Geometric Formulation Research Articles

Geometric decomposition of entropy production in out-of-equilibrium systems

Two qualitatively different ways of driving a physical system out of equilibrium, time-dependent and non-conservative forcing, are reflected by the decomposition of the system's entropy production into excess and housekeeping parts. We show that the difference between these two types of driving gives rise to a geometric formulation in terms of two orthogonal contributions to the currents in the system. This geometric picture in a natural way leads to variational expressions for both the excess and housekeeping entropy, which allow calculating both contributions independently from the trajectory data of the system. We demonstrate this by calculating the excess and housekeeping entropy of a particle in a time-dependent, tilted periodic potential.

Causality and dimensionality in geometric scattering

The scattering matrix which describes low-energy, non-relativistic scattering of spin-1/2 fermions interacting via finite-range potentials can be obtained from a geometric action principle in which space and time do not appear explicitly [1]. In the case of zero-range forces, causality leads to constraints on scattering trajectories in the geometric picture. The effect of spatial dimensionality is also investigated by considering scattering in two and three dimensions. In the geometric formulation it is found that dimensionality is encoded in the phase of the harmonic potential that appears in the geometric action.

Correction to: A geometric formulation of multirotor aerial vehicle dynamics

Correction to: A geometric formulation of multirotor aerial vehicle dynamics

A new geometric approach for sensitivity analysis in linear programming

In this paper, we present a new geometric approach for sensitivity analysis in linear programming that is computationally practical for a decision-maker to study the behavior of the optimal solution of the linear programming problem under changes in program data. First, we fix the feasible domain (fix the linear constraints). Then, we geometrically formulate a linear programming problem. Next, we give a new equivalent geometric formulation of the sensitivity analysis problem using notions of affine geometry which consists write the coefficient vector of the objective function in polar coordinate and determining all the angles for which the solution remains unchanged. Finally, the approach is presented in detail and illustrated with a numerical example.

Explaining Double Split Experiment with Geometrical Algebra Formalism

The Geometric Algebra formalism opens the door to developing a theory upgrading conventional quantum mechanics. Generalizations, stemming from implementation of complex numbers as geometrically feasible objects in three dimensions; unambiguous definition of states, observables, measurements bring into reality clear explanations of conventional weird quantum mechanical features, particularly the results of double split experiments where particles create diffraction patterns inherent to wave diffraction. This weirdness of the double split experiment is milestone of all further difficulties in interpretation of quantum mechanics.

On the closure property of Lepage equivalents of Lagrangians

Lepage equivalents of Lagrangians are a higher order, field-theoretical generalization of the notion of Poincaré-Cartan form from mechanics and play a similar role: they give rise to a geometric formulation (and to a geometric understanding) of the calculus of variations.A long-standing open problem is the determination, for field-theoretical Lagrangians λ of order greater than one, of a Lepage equivalent Φλ with the so-called closure property:Φλ is a closed differential form if and only if λ has vanishing Euler-Lagrange expressions.The present paper proposes a solution to this problem, for general Lagrangians of order r≥1. The construction is a local one; yet, we show that in most of the cases of interest for physical applications, the obtained Lepage equivalent Φλ is actually globally defined.A variant of this construction, which is convenient in the cases when λ is a reducible Lagrangian, is also introduced. In particular, for reducible Lagrangians of order two, the obtained Lepage equivalents are of order one.

A Geometric Model in 3+1D Space-Time for Electrodynamic Phenomena

With the idea to find geometric formulations of particle physics we investigate the predictions of a three-dimensional generalization of the Sine-Gordon model, very close to the Skyrme model and to the Wu-Yang description of Dirac monopoles. With three rotational degrees of freedom of spatial Dreibeins, we formulate a Lagrangian and confront the predictions to electromagnetic phenomena. Stable solitonic excitations we compare with the lightest fundamental electric charges, electrons, and positrons. Two Goldstone bosons we relate to the properties of photons. These particles are characterized by three topological quantum numbers, which we compare to charge, spin, and photon numbers. Finally, we conjecture some ideas for further comparisons with experiments.

Supersonic flutter analysis of variable thickness blades in thermal environment by using isogeometric approach

A modeling procedure is presented for the supersonic flutter analysis of variable thickness (VK) blade subjected to thermal environment in this paper. The considered blades which are made of temperature related FG materials have two different thickness variations. And the linear and nonlinear temperature conditions vary only along the thickness direction. Based on the combination of a refined plate theory (RPT) and isogeometric approach (IGA), the VK blade displacements and geometric formulations are accurately developed. The C1 continuity of RPT for basis functions is easily achieved by the IGA method. The aerodynamic effect is calculated by supersonic piston theory in which the yawed flow angle effect is taken into account. By comparing the present results with some existing data, the accuracy and stability of present modeling are proved. Finally, some parametric studies are carried out to explore the influence of two types of thickness variations, thermal conditions, yawed flow angles and material gradient parameters on the flutter behaviors of the blade.

