The Brachistochrone Problem
The Brachistochrone Problem
Introduction
The word Brachistochrone is a concatenation of two words brachistos that means shortest and chrone that means time. This problem is impressive considering all the number of solutions it attracts. This problem excites while we watch some of the brightest minds wrestle with ideas. This wrestling draws new ideas with the intention of providing a solution. The first person to fight his mind was Galileo. It, however, remained in the works of Galileo until 1696 when Johanna Bernoulli started the conversation of providing solutions to the problem. In his address in Acta Eruditorum, he introduced the issue and asked is listeners to help him solve the problem. Bernoulli posed his question as “Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time.” However, Bernoulli was not the first to think of this problem. Galileo had made the same consideration in 1638.
The Brachistochrone problem requires a person to find a line between two locations of a plane. According to Euclid’s first postulate, a straight line can always be drawn to join any two points. Naturally, a line segment is considered to be the shortest distance. If we want to find the shortest time taken to capture two points as opposed to the shortest distance connecting two places, Brachistochrone curve comes to play — supposing that there is a bead threaded on a string. This bead works on the assumption that they can freely move from point y to point x while the frictional forces are negligible. In this case, we assume that the bead is at constant acceleration and is being acted on by gravity, what will be the shortest path that it can follow to arrive in the shortest time? This puzzle is the whole idea behind the problem. Several mathematicians have spent their time to solve this puzzle, and the first mathematician was Galileo.
Galileo studied this problem in his work the Discourse on the new science. The first version of Galileo’s approach was to establish a straight line from point A to an arbitrary vertical line that would be the quickest to reach. He then calculated the angle to be 45 degrees to the absolute point B. he further demonstrated that it was faster to travel via another point C that was the center of the arc connecting the two points. This assumption was correct; however, Galileo made an erroneous argument that the quickest descent would be through the arc of a circle connecting two points. Bernoulli had the solution to this problem; however, he asked his listeners to aid in solving the problem. These attracted five other solutions, also, Bernoulli’s solution. These solutions included Leibniz, de L’Hopital, Newton and Jacob Bernoulli.
This problem later attracted contribution of other mathematicians like Lagrange and Euler. Their contributions were very significant in solving the problem. Someone like Euler aided in developing a geometric representation that would help determine the shortest graphical distance that was useful in solving the problem.
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Brachistochrone curve
By definition, the Brachistochrone curve is a shaped region joining two points whereby a frictionless bead descends within minimum time. To achieve the required result, the curve should start with a steep gradient. The steep gradient helps in building up the required velocity to enable achieve minimal descent time. However, the steep gradient should not be dominant, and this helps to maintain the gained speed. When the steep slope is dominant, the gained velocity gets lost with an increase in path-length. The figure 1 below illustrates the gradient aspect of Brachistochrone curve.
Figure 1
From the figure 1 above, the cycloid achieves the best results as compared to the straight line and a perfect circle. The balance between gradient and path-length of the cycloid is best described the Euler-Lagrange equation. To explain these phenomena, we must be able to establish the fastest Parabola. The fastest parabola is a region that starts at appoint and ends at a point such as (3,2). To calculate the fastest parabola, we have to consider all the parameters involved. Putting this into consideration, we will have three unknowns in the quadratic equation. We have to then impose one condition to in order to achieve the required results. We shall take the x-coordinate of vertex m to be our known parameter. The figure 2 below shows graphs of m=2, 2.5,3 and 3.5.
Figure 2
A computer simulation establishes that the graph of m=2.5 is the fastest parabola. The graph figure 3 shows the relationship between time of descent versus m.
Figure 3
To establish the fastest parabola, it is essential to find the fastest Nth root. This is the strike of balance between gradient and path length. The figure 4 below represents graphs of equation y=2(x/π)1/n for 2 ≤ n ≥ 6.
Figure 4
The fastest Nth root is n=2, This graph achieves results close to those of a cycloid.
Explanation.
In order to put this into context, we have to consider different variables. We consider kinetic friction and air resistance in modelling the Brachistochrone curve, a typical coefficient is denoted by µ. The frictional coefficient is usually less or equal to 0.1. Approximately a cycloid with a frictional coefficient of 0.1 is usually 3% slower that a frictionless cycloid. This approximation s only true when the cycloid ends at (π,2).
