Geometric Formulation Research Articles

Jet space extensions of infinite-dimensional Hamiltonian systems

We analyze infinite-dimensional Hamiltonian systems corresponding to partial Differential equations on one-dimensional spatial domains formulated with formally skew-adjoint Hamiltonian operators and non-quadratic Hamiltonian density. In various applications, the Hamiltonian density can depend on spatial derivatives of the state such that these systems can not straightforwardly be formulated as boundary port-Hamiltonian system using a Stokes-Dirac structure. In this work, we show that any Hamiltonian system of the above class can be reformulated as a Hamiltonian system on the jet space, in which the Hamiltonian density only depends on the extended state variable itself and not on its derivatives. Consequently, well-known geometric formulations with Stokes-Dirac structures are applicable. Additionally, we provide a similar result for dissipative systems. We illustrate the developed theory by means of the the Boussinesq equation, the dynamics of an elastic rod and the Allen-Cahn equation.

Geometrical wetting boundary condition for complex geometries in lattice Boltzmann color-gradient model

A wetting boundary condition for dealing with moving contact lines on complex surfaces is developed in the lattice Boltzmann color-gradient model. The wetting boundary condition is implemented by combining the geometrical formulation of contact angle and the idea of the prediction–correction wetting scheme, which not only produces the desired contact angles with high accuracy but also avoids the necessity to select an appropriate interface normal vector from multiple solutions that satisfy the contact angle condition. Through the implementation in the framework of color-gradient model, the developed wetting boundary condition is validated against analytical solutions by a series of benchmark cases, including a droplet resting on a cylindrical surface and on a tilt wall, a liquid film migrating between two parallel plates, and the forced imbibition into a pore doublet. The simulation results of static contact angles show that the wetting boundary condition is able to simulate arbitrary values of contact angle and leads to negligible mass leakage across the boundary. For dynamic problems, the wetting boundary condition is found to correctly capture the imbibition dynamics under various flow and viscosity ratio conditions and produce dynamic contact angles that match well with the Cox–Voinov law.

Mechanical design principles in frustrated thin elastic sheets.

Using a geometric formalism of elasticity theory we develop a systematic theoretical framework for shaping and manipulating the energy landscape of slender solids, and consequently their mechanical response to external perturbations. We formally express global mechanical properties associated with non-Euclidean thin sheets in terms of their local rest lengths and rest curvatures, and we interpret the expressions as both forward and inverse problems for designing the desired mechanical properties. We show that by wisely designing geometric frustration, anomalous mechanical properties can be encoded into a material using accessible experimental techniques. To test the methodology we derive a family of ribbon-springs with extreme mechanical behavior such as tunable, anharmonic, and even vanishing rigidities. The presented formalism can be discretized, offering a new methodology for designing mechanical properties and thus opens a new pathway for the design of both continuum and discrete solids and structures.

The Oblate Lambert Problem: Geometric Formulation and Solution of an Unperturbed, Generalized Lambert Problem Governed by Vinti’s Potential

Numerous methods exist for solving the Lambert problem, the two-point boundary value problem (BVP) governed by two-body dynamics. Many applications would benefit from a solution to a perturbed Lambert problem; a few studies have attempted to solve one. Establishing a larger pool of alternative solution methods gives practitioners greater latitude in choosing the solution that best suits their needs. To that end, a novel Lambert-type BVP is constructed in this work that includes oblateness by way of Vinti’s potential, rendering the problem mathematically unperturbed. This BVP is first defined and then converted to a system of equations that is amenable to an iterative solution. The formulation, which is valid for both the zero- and multiple-revolution problems, couples oblate spheroidal (OS) universal variables and OS equinoctial orbital elements together to sow robustness across all orbital regimes, only excepting orbits that are sufficiently rectilinear. For the first time, the solution space is broadly explored, exposing multiple new insights of significant practical use. Initial guess and root-solve techniques are offered to solve the system of equations. When assessed at Earth for robustness, accuracy, and computational efficiency, the zero-revolution algorithm excels across all three performance metrics, with runtimes averaging only about 15 times slower than a typical two-body Lambert solver. The multiple-revolution algorithm, while not yet evaluated as extensively, also exhibits high levels of performance, the formulation generally characterizing the existence of solutions around oblate bodies more accurately than its Keplerian counterpart.

