arXiv subjects

Oliver Gross

Publications and source records attributed to Oliver Gross.

9 recordsLinked to original sources

arXiv2026

Elastic Curves via Geometric Mechanics

Elastic curves are the mathematical shapes of thin elastic rods in equilibrium, with deep connections to mechanics, geometry, and computer graphics. Traditionally described as stationary points of bending energy under length and torsion constraints, their rich theory admits many equivalent characterizations. We develop a new one from the viewpoint of geometric mechanics. Our main contribution relies on a lesser-known isoperimetric characterization: a curve is elastic if and only if it is a critical point of the length functional under fixed area and volume vectors. We show that these constraints transform naturally under orientation-preserving rigid body motions, identifying them as momentum variables for these symmetries. This structure suggests a new discrete theory. We show that the low-order integral quantities length, area, and volume vectors are all naturally defined for polygonal curves, leaving the same transformation laws exactly satisfied. The resulting definition of discrete elastic curves in terms of the isoperimetric characterization restricted to discrete polygonal curves is variational, structure-preserving, and requires no auxiliary discretizations of curvature or material frames. Finally, the same structure carries the Marsden--Weinstein form, a canonical (pre-)symplectic structure on the space of curves, to polygonal curves. This yields novel approaches to Hamiltonian dynamics on discrete space curves, including tangent, vortex-filament, and modified Korteweg--de Vries flows.

math.DG↗

arXiv2026

Vakonomic Fluids

We introduce a novel discretization of the incompressible Euler equations based on their interpretation as geodesic equations on the Lie group of volume-preserving diffeomorphisms. It is well known that encoding diffeomorphisms and their infinitesimal generators through a discretized Koopman representation places a nonholonomic constraint on discrete velocities, for which there is no consensus on a variational treatment. We show that taking the vakonomic perspective, as opposed to the usual perspective of Lagrange--d'Alembert, yields discrete fluid trajectories that remain geodesics on a (sub-)Riemannian manifold. In particular, the resulting vakonomic dynamics are Lie--Poisson and their solutions admit a discrete relabeling symmetry, leading to machine-precision satisfaction of Casimir invariants along with a discrete analogue of Kelvin's Circulation Theorem. Using an efficient momentum map representation based on low-rank Clebsch variables, we show that these vakonomic fluids behave stably and consistently even at low grid resolutions, leading to increased robustness and physical realism in the long term.

math-ph↗

arXiv2026

(Semi-)Invariant Curves from Centers of Triangle Families

We study curves obtained by tracing triangle centers within special families of triangles, focusing on centers and families that yield (semi-)invariant triangle curves, meaning that varying the initial triangle changes the loci only by an affine transformation. We identify four two-parameter families of triangle centers that are semi-invariant and determine which are invariant, in the sense that the resulting curves for different initial triangles are related by a similarity transformation. We further observe that these centers, when combined with the aliquot triangle family, yield sheared Maclaurin trisectrices, whereas the nedian triangle family yields Lima\c{c}on trisectrices.

math.MG↗

arXiv2026

Sub--Riemannian boundary value problems for Optimal Geometric Locomotion

We propose a geometric model for optimal shape-change-induced motions of slender locomotors, e.g., snakes slithering on sand. In these scenarios, the motion of a body in world coordinates is completely determined by the sequence of shapes it assumes. Specifically, we formulate Lagrangian least-dissipation principles as boundary value problems whose solutions are given by sub-Riemannian geodesics. Notably, our geometric model accounts not only for the energy dissipated by the body's displacement through the environment, but also for the energy dissipated by the animal's metabolism or a robot's actuators to induce shape changes such as bending and stretching, thus capturing overall locomotion efficiency. Our continuous model, together with a consistent time and space discretization, enables numerical computation of sub-Riemannian geodesics for three different types of boundary conditions, i.e., fixing initial and target body, restricting to cyclic motion, or solely prescribing body displacement and orientation. The resulting optimal deformation gaits qualitatively match observed motion trajectories of organisms such as snakes and spermatozoa, as well as known optimality results for low-dimensional systems such as Purcell's swimmers. Moreover, being geometrically less rigid than previous frameworks, our model enables new insights into locomotion mechanisms of, e.g., generalized Purcell's swimmers. The code is publicly available.

cs.RO↗

arXiv2025

Isothermal Annealing Effects on $\beta$-Relaxations and Crystallization Behaviors in Amorphous GeTe

