Brachistochrone Research Articles

Geometrical aspects and quantum brachistochrone problem for a collection of N spin-s system with long-range Ising-type interaction

We consider a physical system of N interacting qudits consisting of N spin-s particles coupled via the long-range interaction of Ising-type. We investigate the corresponding dynamics, define the associated quantum state manifold and we give the related Fubini-Study metric. We derive the Gaussian curvature and using the Gauss-Bonnet theorem, we show that the dynamics happen on a two-parametric manifold of spherical topology. We examine the geometrical phase acquired by the system under arbitrary and cyclic evolutions. Further, we study the quantum brachistochrone problem concerning the determination of the smallest possible time to realize a time-optimal evolution. By restricting our study to a two-qubit system under the Ising interaction, a detailed analysis is performed for the Fubini-Study metric, the Gaussian curvature, the geometrical phase and the optimal time in relation with the entanglement of the two qubits.

Brachistochrone on a velodrome

The Brachistochrone problem, which describes the curve that carries a particle under gravity in a vertical plane from one height to another in the shortest time, is one of the most famous studies in classical physics. There is a similar problem in track cycling, where a cyclist aims to find the trajectory on the curved sloping surface of a velodrome that results in the minimum lap time. In this paper, we extend the classical Brachistochrone problem to find the optimum cycling trajectory in a velodrome, treating the cyclist as an active particle. Starting with two canonical cases of cycling on a sloping plane and a cone, where analytical solutions are found, we then solve the problem numerically on the reconstructed surface of the velodrome in Montigny le Bretonneux, France. Finally, we discuss the parameters of the problem and the effects of fatigue.

О структуре оптимальной тяги для «промежуточной» модели летательного аппарата

The paper considers a brachistochrone problem modification including in the objective function a fuel consumption penalty apart from the process time. The material point moves in a vertical plane under gravity, viscous nonlinear friction and traction. The trajectory slope angle and thrust are considered as a control variables. The Pontryagin maximum principle allows reducing the optimal control problem to a boundary value problem for a system of two nonlinear differential equations. Qualitative analysis of the resulting system allows studying the key features of extreme trajectories, including their asymptotic behavior. Extreme thrust control is obtained as a function of the velocity and the trajectory slope angle. The structure of extreme thrust is determined, and the number of switches is analytically determined. The results of numerical solving the boundary value problem are presented, illustrating the analytical conclusions.

Brachistochronic Motion of a Material Point on a Transcendental Surface

A new brachistochrone problem of the motion of a material point on a transcendental surface between two given points in a vertical uniform field of gravity is considered. The surface is a cut horizontal cylinder whose directrix is a cycloid, and generatrix is straight lines parallel to the horizontal axis. Energy dissipation is neglected. A time functional is obtained and a differential equation is derived for a single parameter that describes the form of brachistochrons in the problem. Spatial parametric equations of the brachistochronic motion of a material point on a surface are obtained. The fast response time is determined.

Sensor Coverage Strategy in Underwater Wireless Sensor Networks

This paper mainly describes studies hydrophone placement strategy in a complex underwater environment model to compute a set of "good" locations where data sampling will be most effective. Throughout this paper it is assumed that a 3-D underwater topographic map of a workspace is given as input.Since the negative gradient direction is the fastest descent direction, we fit a complex underwater terrain to a differentiable function and find the minimum value of the function to determine the low-lying area of the underwater terrain.The hydrophone placement strategy relies on gradient direction algorithm that solves a problem of maximize underwater coverage: Find the maximize coverage set of hydrophone inside a 3-D workspace. After finding the maximize underwater coverage set, to better take into account the optimal solution to the problem of data sampling, the finite VC-dimension algorithm computes a set of hydrophone that satisfies hydroacoustic signal energy loss constraints. We use the principle of the maximize splitting subset of the coverage set and the ”dual” set of the coverage covering set, so as to find the hitting set, and finally find the suboptimal set (i.e., the sensor suboptimal coverage set).Compared with the random deployment algorithm, although the computed set of hydrophone is not guaranteed to have minimum size, the algorithm does compute with high network coverage quality.

Analytical and Numerical Solution of the Problem on Brachistochrones in some General Cases

In this paper, we discuss J. Bernoulli’s brachistochrone problem and find its analytical and numerical solutions in the cases where viscous or dry friction are taken into account. We predict the existence of a point of “geometrical phase transition” u0 = ln(1/2k2b); it corresponds to the transition from one class of trajectories to another that qualitatively differs from the initial class. Numerical simulation of the motion in a neighborhood of points of geometric phase transitions is performed. We prove that in the absence of friction forces, the minimization problem for the motion time for any motion along a curvilinear trough under the action of the gravity force can always be reduced to the brachistochrone problem and can be solved without involving methods of calculus of variation, only by general dynamical laws. We find a solution to the classical Bernoulli problem under the condition that the length of the trajectory is fixed. We show that under this isoperimetric condition, the class of trajectories differs from the classical brachistochrone. We also observe the transformation of these trajectories to the cycloid by numerical and analytical analysis.