Geometrical correspondence of the Miura transformation induced from affine Kac–Moody algebras

AbstractThe Miura transformation plays a crucial role in the study of integrable systems. There have been various extensions of the Miura transformation, which have been used to relate different kinds of integrable equations and to classify the bi‐Hamiltonian structures. In this paper, we are mainly concerned with the geometric aspects of the Miura transformation. The generalized Miura transformations from the mKdV‐type hierarchies to the KdV‐type hierarchies are constructed under both algebraic and geometric settings. It is shown that the Miura transformations not only relate integrable curve flows in different geometries but also induce the transition between different moving frames. Moreover, the Miura transformation gives the factorization of generating operators of constraint Gelfand–Dickey hierarchy. Other geometric formulations are also investigated.

A geometric framework for discrete time port‐Hamiltonian systems

AbstractPort‐Hamiltonian systems provide an energy‐based formulation with a model class that is closed under structure preserving interconnection. For continuous‐time systems, these interconnections are constructed by geometric objects called Dirac structures. In this paper, we derive this geometric formulation and the interconnection properties for scattering passive discrete‐time port‐Hamiltonian systems.

Strain, disfigurement and flexure tensors in continuum kinematics

In the differential geometric formulation of nonlinear elasticity, the strain tensor can intrinsically be regarded as a measure of change of the metric of the Riemannian manifold that embodies the continuum undergoing a deformation. Here, this classical concept is revisited and complemented with two newly defined higher order deformation measures, namely the disfigurement and flexure tensors, which respectively measure the change of the Levi-Civita connexion and the Riemann curvature of the continuum. The strain, disfigurement and flexure tensors can, thereby, roughly indicate the alteration of the geometry of a deforming body in the zeroth, first and second orders, respectively. In this light, one may predict that the disfigurement and flexure tensors should be related to the first and second covariant derivatives of the strain tensor. While this is true, the exact strain-disfigurement-flexure relations, as derived in this paper, are found to be involved and non-trivial, offering a deeper understanding of the deformation. In our discussions, distance-preserving, geodesic-preserving and curvature-preserving deformations are studied as three special cases to help better interpret the above-mentioned deformation measures. Some related results have also been found as by-products that might be of current interest or prove useful in future investigations.

A novel approach to egg and math: Improved geometrical standardization of any avian egg profile.

Developing a geometric formulation of any biological object has a number of justifications and applications. Recently, we developed a universal geometric figure for describing a bird's egg in any of the possible basic shapes: spherical, ellipsoidal, ovoid, and pyriform. The formulation proved widely applicable but had a number of drawbacks, including a very obvious join between two parts of the egg. To correct this, we developed the Main Axiom of the universal mathematical formula. This essentially involved making the ordinate of the extremum of the function correspond to half the maximum egg breadth (B), and the abscissa to the reciprocal of the parameter w that reflects the shift of the vertical axis to its coincidence with B. This, in turn, helped us develop a new, simplified mathematical model without a nonbiological join. Experimental verification was performed to confirm the adequacy of the new geometric figure. It accurately described actual avian eggs of various shapes more closely than our previous model. To the best of our knowledge, our new, simplified equation can be applied as a standard for any bird egg that exists in nature. As a rather simple equation, it can be used in a broad range of applications.

Semiclassical quantization of the superstring and Hagedorn temperature

In a recent paper [1], the semiclassical quantization of a string, winding once around the compact Euclidean time circle, on a supergravity background dual to the deep infrared regime of a confining finite temperature gauge theory, was carried out. The string mass-shell condition and, by extrapolation, the Hagedorn temperature to leading order in the holographic limit was deduced. In this work, we improve on those results in three ways. First, we fix some missing details of the related light-cone quantization analysis. Second, we reconsider the problem under the lens of a background-covariant geometrical formalism. This allows us to put the semiclassical mass-shell condition on more solid grounds. Finally, going beyond the semiclassical regime, we compute the Hagedorn temperature at next-to-leading order in the holographic limit. The sub-leading correction turns out to arise entirely from the contribution of the zero modes of the massive worldsheet scalar fields. Our result matches that of a recent analysis in the literature based on the Horowitz-Polchinski stringy star effective model.