A secondary $\beta$-relaxation process is often the dominant source of atomic dynamics below $T_\mathrm{g}$ in many glass forming systems. Recent studies reported the presence of $\beta$-relaxations in amorphous phase-change materials (PCMs) and showed that suppressing the $\beta$-relaxation via annealing in Ge$_{15}$Sb$_{85}$ can effectively slow down its crystallization kinetics. Yet, when Sb is replaced by Te, similar annealing protocol has little effect on the Te-rich alloy Ge$_{15}$Te$_{85}$. Here, we investigate amorphous GeTe that is a Sb-free PCM, but with faster crystallization kinetics than Ge$_{15}$Te$_{85}$. Using powder mechanical dynamic spectroscopy, we observe a clear reduction of the excess-wing in the loss modulus upon isothermal annealing, indicating a suppression of its $\beta$-relaxation. Ultrafast calorimetric analysis and time-resolved optical reflectivity measurements show that, whereas as-deposited GeTe exhibit stochastic crystallization behaviors, annealed samples crystallize more slowly with reduced stochasticity. Synchrotron X-ray scattering experiments reveal reinforced Peierls-like distortions in the amorphous structure after annealing, and demonstrate that, even if annealing introduces nucleation sites, it nonetheless slows down crystallization kinetics. These finding suggests that, in annealed GeTe, crystallization is limited by crystal growth rate, which is retarded through the suppression of $\beta$-relaxation.

cond-mat.mtrl-sci↗

arXiv2024

A variational approach to geometric mechanics for undulating robotic locomotion

Limbless organisms of all sizes use undulating patterns of self-deformation to locomote. Geometric mechanics, which maps deformations to motions, provides a powerful framework to formalize and investigate the theoretical properties and limitations of such modes of locomotion. However, the inherent level of abstraction poses a challenge when bridging the gap between theory or simulations and laboratory experiments. We investigate the challenges of modeling motion trajectories of an undulating robotic locomotor by comparing experiments and simulations performed with a variational integrator. Despite the extensive simplifications that the model based on a geometric variation principle entails, the simulations show good agreement on average. Notably, our approach merely requires the knowledge of the \emph{dissipation metric} -- a Riemannian metric on the configuration space, which can in practice be approximated by means closely resembling \emph{resistive force theory}.

cs.RO↗

arXiv2024

Elastic Curves with Variable Bending Stiffness

We study stationary points of the bending energy of curves $\gamma\colon[a,b]\to\mathbb{R}^n$ subject to constraints on the arc-length and the curve's holonomy while simultaneously allowing for a variable bending stiffness along the arc-length of the curve. Physically, this can be understood as a model for an elastic wire with isotropic cross-section of varying thickness. We derive the corresponding Euler-Lagrange equations for variations that are compactly supported away from the end points thus obtaining characterizations for elastic curves with variable bending stiffness. Moreover, we provide a collection of alternative characterizations, e.g., in terms of the curvature function. Adding to numerous known results relating elastic curves to dynamics, we explore connections between elastic curves with variable bending stiffness, variable length pendulums and the flow of vortex filaments with finite thickness.

math.DG↗

arXiv2023

Force-Free Fields are Conformally Geodesic

In this paper, we establish an equivalence between force-free fields and conformally geodesic fields, and between harmonic fields and conformally eikonal fields in the context of conformal geometry. In contrast to previous work, our approach and equivalence results generalize to arbitrary dimensions. In accordance with three-dimensional theory, our defining equations emerge as the Euler-Lagrange equations of hierarchies of variational principles - distinguished by the topological constraints they impose - and retain the known inclusions of the special cases from each other. Specifically, we relate stationary points of hierarchies of $L^2$ resp. $L^1$-optimization problems by a conformal change of metric, provide an explicit construction of the conformal factors relating the relevant metrics and identify the field lines of physical vector fields fields as conformal geodesics. Despite the allowed topological complexity of the fields under consideration, these observations reveal geometric order which is obtained by merely pointwise rescaling of the metric.

math.DG↗

arXiv2022

Disentangling structural and kinetic components of the α-relaxation in supercooled metallic liquids

The particle motion associated to the α-relaxation in supercooled liquids is still challenging scientists due to its difficulty to be probed experimentally. By combining synchrotron techniques, we found the existence of microscopic structure-dynamics relationships in Pt42.5Cu27Ni9.5P21 and Pd42.5Cu27Ni9.5P21 liquids which allows us to disentangle structural and kinetic contributions to the α-process. While the two alloys show similar kinetic fragilities, their structural fragilities differ and correlate with the temperature dependence of the stretching parameter describing the decay of the density fluctuations. This implies that the evolution of dynamical heterogeneities in supercooled alloys is determined by the rigidity of the melt structure. We find also that the atomic motion not only reflects the topological order but also the chemical short-range order, which can lead to a surprising slowdown of the α-process at the mesoscopic length scale. These results will contribute to the comprehension of the glass transition, which is still missing.

cond-mat.mtrl-sci↗