Frenet-Serret Curvature Research Articles

Hamiltonian dynamics of gonihedric string theory

We develop in a consistent manner the Ostrogradski–Hamilton framework for gonihedric string theory. The local action describing this model, being invariant under reparametrizations, depends on the modulus of the mean extrinsic curvature of the worldsheet swept out by the string, and thus we are confronted with a genuine second-order in derivatives field theory. In our geometric approach, we consider the embedding functions as the field variables and, even though the highly nonlinear dependence of the action on these variables, we are able to complete the classical analysis of the emerging constraints for which, after implementing a Dirac bracket, we are able to identify both the gauge transformations and the proper physical degrees of freedom of the model. The Ostrogradski–Hamilton framework is thus considerable robust as one may recover in a straightforward and consistent manner some existing results reported in the literature. Further, in consequence of our geometrical treatment, we are able to unambiguously recover as a by-product the Hamiltonian approach for a particular relativistic point-particle limit associated with the gonihedric string action, that is, a model linearly depending on the first Frenet–Serret curvature.

Curvature-dependent energies minimizers and visual curve completion

Geometrical actions often used to describe elastic properties of elastic rods and fluid membranes have been proposed recently to explain functional mechanism of the primary visual cortex V1. These energies are defined in terms of functionals depending on the Frenet–Serret curvatures of a curve (profile curve, for axisymmetric membranes) and are relevant in image restoration by curve completion. In this context, extremals of length, total squared curvature (bending energy) and total squared torsion, acting on spaces of curves of the unit tangent bundle of the plane, are studied here. We first see that Sub-Riemannian geodesics in \({\mathbb {R}}^2\times {\mathbb {S}}^1\) project down to minimizers of a total curvature type energy in the plane. This motivates us to analyze the associated variational problem in Euclidean space under different boundary conditions. Although, as we show, parametrized extremals can be obtained by quadratures, their concrete explicit determination faces technical difficulties which can be overcome numerically. We use a numerical approach, based on a gradient descent method, to determine both critical trajectories for these three energies and their projection into the image plane under different boundary and isoperimetric constraints.

On the Accelerated Observer’s Proper Coordinates and the Rigid Motion Problem in Minkowski Spacetime

Physicists have been interested in accelerated observers for quite some time. Since the advent of special relativity, many authors have tried to understand these observers in the framework of Minkowski spacetime. One of the most important issues related to these observers is the problematic definition of rigid motion. In this paper, I write the metric in terms of the Frenet-Serret curvatures and the proper coordinate system of a general accelerated observer. Then, I use this approach to create a systematic way to construct a rigid motion in Minkowski spacetime. Finally, I exemplify the benefits of this procedure by applying it to two well-known observers, namely, the Rindler and the rotating ones, and also by creating a set of observers that, perhaps, may be interpreted as a rigid cylinder which rotates while accelerating along the axis of rotation.

Measurement Properties of Simple Biomechanical Measures of Walking Effort

A simple measure of overall walking effort would be valuable to patients and clinicians to select suitable treatment interventions and to monitor progress. In this paper, the reproducibility and responsiveness of seven potentially useful clinical measures of walking effort are presented. These walking effort outcomes were derived from a compass gait model and space curve displacement, acceleration and differential geometry theory. The walking effort outcomes were primarily calculated from the motion of a point on or in the rigid body pelvis as a patient walked cyclically. These motion data were collected from eight healthy volunteers who each walked on a treadmill for 4 min, at four different speeds, repeated twice. Four of the seven walking effort outcomes clearly had better measurement properties. The path length ratio, acceleration ratio, Frenet–Serret torsion and Frenet–Serret energy had excellent reproducibility (ICC>0.8) and responded to a small change in walking speed (< 0.03 m/s) compared to two versions of the biomechanical efficiency quotient and the Frenet–Serret curvature. The measurement properties of most outcomes were not consistently improved using a point in versus on the pelvis. This study presents four biomechanical walking effort outcomes that have good theoretical underpinnings, excellent reproducibility and responsiveness, are simple and easy to administer with relatively inexpensive equipment, and can be used in real world environments. However, future work must investigate the minimal clinically important change of these outcome measures before they can be used in clinical practice.

Hamiltonian Frenet–Serret dynamics

The Hamiltonian formulation of the dynamics of a relativistic particle described by a higher derivative action that depends both on the first and the second Frenet–Serret curvatures is considered from a geometrical perspective. We demonstrate how reparametrization covariant dynamical variables and their projections onto the Frenet–Serret frame can be exploited to provide not only a significant simplification of but also novel insights into the canonical analysis. The constraint algebra and the Hamiltonian equations of motion are written down and a geometrical interpretation is provided for the canonical variables.

Frenet–Serret dynamics

We consider the motion of a particle described by an action that is a functional of the Frenet–Serret (FS) curvatures associated with the embedding of its worldline in Minkowski space. We develop a theory of deformations tailored to the FS frame. Both the Euler–Lagrange equations and the physical invariants of the motion associated with the Poincaré symmetry of Minkowski space, the mass and the spin of the particle, are expressed in a simple way in terms of these curvatures. The simplest non-trivial model of this form, with the Lagrangian depending on the first FS (or geodesic) curvature, is integrable. We show how this integrability can be deduced from the Poincaré invariants of the motion. We go on to explore the structure of these invariants in higher-order models. In particular, the integrability of the model described by a Lagrangian, that is, a function of the second FS curvature (or torsion), is established in a three-dimensional ambient spacetime.

Quasi-harmonicity and power spectra in the FPU model

We evaluate the power spectra of the time series for the following simple observables in the Fermi-Pasta-Ulam model: harmonic energy, kinetic energy, microcanonical density, Frenet-Serret curvature and the Lyapunov variable. For some of these observables, also in the stochastic regime, the spectra show a well defined quasi-harmonic structure, with harmonic frequencies shifted with a single rescaling factor, as calculated in a previous paper. Even higher frequencies are excited: as replicas of the harmonic window at low energy, to end up with a smooth distribution at high energy, showing a power law behaviour (flicker noise). In the intermediate region the shape depends on the observables, but in all cases the crossover is the maximum of the shifted harmonic spectrum. This establishes an intrinsic short-time scale depending only on the energy density, as does the frequency rescaling factor. For the curvature, we also evaluate the standard deviation: above threshold, at increasing energy, it decreases exactly as the inverse of the rescaling factor. This can be interpreted as a focalization around `effective tori' of a harmonic-like regime which apparently coexist with the chaotic motion.