Classical Brachistochrone Problem Research Articles

Minimum time controller in a class of chemical reactors based on Lagrangian approach

Abstract The main goal of this work is the construction of a class of controller, which employs directly a Lagrangian formulation to resolve the classical brachistochrone problem, this allows to obtain an optimal controller which reaches in a minimum time the stabilization of an isothermal continuous stirred tank reactor, whose chemical kinetic model is based on the power law. The proposed methodology is compared with an input/output linearizing which achieve asymptotic and exponential closed-loop convergence, sliding-mode controller with a finite time convergence and an exact gradient optimal control to compare the time convergence performance. Numerical experiments show the satisfactory performance of the proposed controller, despite sustained disturbances in the concentration input feed.

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.

Modern explorations of the Brachistochrone-related problem: using the Udwadia–Kalaba approach

Modern explorations regarding the search for a curve subject to a minimization principle and its inverse, namely the search for the minimization object based on a given curve, are made. By using the Udwadia–Kalaba theory, which subsumes the Coulomb friction force as a non-ideal constraint, we obtain the analytic expression of the equation of motion for a particle moving along the Brachistochrone cycloid curve under Coulomb friction. Then we perform the inverse problem for the moving particle. That is, while the Brachistochrone cycloid curve is given, we seek the corresponding minimization object. Both the situations with and without friction are addressed. Finally, we return to the search for a curve subject to a minimization principle to complete the loop. However, this time we presume the minimization object is the total travel time, which was addressed in the classical Brachistochrone problem (hence, frictionless), while recognizing the presence of Coulomb friction. All three analyses come to meet at the special case when there is no friction. Our research reveals profound insights that have not been reported previously. The loop analysis also suggests a new angle for the study of dynamic systems.

Approaching the brachistochrone using inclined planes—striving for shortest or equal travelling times

The classical brachistochrone problem asks for the path on which a mobile point M just driven by its own gravity will travel in the shortest possible time between two given points A and B. The resulting curve, the cycloid, will also be the tautochrone curve, i.e. the travelling time of the mobile point will not depend on its starting position. We discuss three similar problems of increasing complexity that restrict the motion to inclined planes. Without using calculus we derive the respective optimal geometry and compare the theoretical values to measured travelling times. The observed discrepancies are quantitatively modelled by including angular motion and friction. We also investigate the correspondence between the original problem and our setups. The topic provides a conceptually simple yet non-trivial problem setting inviting for problem based learning and complex learning activities such as planing suitable experiments or modelling the relevant kinematics.

The speed graph method: pseudo time optimal navigation among obstacles subject to uniform braking safety constraints

This paper considers the synthesis of pseudo time optimal paths for a mobile robot navigating among obstacles subject to uniform braking safety constraints. The classical Brachistochrone problem studies the time optimal path of a particle moving in an obstacle free environment subject to a constant force field. By encoding the mobile robot's braking safety constraint as a force field surrounding each obstacle, the paper generalizes the Brachistochrone problem into safe time optimal navigation of a mobile robot in environments populated by polygonal obstacles. Convexity of the safe travel time functional, a path dependent function, allows efficient construction of a speed graph for the environment. The speed graph consists of safe time optimal arcs computed as convex optimization problems in $$O(n^3 \log (1/\epsilon ))$$O(n3log(1/∈)) total time, where n is the number of obstacle features in the environment and $$\epsilon $$∈ is the desired solution accuracy. Once the speed graph is constructed for a given environment, pseudo time optimal paths between any start and target robot positions can be computed along the speed graph in $$O(n^2\log n)$$O(n2logn) time. The results are illustrated with examples and described as a readily implementable procedure.

Time-optimal control of rolling bodies

The brachistochrone problem is usually solved in classical mechanics courses using the calculus of variations, although it is quintessentially an optimal control problem. In this paper, we address the classical brachistochrone problem and two vehicle-relevant generalisations from an optimal control perspective. We use optimal control arguments to derive closed-form solutions for both the optimal trajectory and the minimum achievable transit time for these generalisations. We then study optimal control problems involving a steerable disc rolling between prescribed points on the interior surface of a hemisphere. The effects of boundary and control constraints are examined. For three-dimensional problems of this type, which involve rolling bodies and nonholonomic constraints, numerical solutions are used.

Brachistochrone for a rigid body sliding down a curve

Extremality conditions in the brachistochrone problem for a perfectly rigid body sliding without friction down an unknown (to be determined) curve in a vertical plane are found. In this case, the body has to track the tangent to the trajectory. According to the principle of constraint release, the reaction of the support creates a torque used as control. For dimensionless Appell's equations with a single similarity coefficient, the standard problem of the fastest descent from a given initial point to a given terminal point assuming that the initial velocity is zero is formulated. The Okhotsimskii-Pontryagin method is used to analyze the differential of the objective function. Necessary optimality conditions are found, and a formula for the optimal control that does not involve adjoint variables is derived from them. Properties of the optimal trajectories are investigated analytically both in the general case and for the limiting (zero and infinite) values of the similarity coefficient. It is found that cycloid-shaped brachistochrones occur as the similarity coefficient tends to infinity. For some values of the similarity coefficient, numerical results are presented that demonstrate the shape of the corresponding brachistochrones and the optimal time of motion. The results are compared with those obtained by solving the classical brachistochrone problem.