10.31673/2412-9070.2020.050311

The article considers an artificial neural network — a multilayer perceptron, as well as the Kohonen network, as tools for visual image recognition. The work is devoted to the analysis of the main functional tasks that can be solved by our chosen neural networks, with maximum efficiency and expediency. We will also consider these neural networks as tools for primary analysis of visual images for further use both for scientific purposes and for their substantive implementation in business areas where neural networks are most applicable and will bring maximum effect to ordinary users. One of the most common algorithms for learning multilayer neural networks is the error backpropagation algorithm. This algorithm calculates the gradient surface gradient vector. Then we move by some value in the direction of the vector (it will show us the direction of the fastest descent), where the error value will be less. This consistent progress will gradually minimize the error. Here there are difficulties with definition of size on which it is necessary to advance. If the size of the step is relatively large, it will lead to the fastest descent, but there is a chance to "jump" the desired point or go in the wrong direction, if the surface has a fairly complex shape. For example, if the surface is a narrow ravine with steep slopes, the algorithm will move very slowly, jumping from one slope to another. If the step size is small, it will lead to finding the most optimal direction, but can significantly increase the number of iterations. To achieve the most optimal result, the step size is taken in proportion to the steepness of the slope with some constant - the speed of learning. The choice of this constant is made experimentally and depends on the conditions of a particular problem. Having conducted the work, it should be noted that the multilayer perceptron is successfully used to solve various problems, in particular as effectively as possible for the problem of pattern recognition.

Algorithm for solving unconstrained unitary quantum brachistochrone problems

We introduce an iterative method to search for time-optimal Hamiltonians that drive a quantum system between two arbitrary, and in general mixed, quantum states. The method is based on the idea of progressively improving the efficiency of an initial, randomly chosen, Hamiltonian, by reducing its components that do not actively contribute to driving the system. We show that our method converges rapidly even for large dimensional systems, and that its solutions saturate any attainable bound for the minimal time of evolution. We provide a rigorous geometric interpretation of the iterative method by exploiting an isomorphism between geometric phases acquired by the system along a path and the Hamiltonian that generates it, and discuss resulting similarities with Grover's quantum search algorithm. Our method is directly applicable as a powerful tool for state preparation and gate design problems.

High-performance cycloid inspired wearable electromagnetic energy harvester for scavenging human motion energy

Eco-friendly and wearable power sources are in high demand because of the hasty growth in smart wearable electronic devices including health care monitoring sensors. Here, we successfully designed and fabricated a high-performance cycloid-inspired wearable electromagnetic energy harvester (CEEH) for scavenging low frequency (≤5 Hz) human motion energy. The proposed CEEH introduces a cycloid curved structure as an energy harvester for the first time which provides the fastest descent for the freely rolling spherical magnet in the curve path, resulting an increment in the rate of cutting magnetic flux. In order to demonstrate the capability of the proposed device for harvesting electrical energy from the natural human motions such as arm swinging and vibration, hand-shaking vibration motion tests and custom-made swinging arm tests were performed. The as-fabricated harvester can deliver an average power of 8.8 mW under the excitation vibration of 5 Hz at an optimum load resistance of 104.7 Ω. Moreover, it can continuously power a commercial sporty stopwatch for more than 16 min and a wristwatch for more than 34 min from just 5 s of hand motion vibration. The proposed device exhibits much enhanced performance which is more than 1.45 times higher in comparison with other different geometric structures such as straight and circular design. The outstanding energy harvesting capability of the proposed energy harvester shows huge potential as a wearable energy harvester for wrist and foot worn applications and as a sustainable power source for powering smart wearable or portable electronic devices and systems.

Restricted Brachistochrone

The motion of a bead along a path restricted to straight lines (restricted brachistochrone), sliding without friction from rest and accelerated by gravity, is considered. For two shapes of path, the geometry of the route optimized to provide the least travel time from one point to another is obtained. The bead’s travel times, path lengths, and average velocities are compared between the two presented models, and with travel along a cycloid path, which (as the solution to the original brachistochrone problem) provides the lowest possible travel time. The calculations are made with and without the use of calculus, and therefore the problems presented are comprehensible for a large variety of students.

Selected Applications of the Theory of Connections: A Technique for Analytical Constraint Embedding.