An Intrinsic Version of the k-Harmonic Equation

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.

Simplified wetting boundary scheme in phase-field lattice Boltzmann model for wetting phenomena on curved boundaries.

In this work, a simplified wetting boundary scheme in the phase-field lattice Boltzmann model is developed for wetting phenomena on curved boundaries. The proposed scheme combines the advantages of the fluid-solid interaction scheme and geometric scheme-easy to implement (no need to interpolate the values of parameters exactly on solid boundaries and find proper characteristic vectors), the value of contact angle can be directly prescribed, and no unphysical spurious mass layer-and avoids mass leakage. Different from previous works, the values of the order parameter gradient on fluid boundary nodes are directly determined according to the geometric formulation rather than indirectly regulated through the order parameters on ghost solid nodes (i.e., ghost contact-line region). For this purpose, two numerical approaches to evaluate the order parameter gradient on fluid boundary nodes are utilized, one with the prevalent isotropic central scheme and the other with a local gradient scheme that utilizes the distribution functions. The simplified wetting boundary schemes with both numerical approaches are validated and compared through several numerical simulations. The results demonstrate that the proposed model has good ability and satisfactory accuracy to simulate wetting phenomena on curved boundaries under large density ratios.

Constraints on sequential discontinuities from the geometry of on-shell spaces

We present several classes of constraints on the discontinuities of Feynman integrals that go beyond the Steinmann relations. These constraints follow from a geometric formulation of the Landau equations that was advocated by Pham, in which the singularities of Feynman integrals correspond to critical points of maps between on-shell spaces. To establish our results, we review elements of Picard-Lefschetz theory, which connect the homotopy properties of the space of complexified external momenta to the homology of the combined space of on-shell internal and external momenta. An important concept that emerges from this analysis is the question of whether or not a pair of Landau singularities is compatible — namely, whether or not the Landau equations for the two singularities can be satisfied simultaneously. Under conditions we describe, sequential discontinuities with respect to non-compatible Landau singularities must vanish. Although we only rigorously prove results for Feynman integrals with generic masses in this paper, we expect the geometric and algebraic insights that we gain will also assist in the analysis of more general Feynman integrals.

Analytic plenoptic camera diffraction model and radial distortion analysis due to vignetting.

Using a mathematical approach, this paper presents a generalization of semi-analytical expressions for the point spread function (PSF) of plenoptic cameras. The model is applicable in the standard regime of the scalar diffraction theory while the extension to arbitrary main lens transmission functions generalizes a priori formalism. The accuracy and applicability of the model is well verified against the exact Rayleigh-Sommerfeld diffraction integral and a rigorous proof of convergence for the PSF series expression is made. Since vignetting can never be fully eliminated, it is critical to inspect the image degradation it poses through distortions. For what we believe is the first time, diffractive distortions in the diffraction-limited plenoptic camera are closely examined and demonstrated to exceed those that would otherwise be estimated by a geometrical optics formalism, further justifying the necessity of an approach based on wave optics. Microlenses subject to the edge diffraction effects of the main lens vignetting are shown to translate into radial distortions of increasing severity and instability with defocus. The distortions due to vignetting are found to be typically bound by the radius of the geometrical defocus in the image plane, while objects confined to the depth of field give rise to merely subpixel distortions.