On Differential Geometric Formulations of Slow Invariant Manifold Computation: Geodesic Stretching and Flow Curvature

The theory of slow invariant manifolds (SIMs) is the foundation of various model-order reduction techniques for dissipative dynamical systems with multiple time-scales, e.g. in chemical kinetic models. The construction of SIMs and many approximation methods exploit the restrictive requirement of an explicit time-scale separation parameter. Most of those methods are also not formulated covariantly, i.e. in terms of tensorial constructions. We propose an intrinsically coordinate-free differential geometric approximation criterion approximating normally attracting invariant manifolds (NAIMs). We translate some ideas behind existing approximation approaches, the stretching based diagnostics (SBD) and the flow curvature method (FCM) to tensors of Riemannian geometry, specifically to spacetime curvature in extended phase space. For that purpose we derive from flow-generating smooth vector fields a metric tensor such that the original dynamical system is a geodesic flow on a Riemannian manifold. We apply the resulting method to test models.

Geometry of information structures, strategic measures and associated stochastic control topologies

In many areas of applied mathematics, decentralization of information is a ubiquitous attribute affecting how to approach a stochastic optimization, decision and estimation, or control problem. In this review article, we present a general formulation of information structures under a probability theoretic and geometric formulation. We define information structures, place various topologies on them, and study closedness, compactness and convexity properties on the strategic measures induced by information structures and decentralized control/decision policies under varying degrees of relaxations with regard to access to private or common randomness. Ultimately, we present existence and tight approximation results for optimal decision/control policies. We discuss various lower bounding techniques, through relaxations and convex programs ranging from classically realizable and classically non-realizable (such as quantum theoretic and non-signaling) relaxations. For each of these, we establish closedness and convexity properties and also a hierarchy of correlation structures. As a further theme, we review and introduce various topologies on decision/control strategies defined independent of information structures, but for which information structures determine whether the topologies entail utility in arriving at existence, compactness, convexification or approximation results. These approaches, which we term as the strategic measures approach and the control topology approach, lead to complementary results on existence, approximations and upper and lower bounds in optimal decentralized stochastic decision, estimation, and control problems.

The 3-Sphere Instead of Hilbert Space

The Geometric Algebra formalism opens the door to developing a theory replacing conventional quantum mechanics. Generalizations, stemming from changing of complex numbers by geometrically feasible objects in three dimensions, followed by unambiguous definition of states, observables, measurements, bring into reality clear explanations of weird quantum mechanical features, for example primitively considering atoms as a kind of solar system. The three-sphere S3 becomes the playground of the torsion kind states eliminating abstract Hilbert space vectors. The S3 points evolve, governed by updated Schrodinger equation, and act in measurements on observable as operators.

Estimation of breakage position in wire rope cross-section by geometrical formulation of wire shape

Estimation of breakage position in wire rope cross-section by geometrical formulation of wire shape

A geometrical formulation for the workspace of parallel manipulators

AbstractIn study this paper, a geometric formulation is proposed to describe the workspace of parallel manipulators by using a recursive approach as an extension of volume generation for solids of revolution. In this approach, the workspace volume and boundary for each limb of the parallel manipulator is obtained with an algebraic formulation derived from the kinematic chain of the limb and the motion constraints on its joints. Then, the overall workspace of the mechanism can be determined as the intersection of the limb workspaces. The workspace of different kinematic chains is discussed and classified according to its external shape. An algebraic formulation for the inclusion of obstacles in the computation is also proposed. Both analytical models and numerical simulations are reported with their advantages and limitations. An example on a 3-SPR parallel mechanism illustrates the feasibility of the formulation and its efficiency.

NON-UNIFORM RATIONAL B-SPLINE BASED ISO-GEOMETRIC ANALYSIS FOR A CLASS OF HYDRODYNAMIC PROBLEMS

Herein, we present the design and development of a ‘Non-uniform Rational B-spline (NURBS)’ based iso-geometric approach for the analysis of a number of ‘Boundary Value Problems (BVPs)’ relevant in hydrodynamics. We propose a ‘Potential Function’ based ‘Boundary Element Method (BEM)’ and show that it holds the advantage of being computationally efficient over the other known numerical methods for a wide range of external flow problems. The use of NURBS is consistent, as inspired by the ‘iso-geometric analysis’, from geometric formulation for the body surface to the potential function representation to interpolation. The control parameters of NURBS are utilised and they have been explored to arrive at some preferable values and parameters for parameterization and the knot vector selection. Also, the present paper investigates the variational strength panel method, and its computational performance is analyzed in comparison with the constant strength panel method. The two variations have been considered, e.g. linear and quadratic. Finally, to illustrate the effectiveness and efficiency of the proposed NURBS based iso-geometric approach for the analysis of boundary value problems, five different problems (i.e. flow over a sphere, effect of the knot vector selection on analysis, flow over a rectangular wing section of NACA 0012 aerofoil section, performance of DTMB 4119 propeller (un-skewed), performance of DTNSDRC 4382 propeller (skewed)) are considered. The results show that in the absence of predominant viscous effects, a ‘Potential Function’ based BEM with NURBS representation performs well with very good computational efficiency and with less complexity as compared to the results available from the existing approaches and commercial software programs, i.e. low maximum errors close to 110−3 , faster convergence with even up to 75 % reduction in the number of panels and improvements in the computational efficiency up to 32.5 % even with low number of panels.