In this paper, we consider several new applications of the recently introduced mathematical framework of the Theory of Connections (ToC). This framework transforms constrained problems into unconstrained problems by introducing constraint-free variables. Using this transformation, various ordinary differential equations (ODEs), partial differential equations (PDEs) and variational problems can be formulated where the constraints are always satisfied. The resulting equations can then be easily solved by introducing a global basis function set (e.g., Chebyshev, Legendre, etc.) and minimizing a residual at pre-defined collocation points. In this paper, we highlight the utility of ToC by introducing various problems that can be solved using this framework including: (1) analytical linear constraint optimization; (2) the brachistochrone problem; (3) over-constrained differential equations; (4) inequality constraints; and (5) triangular domains.

Spline collocation approach to study Brachistochrone problem

In this paper authors discussed a problem of quickest descent, the Brachistochrone curve. Spline collocation method is used to solve the non-linear boundary value problem. The numerical results obtained are compared with the transformation method to show effectiveness and accuracy of this method.

Duality Gap Estimates for Weak Chebyshev Greedy Algorithms in Banach Spaces

The paper studies weak greedy algorithms for finding sparse solutions of convex optimization problems in Banach spaces. We consider the concept of duality gap, the values of which are implicitly calculated at the step of choosing the direction of the fastest descent at each iteration of the greedy algorithm. We show that these values give upper bounds for the difference between the values of the objective function in the current state and the optimal point. Since the value of the objective function at the optimal point is not known in advance, the current values of the duality gap can be used, for example, in the stopping criteria for the greedy algorithm. In the paper, we find estimates of the duality gap values depending on the number of iterations for the weak greedy algorithms under consideration.

BRAHISTOCHRONOUS MOTION OF THE MATERIAL POINT ON AN INCLINED PLANE IN A UNIFORM GRAVITATIONAL FIELD

Background. The variational problem, which is posed and solved in this work, is a natural generalization of the classical problem of I. Bernoulli about the search for brachistochrones in a vertical plane. In the proposed formulation, it is new and relevant from a practical point of view in such areas as engineering, transport and logistics, sports events, etc. Objective. The aim of the paper is to find such a curve on an inclined plane, moving along which, without an initial velocity in a uniform gravitational field, from one given point of the plane to another, the material point will make such a transition in the shortest time. Methods. To achieve this goal, the classical methods of the calculus of variations were used, namely, the Euler equation. Results. A time functional is constructed, using which the differential equation of the spatial brachistochrone, which lies on an inclined plane, is analytically derived. After its integration in a closed form, an algebraic brachistochrone equation is obtained. The results of the study are illustrated graphically. At the starting point M of the brachistochrone, the direction of the initial velocity of the material point is established. A comparative analysis of the transition time for the optimal brachistochrone curve and two alternative paths of motion of the material point is carried out. Conclusions. It is proved that the projection of the brachistochrone on the plane is not a cycloid. It is shown that the vector of the initial velocity of the material point at the starting point M of the brachistochrone is perpendicular to the x -axis. It was established that the minimum time of transition depends on the parameter a of the inclined plane, the energy dissipation coefficient k , and also on the coordinates of the starting M and finishing N points through which the brachistochrone passes.

The remarkable properties of the discrete brachistochrone

We present a new solution to the discrete brachistochrone problem, based on the variational principle. There are two interesting and surprising properties of the solution. First, the sliding times along all the straight segments are equal and are independent of the initial velocity. Secondly, we find a simple relation between the angle of the kth segment to the vertical and the angle to the vertical of the first segment. Based on these results, a Matlab code was written that calculates efficiently and accurately the solution for N segments between arbitrary end points.

On the brachistochrone of a fluid-filled cylinder

We discuss a fluid dynamic variant of the classical Bernoulli’s brachistochrone problem. The classical brachistochrone for a non-dissipative particle is governed by maximization of the particle’s kinetic energy, resulting in a cycloid. We consider a variant where the particle is replaced by a cylinder (bottle) filled with a viscous fluid and attempt to identify the shape of the curve connecting two points along which the bottle would move in the shortest time. We derive the system of integro-differential equations governing system dynamics for a given shape of the curve. Using these equations, we pose the brachistochrone problem by invoking an optimal control formalism and show that (in general) the curve deviates from a cycloid. This is due to the fact that increasing the rate of change of the bottle’s kinetic energy is accompanied by increased viscous dissipation. We show that the bottle motion is governed by a balance between the desire to minimize travel time and the need to reach the end point in the face of increased dissipation. The trade-off between these two physical forces plays a vital role in determining the brachistochrone of a fluid-filled cylinder. We show that in the two limits of either vanishing or high viscosity, the brachistochrone for this problem reduces to a cycloid. An intermediate viscosity range is identified where the fluid brachistochrone is non-cycloidal. Finally, we show the relevance of these results to the dynamics of a rolling liquid marble.