Quintessence like behavior of symmetric teleparallel dark energy: Linear and nonlinear model

In Einstein’s General Relativity (GR), the gravitational interactions are described by the spacetime curvature. Recently, other alternative geometric formulations and representations of GR have emerged in which the gravitational interactions are described by the so-called torsion or non-metricity. Here, we consider the recently proposed modified symmetric teleparallel theory of gravity or [Formula: see text] gravity, where [Formula: see text] represents the non-metricity scalar. In this paper, motivated by several papers in the literature, we assume the power-law form of the function [Formula: see text] as [Formula: see text] (where [Formula: see text], [Formula: see text], and [Formula: see text] are free model parameters) that contains two models: Linear ([Formula: see text]) and nonlinear ([Formula: see text]). Further, to add constraints to the field equations we assume the deceleration parameter form as a divergence-free parametrization. Then, we discuss the behavior of various cosmographic and cosmological parameters such as the jerk, snap, lerk, [Formula: see text] diagnostic, cosmic energy density, isotropic pressure, and equation of state (EoS) parameter with a check of the violation of the strong energy condition (SEC) to obtain the acceleration phase of the Universe. Hence, we conclude that our cosmological [Formula: see text] models behave like quintessence dark energy (DE).

Automatic parking trajectory planning based on random sampling and nonlinear optimization

The current parking trajectory planning algorithms based on geometric connections or formulation of optimization problems in automatic parking systems have strict requirements on the starting position, lower planning efficiency and discontinuous curvature of the reference trajectory. In order to solve these problems, a hierarchical planning algorithm which is combined with nonlinear optimization and the improved RRT* algorithm (Rapidly-exploring Random Tree Star) with Reeds-Shepp curve is proposed in this paper. First, the improved RRT*RS algorithm with the rapid repulsion-straddle experiment is designed for enhancing the efficiency of path planning. Second, because of the shortcomings of the Reeds-Shepp curve that can meet the minimum turning radius but not realize the continuous curvature of the path, a nonlinear optimization problem based on convex-set obstacle constraints is formulated and solved. Finally, simulation results show that the proposed parking trajectory planning algorithm in this paper can plan an effective parking trajectory with continuous curvature in different starting positions and multiple parking scenarios.

Solutions of the Ortvay Rudolf International Competition in Physics: Mass of a Force Carrier Particle (2014/24 Problem)

This paper is the fourth in a series of published solutions discussing problems of the Ortvay Rudolf international competition. The problem treated below is an exercise in special relativity, applied to classical description of particle-exchange during scattering. As it is known, introducing rapidity can radically simplify the calculations in special relativity, as well as offer a simple geometric intuition. However, students often prefer doing calculations directly through formulas involving velocities, resulting in lengthy calculations, often increasing the chance of errors. It is our belief that giving the students simple, but not trivial problems before demonstrating the short and elegant solution presented below can help students to appreciate and understand the geometrical formulation of special relativity. Our calculation does not use mathematical techniques beyond those expected of high-school students, and the first half of this work may be used in special high-school classes.

A Hamiltonian and geometric formulation of general Vlasov-Maxwell-type models

A Hamiltonian and geometric formulation of general Vlasov-Maxwell-type models

Remarks on the Integral Form of D=11 Supergravity

We make some considerations and remarks on D = 11 supergravity and its integral form. We start from the geometrical formulation of supergravity and by means of the integral form technique we provide a superspace action that reproduces (at the quadratic level) the recent formulation of supergravity in pure spinor framework. We also make some remarks on Chevalley-Eilenberg cocycles and their Hodge duals.

Painlevé equations, integrable systems and the stabilizer set of Virasoro orbit

We study a geometrical formulation of the nonlinear second-order Riccati equation (SORE) in terms of the projective vector field equation on [Formula: see text], which in turn is related to the stability algebra of Virasoro orbit. Using Darboux integrability method, we obtain the first integral of the SORE and the results are applied to the study of its Lagrangian and Hamiltonian descriptions. Using these results, we show the existence of a Lagrangian description for SORE, and the Painlevé II equation is analyzed.

Teleparallel Newton–Cartan gravity

We discuss a teleparallel version of Newton–Cartan gravity. This theory arises as a formal large-speed-of-light limit of the teleparallel equivalent of general relativity (TEGR). Thus, it provides a geometric formulation of the Newtonian limit of TEGR, similar to standard Newton–Cartan gravity being the Newtonian limit of general relativity. We show how by a certain gauge-fixing the standard formulation of Newtonian gravity can be recovered.