300 years ago
Johann Bernoulli solved the problem of brachistochrone (the problem of
finding the fastest travel curve's form) using the optical Fermat concept. In
the same way we solved some generalisations of this problem. We obtained the
fastest travel curve's form in a gravitational field for a point-like mass.
When moving in a uniform gravitational field we considered the influence of
the dry friction or drag.
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1. Introduction
Last year there was the 300th anniversary of solving the problem of curve of
the fastest travel. It was the first step in the development of variational
calculus. In spring 1696 Johann Bernoulli proposed a brachistochrone problem
(from the Greek
-- short,
-- time) in the famous scientific magazine Acta Eruditorum. A
material particle moves without friction along a curve. This curve connects
point A with point B (point A is placed above point B). No forces
affect it, except the gravitational attraction. The time of travel from A
to B must be the smallest. This brings up the question: what is the form of
this curve?
Bernoulli announced that he had found an excellent solution of this problem, but he did not want to publish it in order to prompt prominent mathematicians to apply their skill to the mathematical problems of a new type. In particular, he challenged to a competition his elder brother Jacob. At that time they were in hostile relations, and Johann openly called Jacob an ignorant. Jacob Bernoulli published his solution of the problem in the same year. It is intriguing that the geometrical optics' laws were applied to a mathematical problem dealing with mechanical motion.
The main ideas of this solution are given in the book [1]. On this basis a differential equation of a brachistochrone is built and solved in the next section of this article. It will be shown that the fastest travel curve is an arc of a cycloid. Some generalisations of the problem are considered in the next sections. There is a friction affecting the particle in them. There can be the dry friction in accordance with the Coulomb-Amontons law, , as well as the fluid friction proportional to velocity. We also considered the fastest travel trajectories of a point-like or spherical mass moving in a gravitational field.
2. A solution of the classical brachistochrone problem
During the motion of the particle along brachistochrone its velocity v and
angle between the direction of this velocity and the vertical
changes. In order to find the connection between these quantities, Jacob
Bernoulli used the analogy with the discovered in the XVII century Fermat
concept. It states that the light moves in the medium with varying
refractivity in such a way that its travel time is minimal. In the case when
the refraction index n depends only on y coordinate the Snellius (Snel
van Royen) law states that the quantity
is constant. Here
is the angle between the y axis and light path. The light velocity
in medium is v=C/n and the relationship
is fulfilled
for ray's trajectory. Therefore, the equation
Value corresponds to the point A and
to the
point B. We can obtain from the conditions
Figure 2 shows the dependence of on x0/y0 which was
calculated using a computer. Parameter C1 can be obtained from (9) using
the calculated value of . Using (3), (7) and (8), we obtain
3. The generalisations of the brachistochrone problem
Below we will consider some generalisations of the brachistochrone problem.
When generalising, we tried to adhere to the following rules:
As a result we choose the following problems:
The giant brachistochrone. If curve's size becomes comparable to Earth's one, one has to take into account the nonuniformity of the gravitational field. This problem will be considered in Sec. 4.
Brachistochrone with friction. Friction was added when moving. A dry friction with the friction index is considered in Sec. 5. The affection of medium's drag, proportional to the velocity (low Reynolds numbers) is considered in Sec. 6.
4. The giant brachistochrone
In this problem one has to find the form of the fastest travel curve of a
point-like or spherical mass M in a gravitational field. It would be
reasonable to use the polar coordinates with an origin placed at the centre
of the attracting body. We denote the radial coordinate of the point A by
r1; r2 is the same for the point B. The angle coordinate will
be reckoned from the direction of point A. The condition of the
conservation of energy will take the form
Combining Eqs. (12)-(14) and setting n equal to C/v we obtain the
sought-for differential equation
A set of trajectories which start from the same point, corresponding to the
values of parameters equal to
is shown in Fig. 3. One can
see that the particle moves nearly radially at the start, deflecting
increasingly during the motion. From Eq. (13) one can see that when
the particle cannot reach the centre. It must reach the minimal
radius r1z0 and begin receding from the attracting body. If the minimal
distance accords to the azimuthal angle , the trajectory will be
symmetrical about the
ray. The trajectory ends on r1
distance from the centre of attraction in the point in which the azimuthal
angle is equal to and the particle's velocity vanishes. Let us
consider the relation . Introducing new variable
U=(z0/z)2/3, we obtain from (17)
5. Brachistochrone with friction
Let us return to the brachistochrone problem in an uniform gravitational
field. We set the friction coefficient of the particle with the surface equal
to . Equations (1) and (2) will be still valid, but Eq. (3) will be
violated because of the energy losses for the friction. We will find the
equation, generalising it. The supporting force N consists of
the weights component and the centrifugal force
mv2/R=mv2
k, where R and k are the radius of curvature and the curvature of the
trajectory
Let us suppose that the trajectory is curved in the same way as the cycloid
does (y''<0), so the centrifugal force is oriented in the same direction as
the normal component of the weight is. Otherwise, the body can break away
from the trajectory. For the infinitesimal shift dS along the trajectory
the work of friction forces is equal to
It is easy to test that for this expression reduces to the equation
of the cycloid with radius 1/4. When , the shape of the
trajectories dramatically changes. From (25) one can see that at y''<0
expressions and must have the same signs, therefore . Contrary to a cycloid, brachistochrone with friction cannot hook up and
even becomes horizontal. In Eq. (28) x can tend to infinity only if one of
terms in the right part diverges. It is possible only if
.
Thus, if brachistochrone with friction can have infinite length, it tends
asymptotically to the friction angle as
Expanding the right part of (28) into a series in reciprocal powers of y'
at
, we obtain
The trajectories of the body at , obtained using the numeric integration of Eq. (28) are shown in Fig. 4. The details of calculations are given in Sec. 7.
6. Brachistochrone with drag
If the drag force is proportional to velocity
One can see that the condition must be valid. For
as
This equation differs from (30) with the expression for and the change of scale of x. From this one can see that the brachistochrone with drag differs from a cycloid very slightly in the start of falling, so we do not give its plot.
Let us demonstrate that the qualitative behaviour is valid in the case when
the forces of friction and of drag F(v), which increases when the velocity
grows, affect the particle. The acceleration of particle is equal to
7. Calculations
Let us give the details of the calculation of integral (28) which has the
form
The numeric calculations were done on the computer using the programming language FORTRAN. When studying the form of the trajectories at the start of movement we used the approximate formula (31) or similar to it. We also used expansion of the integrand near its singularity point z=z0 when calculating the integrals (17) and (18).
References
1. R. Courant, H. Robbins, What is Mathematics?, Oxford University Press, London 1948
2. G. Korn, T. Korn, Mathematical Handbook, McGraw-Hill, New York 1968