Solution of variational problems via hybrid of block-pulse and Lagrange interpolating

A direct method for solving variational problems using the hybrid of block-pulse functions and Lagrange interpolating polynomials is presented. The method is based upon hybrid functions approximations. The properties of hybrid functions, which consist of block-pulse functions and Lagrange interpolating polynomials, are presented. These properties are then utilised to reduce the variational problems to the solution of non-linear equations. Three examples are considered, in the first example the classical brachistochrone problem is studied and in the second and third examples, two non-linear problems are examined. The method is general, easy to implement and yields very accurate results.

The brachistochrone problem—an introduction to variational calculus for undergraduate students

The importance of variational calculus is usually not reflected in the basic, undergraduate engineering curriculum. In this paper, we present the result of a pedagogical project aimed at introducing variational calculus at an early stage in the engineering education. This introduction is based on the classical brachistochrone problem and combines both theoretical and experimental analysis.

Single-Term Walsh Series Direct Method for the Solution of Nonlinear Problems in the Calculus of Variations

A direct method for solving variational problems using single-term Walsh series is presented. Two nonlinear examples are considered. In the first example the classical brachistochrone problem is examined. and in the second example a higher-order nonlinear problem is considered. The properties of single-term Walsh series are given and are utilized to reduce the calculus of variations problems to the solution of algebraic equations. The method is general, easy to implement and yields accurate results.

Evolutionary solutions to the brachistochrone problem with Coulomb friction

The classical brachistochrone problem was originally proposed by Johann Bernoulli more than 300 years ago, and is still one of the most elegant problems in modern mathematics. This paper examines a generalisation of the original brachistochrone problem by the inclusion of a non-conservative force in the form of Coulomb friction. Solutions to this problem are found using an evolutionary computational technique based on Darwinian natural selection and survival of the fittest. These evolutionary solutions tend to indicate that as friction is introduced the optimum curves develop from cycloidal (the analytical solution for the conservative problem) toward straight lines. The flexibility of the evolutionary method to optimise for other objectives is then discussed.

The arrival time brachistochrones in general relativity

We consider a general relativistic version of the classical brachistochrone problem, whose solutions are causal curves, parameterized by a constant multiple of their proper time and with 4-acceleration perpendicular to a given observer field, that extremize the arrival time measured by an observer at the final endpoint. This kind of brachistochrones presents characteristics different from the travel time brachistochrones, that were studied in [8, 9, 10]. In this article we formulate the variational problem in a general context; moreover, in the case of a stationary metric, we prove two variational principles and we determine the second order differential equation satisfied by the arrival time brachistochrone. Using these variational principles and techniques from Critical Point Theory we establish some results concerning the existence and the multiplicity of travel time brachistochrones with a given energy between an event and an observer.

The brachistochrone problem with frictional forces

In this paper we show the existence of the solution for the classical brachistochrone problem under the action of a conservative eld in presence of frictional forces. Assuming that the frictional forces and the potential grow at most linearly, we prove the existence of a minimizer on the travel time between any two given points, whenever the initial velocity is great enough. We also prove the uniqueness of the minimizer whenever the given points are suciently close.

On the analytical solution of the brachistochrone problem in a non-conservative field

A new approach to obtain an analytical solution of the brachistochrone problem in a non-conservative velocity-dependent frictional resistance field is presented. Geometrical and energy constraints are incorporated into a time functional through Lagrangian multipliers and the Euler-Lagrange equations in a natural coordinate system are derived. The novelty of the present approach is a parametrization of the Euler-Lagrange equations by the slope angle of the trajectory. By exploiting a special structure of the governing equations of the problem, all function-variables are eliminated and the remaining two unknown parameters are eventually determined from the two non-linear equations. This approach offers several advantages over the well-known solution by Bolza, and establishes an analogy of the brachistochrone problem with other mechanical problems, in particular with a bending of a planar beam. The solution of the classical (Bernoulli's) brachistochrone problem is derived in explicit, yet alternative formulae. A numerical example assuming the linear resistance law (Newtonian fluid) is presented, and the influence of the coefficient of viscous friction, k, on the brachistochrone motion is analyzed. The limiting case k → ∞ is also dealt with.

Brachystochronen als zeitkürzeste Fahrspuren von Bobschlitten

Bobsled move on tracks with a known geometry, i.e. with a given cross-sectional shape, turn angle, and elevation drop. The problem is to find the time-shortest path through a turn. To investigate this problem the classical brachistochrone problem is generalized to arbitrarily twofold curved surfaces in the 3-dimensional Euclidean space. The corresponding equation of a minimum time path involves two other special extremals: the geodesics and the Jacobi geodesics.

Investigating the brachistochrone with a multistage Monte Carlo method

Abstract Multistage Monte Carlo methods are capable of producing approximate solutions to the classical brachistochrone problem. The history and formulation of the classical brachistochrone problem is briefly discussed, and a mathematical programming problem for an approximate brachistochrone problem is derived. Multistage Monte Carlo methods are outlined, and an ANSI-standard C language program which implements a multistage Monte Carlo solution to this mathematical programming problem is presented. Experimental results for ten-segment and twenty-segment brachistochrones are tabulated, and extensions and opportunities for further investigations are suggested

An optimization example: The brachistochrone problem

In the teaching of optimization methods such as in Control courses, a frequently used introductory example is the classical brachistochrone problem. The Calculus of Variations solution is usually obtained by introducing a new parameter to solve the nonlinear differential equation. This letter presents an alternative method which introduces as a parameter the instantaneous angular direction of the falling particle and obtains a solution in which a nonlinear differential equation does not arise.