Brachistochrone curve

The brachistochrone ( gr brachistos shortest, chronos time ) is a friction-free path between a start and an equal or deeper endpoint on which a mass point at the fastest slides to the end point under the influence of gravitational force. The low point of the web may be lower than the end point.

The body sliding on such a rail goal faster than, for example, on a rectilinear path, even though it is shorter.

At the same time, this curve is a tautochrone, i.e. each point of the curve needs the same time to move to the lowest point. This fact is exploited in so-called Zykloidenpendel in which the pendulum mass swings on a tautochrone.

Form

The brachistochrone is part of a cycloid.

History

Johann Bernoulli has dealt with the problem of the fastest case. In 1696, he finally found the solution to the brachistochrone. Today, this is often seen as the birth of the calculus of variations.

Christiaan Huygens published in 1673 in his treatise Horologium Oscillatorium a gang accurate pendulum clock with a Zykloidenpendel, in which he took advantage of the fact that the evolute of the cycloid is itself a cycloid again. The advantage of the gear accuracy is offset by the increased friction.

Function

The brachistochrone slips easily into a parametric representation describe, that is, you can display its points as a vector, which varies with a parameter. : And coordinates - As a function of the angle (in radians), around which the wheel has turned with radius during rolling, which are

It is helpful for the understanding of this curve: the radius times the angle " point of contact of the circle - circle center - Brachistochronenpunkt " is already unrolled circuit.

Derivation

Consider in the - plane, a curve along which the mass point from the start glide with continuous time to the destination.

He has the kinetic energy

And the potential energy

It is the height in the gravitational field and the gravitational acceleration.

Slips initially quiescent mass point from the origin going on, so along its path, the total energy received and has the initial value of zero,

This can be gradually dissolved. The derivative of the inverse function that indicates what time it is when the particle passes through the place, this is inversely

If we integrate over the range from 0 to, we obtain the quantity to be minimized runtime as functional of the trajectory

To connect to the usual physical variational problems designations, we call the integration variable, denoted by and minimize the sake of simplicity with functional multiplied. So we minimize the effect

With Lagrangian

Since the Lagrangian is not the integration parameters, depends of the time, the associated after the Noether theorem energy

Received on the path for which is minimal. Thus, the function fulfilled by a positive constant, the equation

As a particle falls vertically from the peak height in the Kepler potential.

Rather than solve for this equation with separate variables and integrate, you just confirmed that

A parametric solution of this equation is, whereby

Exploits. So the desired path is given parametrically by

In this case, at the last decomposition clear that the rail is composed of the position vectors of the hub of a wheel with a radius, the rolls of the axis plus the spoke vector initially points upward and is rotated by the angle. The curve is the path of a boundary point of a rolling wheel.

Special properties of the web

• The track is independent of the mass and the weight of the body, so regardless of the size of the acceleration due to gravity.
• Likewise, changes a rolling ball, the rotational energy picks up nothing to the ideal curve.
• The tangent at the point A is vertical.
• If two brachistichrone the same slope between A and B, they are similar.
• If the gap is not smaller than 1 / π ( 31.83 %), so B is the lowest point on the curve with a smaller slope is the low point between A and B.
• Is the gradient 0, that A and B at the same height, the curve is symmetrical.

Pictures

Phase 2: Front ball accelerated by stronger fall.

Phase 3: Front ball, despite longer path forward.