean in a multivariate mixed linear model and the best quadratic unbiased estimators (BQUE) of the covariance components in a new linear model built from the initial one. The results are expressed in a coordinate-free approach in order to accord them an intuitive representation in finite dimensional Hilbert spaces.

Some properties of the best linear unbiased estimators in multivariate growth curve models

The purpose of this article is to build a class of the best linear unbiased estimators (BLUE) of the linear parametric functions, to prove some necessary and sufficient conditions for their existence and to derive them from the corresponding normal equations, when a family of multivariate growth curve models is considered. It is shown that the classical BLUE known for this family of models is the element of a particular class of BLUE built in the proposed manner. The results are expressed in a convenient computational form by using the coordinate-free approach and the usual parametric representations.

Effects of three-body atomic interaction and optical lattice on solitons in quasi-one-dimensional Bose-Einstein condensate

We make use of a coordinate-free approach to implement Vakhitov-Kolokolov criterion for stability analysis in order to study the effects of three-body atomic recombination and lattice potential on the matter-wave bright solitons formed in Bose-Einstein condensates. We analytically demonstrate that (i) the critical number of atoms in a stable BEC soliton is just half the number of atoms in a marginally stable Townes-like soliton and (ii) an additive optical lattice potential further reduces this number by a factor of √1 − bg3 with g3 the coupling constant of the lattice potential and b = 0.7301.

Directed bondgraphs and integral matroids

The presentation of a theory for the mathematical foundations of the bondgraph modelling method is continued by this analysis of the combinatorial properties of directed bondgraphs (di-bondgraphs). This theory also provides a means to explore the relationship between a bondgraph model and a graph-theoretic model of the same system, as well as establishing a framework for developing a generalized coordinate-free approach which encompasses both methods for modelling a spatially discrete physical system. A di-bondgraph, B ¯ , is formed by adding a half-arrow on each bond of a bondgraph, called the underlying bondgraph. Only junctions and bonds are included, and no physical components or effort and flow variables. The junctions in B ¯ define a set of elementary integral chains which generate a chain group; its restriction to external bonds, the cycle chain group, provides a set of combinatorial constraints with integral coefficients. These are essential for representing polarities for physical measurements in expressing the spatial constraints (generalized Kirchhoff laws) that relate the physical variables in the model. The cycle chain group also provides an integral representation of the cycle matroid of the di-bondgraph, which represents the interconnections between components. Duals of these structures, the cocycle chain group and matroid, can be constructed for a di-bondgraph in the same way. Orthogonality over the integers is established for the cycle and cocycle chain groups, a crucial property which ensures that generalized energy, the product of each pair of dual variables, will be a conserved quantity over the model (Tellegen’s theorem or equivalent). Causality for di-bondgraphs is analyzed using pseudo-base colouring of the bonds, a procedure equivalent to the addition of causal strokes in a conventional bondgraph model. It is established that the pseudo-base set obtained from a pseudo-base colouring with no even causal loop always provides a base of the cycle matroid of a simple di-bondgraph with no redundant junction. Formulas for the rank and corank of such a di-bondgraph in terms of junction and external bond counts are provided.

Lax-Phillips scattering theory and well-posed linear systems: a coordinate-free approach

We give a further elaboration of the fundamental connections between Lax-Phillips scattering, conservative input/state/output linear systems and Sz.-Nagy-Foias model theory for both the discrete- and continuous-time settings. In particular, for the continuous-time setting, we show how to locate a scattering-conservative L2-well-posed linear system (in the sense of Staffans and Weiss) embedded in a Lax-Phillips scattering system presented in axiomatic form; conversely, given a scattering-conservative linear system, we show how one can view the space of finite-energy input-state-output trajectories of the system as the ambient space for an associated Lax-Phillips scattering system. We use these connections to give a simple, conceptual proof of the identity of the scattering function of the scattering system with the transfer function of the input-state-output linear system. As an application we show how system-theoretic ideas can be used to arrive at the spectral analysis of the scattering function.

Nonlinear shell theory: a duality approach

The nonlinear kinematics of thin shells is developed in full generality according to a duality approach in which kinematics plays the basic role in the definition of the model. The Kirchhoff‐Love shell model is the central issue and is discussed in detail but shear deformable and polar models are also considered and critically reviewed. The analysis is developed with a coordinate-free approach which provides a direct geometrical picture of the shell model. The finite and tangent Green strains of the foliated continuum are explicitly expressed in terms of middle surface kinematics. The new expressions contributed here do not require the splitting of the velocity into parallel and normal components to the middle surface, and provide a computationally convenient context. Finite strain measures for the shell and their tangent and secant rates are analyzed and consistency and nonredundancy properties are discussed. The relations between the finite Green strain, its tangent and secant rates and the corresponding shell strains, are provided. The differential and boundary equilibrium equations of the shell are given in variational terms, both in unsplit and split form. A new expression of the boundary equilibrium equations is contributed and its mechanical soundness with respect to the classical one is emphasized. Equilibrium in a reference placement for the shell model is briefly discussed. The theory of shells is intimately connected with the geometry of Riemannian manifolds and elements of the relevant theory are summarized in the paper. The middle surface of a shell model is a two-dimensional submanifold of the three-dimensional euclidean space (the classical physical space). The treatment of large configuration changes of the Kirchhoff‐Love shell is conveniently carried out by considering the shell as a foliation of the middle surface induced by a distance function. The description of the shell kinematics in terms of that of the middle surface is carried out by considering the nonlinear projection of the points of the foliation onto the middle surface and its derivative which is a symmetric linear operator expressible in terms of the linear projector onto the tangent plane and of the middle surface shape operator. Remarkably, this result is also useful in elastoplasticity for conveniently estimating the algorithmic tangent stiffness [Romano et al. 2005a; Romano et al. 2006]. The properties of consistency and nonredundancy to be fulfilled by a well defined strain measure are briefly illustrated to provide a basis for the analysis of shell models. The nonlinear analysis of shell deformation is developed in detail with a coordinate-free exposition. The treatment is founded on a variational approach in which kinematics plays a primary role while statics is introduced by duality. Stress fields are envisaged as Lagrange multipliers of the implicit representation of the rigidity constraint provided by a strain measure. In the classical three-dimensional continuum mechanics the Green strain measure is the most natural one. Accordingly,