Related papers: A detailed analysis of the brachistochrone problem

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…

We consider the problem of finding paths of shortest transit time between two points (popularly known as Brachistochrone) for cylinders with off-centered center of mass, rolling down without slip, subject solely to the force of gravity.…

Consider a frictionless surface S in a gravitational field that need not be uniform. Given two points A and B on S, what curve is traced out by a particle that starts at A and reaches B in the shortest time? This paper considers this…

The classical problem of the brachistochrone asks for the curve down which a body sliding from rest and accelerated by gravity will slip (without friction) from one point to another in least time. In undergraduate courses on classical…

This paper studies brachistochrone trajectories. Four rules are formulated as sufficient conditions. Two rules apply for a general conservative force. Two rules apply for a central force. A central force allows wire replacement. The wire is…

We solve the brachistochrone problem for a particle travelling through a spherical mass distribution of uniform density. We examine the connection between this problem and the popular "gravity elevator" result. The solution is compared to…

The brachistochrone, the curve of fastest descent under gravity, is a cycloid when friction is absent. Underwater, however, buoyancy, viscous drag, and the added mass of entrained fluid fundamentally alter the problem. We formulate and…

We analyze the motion of a particle in the gravity field along a family of differentiable curves taking into account the Coulomb friction forces. A parametric equation of the optimal curves is given that generalizes the cycloid one in this…

This paper presents and proves an equation for the time horizon of symmetric trajectories with zero boundary conditions and bounded derivatives of arbitrary order. This equation holds regardless of the number of phases comprising the…

The Lambert problem is to determine the gravitational orbit between two points that has a specified time of flight, allowing the second point to be a moving target such as a satellite. After a review of gravitational orbits, a solution of…

Influence diagrams are a decision-theoretic extension of probabilistic graphical models. In this paper we show how they can be used to solve the Brachistochrone problem. We present results of numerical experiments on this problem, compare…

We consider different generalizations of the Brachistochrone Problem in the context of fundamental concepts of classical mechanics. The correct statement for the Brachistochrone problem for nonholonomic systems is proposed. It is shown that…

If a particle has to fall first vertically 1 m from A and then move horizontally 1 m to B, it takes a time $t(=\tau_1+\tau_2=\tau_3=3/\sqrt{2g})=0.67$ s. Under gravity and without friction, if it sides down on a linear track inclined at…

Many combinatorial optimization problems can be formulated as the search for a subgraph that satisfies certain properties and minimizes the total weight. We assume here that the vertices correspond to points in a metric space and can take…

We consider the problem of optimizing the product of the distances from a given point in a triangle to each vertex. There are two possible cases in general. For isosceles triangles, we explicitly show exactly when both cases occur.

We study a time minimization problem on the group of motions of a plane with admissible control in a half-disk. The considered control system describes a model of a car that can move forward on a plane and turn in place. Optimal…

The control problem of the working tool movement along a predefined trajectory is considered. The integral of kinetic energy and weighted inertia forces for the whole period of motion is considered as a cost functional. The trajectory is…

In recent papers it has been suggested that human locomotion may be modeled as an inverse optimal control problem. In this paradigm, the trajectories are assumed to be solutions of an optimal control problem that has to be determined. We…

We study the tracking of a trajectory for a nonholonomic system by recasting the problem as a constrained optimal control problem. The cost function is chosen to minimize the error in positions and velocities between the trajectory of a…

The optimal visiting problem is the optimization of a trajectory that has to touch or pass as close as possible to a collection of target points. The problem does not verify the dynamic programming principle, and it needs a specific…