Brachistochrone – The Path of Quickest Descent

It is now more than three centuries since Johann Bernoulli solved one of the most intriguing problems in the history of the development of mathematics. Adapting Fermat’s principle of least time, applicable for the path followed by a ray of light as it passes through a series of media with decreasing values of refractive index, to the motion of a point mass under the influence of gravity alone, Bernoulli solved the problem of quickest descent of a point mass in a vertical plane from a point to a lower point, but not vertically below it. It is said to be one of the most important problems in mathematics as it paved the way for many branches of modern mathematics, including calculus of variations. In this article, we discuss the historical development of Bernoulli’s challenge problem, its solution, and several anecdotes connected with the story of brachistochrone. We conclude the article with an important property: the ‘tautochronous property’ of the brachistochrone curve, discovered by Huygens and used by himin making clocks. The spirit with which the business of mathematics was transacted in the centuries gone by is highlighted.

Optimization based trajectory planning for real-time 6DoF robotic patient motion compensation systems.

PurposeRobotic stabilization of a therapeutic radiation beam with respect to a dynamically moving tumor target can be accomplished either by moving the radiation source, the patient, or both. As the treatment beam is on during this process, the primary goal is to minimize exposure of normal tissue to radiation as much as possible when moving the target back to the desired position. Due to the complex mechanical structure of 6 degree-of-freedom (6DoF) robots, it is not intuitive as to what 6 dimensional (6D) correction trajectory is optimal in achieving such a goal. With proportional-integrative-derivative (PID) and other controls, the potential exists that the controller may generate a trajectory that is highly curved, slow, or suboptimal in that it leads to unnecessary exposure of healthy tissue to radiation. This work investigates a novel feedback planning method that takes into account a robot’s mechanical joint structure, patient safety tolerances, and other system constraints, and performs real-time optimization to search the entire 6D trajectory space in each time cycle so it can respond with an optimal 6D correction trajectory.MethodsComputer simulations were created for two 6DoF robotic patient support systems: a Stewart-Gough platform for moving a patient’s head in frameless maskless stereotactic radiosurgery, and a linear accelerator treatment table for moving a patient in prostate cancer radiation therapy. Motion planning was formulated as an optimization problem and solved at real-time speeds using the L-BFGS algorithm. Three planning methods were investigated, moving the platform as fast as possible (platform-D), moving the target along a straight-line (target-S), and moving the target based on the fastest descent of position error (target-D). Both synthetic motion and prior recorded human motion were used as input data and output results were analyzed.ResultsFor randomly generated 6D step-like and sinusoidal synthetic input motion, target-D planning demonstrated the smallest net trajectory error in all cases. On average, optimal planning was found to have a 45% smaller target trajectory error than platform-D control, and a 44% smaller target trajectory error than target-S planning. For patient head motion compensation, only target-D planning was able to maintain a ≤0.5mm and ≤0.5deg clinical tolerance objective for 100% of the treatment time. For prostate motion, both target-S planning and target-D planning outperformed platform-D control.ConclusionsA general 6D target trajectory optimization framework for robotic patient motion compensation systems was investigated. The method was found to be flexible as it allows control over various performance requirements such as mechanical limits, velocities, acceleration, or other system control objectives.

Numerical optimization of the transfer function of the intelligent building management system

The paper deals with structural-parametric models for describing dynamic processes of technical systems of an intelligent building. The task of searching for the transfer function of the synthesized elements and devices of its information-measuring and control systems based on the Mason method is formalized. The components of the transfer function are presented in the form of characteristic polynomials in the structural scheme of the energy-information model of the circuit. The results of a comparative analysis of search methods for multiple real and complex conjugate polynomial roots are presented. To organize their search, an iterative method of unconditional optimization of Fletcher-Reeves was chosen. This conjugate gradient method allows to solve the problem of numerical optimization in a finite number of steps and shows the best convergence in comparison with the methods of the fastest descent, with the same order of difficulty of performing the steps of the algorithm.

Analytical solution of gravity tunnels through an inhomogeneous Earth

A piecewise linear approximation of the Preliminary reference Earth model (PREM) is used for the Earth's velocity profile. The analytical solution derived for the motion on the shortest path and the path of fastest descent corresponds to the direct numerical integration of the PREM. This explains why traversal times can be approximated by the assumption of constant gravity, while the gravity of the Earth is certainly not constant.