Ковариантный формализм гидродинамики гамильтоновых систем

The theory of hydrodynamic reduction of non-autonomous Hamiltonian mechanics (V. Kozlov, 1983) is presented in the geometric formalism of bundles over the time axis R. In this formalism, time is one of the coordinates, not a parameter; the connections describe reference frames and velocity fields of mechanical systems. The equations of the theory are presented in a form that is invariant with respect to time-dependent coordinate transformations and the choice of reference frames.

Consistent higher order sigma left(mathcal{GG}to hright) , Gamma left(hto mathcal{GG}right) and \u0393(h \u2192 \u03b3\u03b3) in geoSMEFT

We report consistent results for Γ(h → γγ), sigma left(mathcal{GG}to hright) and Gamma left(hto mathcal{GG}right) in the Standard Model Effective Field Theory (SMEFT) perturbing the SM by corrections mathcal{O}left({overline{upsilon}}_T^2/16{pi}^2{Lambda}^2right) in the Background Field Method (BFM) approach to gauge fixing, and to mathcal{O}left({overline{upsilon}}_T^4/{Lambda}^4right) using the geometric formulation of the SMEFT. We combine and modify recent results in the literature into a complete set of consistent results, uniforming conventions, and simultaneously complete the one loop results for these processes in the BFM. We emphasize calculational scheme dependence present across these processes, and how the operator and loop expansions are not independent beyond leading order. We illustrate several cross checks of consistency in the results.

The one-loop tadpole in the geoSMEFT

Making use of the geometric formulation of the Standard Model Effective Field Theory we calculate the one-loop tadpole diagrams to all orders in the Standard Model Effective Field Theory power counting. This work represents the first calculation of a one-loop amplitude beyond leading order in the Standard Model Effective Field Theory, and discusses the potential to extend this methodology to perform similar calculations of observables in the near future.

A geometric formulation of multirotor aerial vehicle dynamics

A geometric dynamic modeling framework for generic multirotor aerial vehicles (MAV), based on a modern Lie group formulation of classical screw theory, is presented. Our framework allows for a broad range of rotor-wing configurations: any number of rotors can be attached in arbitrary configurations to either the body or wings, with the rotors and wings also tiltable. Our framework takes into account all masses and inertias of the MAV body and rotors, and accounts for both rotor thrust forces and moments as well as external aerodynamic and other forces. Compared to existing methods, our Lie group framework possesses several practical advantages useful for applications ranging from design optimization to model identification and trajectory optimization: (1) the dynamic equations can be easily transformed to coordinates of any reference frame; (2) kinematic and mass–inertial parameters can be easily factored from the dynamic equations; (3) exact, closed-form analytic derivatives of the dynamics with respect to the configuration variables are easily derived. We demonstrate our systematic modeling procedure on examples of fixed-tilt, variable-tilt and hybrid MAVs with wings.

A geometric formulation of linear elasticity based on discrete exterior calculus

A direct formulation of linear elasticity of cell complexes based on discrete exterior calculus is presented. The primary unknowns are displacements, represented by a primal vector-valued 0-cochain. Displacement differences and internal forces are represented by a primal vector-valued 1-cochain and a dual vector-valued 2-cochain, respectively. The macroscopic constitutive relation is enforced at primal 0-cells with the help of musical isomorphisms mapping cochains to smooth fields and vice versa. The balance of linear momentum is established at primal 0-cells. The governing equations are solved as a Poisson’s equation with a non-local and non-diagonal material Hodge star. Numerical simulations of several classical problems with analytic solutions are presented to validate the formulation. Excellent agreement with known solutions is obtained. The formulation provides a method to calculate the relations between displacement differences and internal forces for any lattice structure, when the structure is required to follow a prescribed macroscopic elastic behaviour. This is also the first and critical step in developing formulations for dissipative processes in cell complexes.

Geometry and topology of anti-BRST symmetry in quantized Yang–Mills gauge theories

The entire geometric formulations of the BRST and the anti-BRST structures are worked out in presence of the Nakanishi–Lautrup field. It is shown that in the general form of gauge fixing mechanisms within the Faddeev–Popov quantization approach, the anti-BRST invariance reflects thoroughly the classical symmetry of the Yang–Mills theories with respect to gauge fixing methods. The Nakanishi–Lautrup field is also defined and worked out as a geometric object. This formulation helps us to introduce two absolutely new topological invariants of quantized Yang–Mills theories, so-called the Nakanishi–Lautrup invariants. The cohomological structure of the anti-BRST symmetry is also studied and the anti-BRST topological index is derived accordingly.