The effect of kinetic friction on travel time is negligible when it comes to the Brachistochrone curve. The question is to what extent the shape of the cycloid affects time descent when friction is factored into a Brachistochrone curve. This question raises issues to whether with the introduction of friction there is a new fastest curve introduced or the cycloid still remains to be the fastest curve. Incase there is a newer fastest curve introduced, where does this place the cycloid. This implies that the cycloid can be either placed above or below the new system.
The introduction of frictional force introduces a different graphical representation and it is a nice cycloid generalization. It is important to note that the graphical representation as well as the physical representation is as a result of several factors which includes frictional force. These representations can be derived or be graphically compared to other systems. During the representations there are certain assumptions that need to be carried out. The first assumption is that the frictional force is directly proportional to the normal component of the weight the bead. This force is further assumed to be acting tangentially in the opposite direction. This is a very important assumption to conduct. As a result of the shape of the cycloid, that is the curvature of the path, the normal component as well contributes to the frictional force. The figure 5 below shows the representation of the forces.
Figure 5
Generally, this component that contributes to the frictional force is neglected in differential equations derivations. This negligence of the component is because most students are mainly familiarized with the weight component. The negligence is a challenge in establishing a realistic Brachistochrone curve. To achieve desired results, we use graphical representations that yield a more realistic solution. This solution differs from the new Brachistochrone curve as well as the cycloid. Its solutions are mainly attributed to its ability to incorporate qualitative changes as well as the use of differential equations. A constrained variation technique is useful in attaining desired results as it is solvable despite having messy equations.
In order to under the derivation of the differential equations, it is essential to have some understanding on the physical analysis. The analysis will entail comparison of forces. These forces are force with friction as compared to force without friction. This analysis is based on a cycloid. If we neglect the curvature of the cycloid, then the magnitude of frictional force is less at steep gradients. This frictional force ranges from zero at vertical tangents to its total weight at its horizontal tangent. This hypothesis illustrates that the gradient contributes to more weight when compared to length and optimal curve. The initial gradient should be more or less steep than the gradient of the cycloid. It is important to note that the normal component of acceleration is proportional to the speed squared. This implies that when factored in the frictional modelling, the results must be different. When a cycloid as a steeper gradient, it will require a more curvature for its latter path whereas there is a greater velocity.
Deriving the fastest curve with kinetic friction.
Let us take our starting point to be the origin of a curve that is oriented in the positive of the y-axis downwards. We want to establish the fastest curve y(x). this curve starts at (0,0) and ends at an arbitrary point of (a,b). if we use the assumption that there is no friction, we can use the equation of conservation of energy. We can as well equate work with the change in kinetic energy. By equating work with change kinetic energy, we get v=√2gy. In this equation velocity is given as ∂s/∂t. We can as well introduce frictional forces that can help us use do line integration to establish work. The other alternative is starting with the equations of motion. At location (x, y) on the curve in figure 5 It is possible to express figure 5 considering unit tangent and normal vectors in form of arc-length s as,
T=□(24&d x)/(d s) i+(d y)/(d s) j and N=-□(24&d y)/(d s) i+(d x)/(d s) j
The forces of friction and gravity are expressed by
F_gravity=m g j And F_friction=-μ (F_gravity.N) T=-μ m g □(24&d x)/(d s) T
The curve’s components, in T’s direction, are given as
F_gravity.T=m g □(24&d y)/(d s) And F_friction.T=-μ m g □(24&d x)/(d s)
When the components are used in newton law, they give
m □(24&d v)/(d t)=m g □(24&d y)/(d s)-μ m g □(24&d x)/(d s) ………. (1)
And substituting
□(24&dv)/dt=v □(24&dv)/dt=□(24&1)/2 □(24&d)/(d s)(v^2)
Integrating (1) w.r.t s gives
□(24&1)/2 v^2=g(y-μx) or v=√(2g(y-μx))
Applying the chain rule to the equation v=ds⁄dt and using the length of the arc formulae for the ds⁄dx to answer for the dx⁄dt as an x function that can be inverted to obtain the total time,
T(x,y,y^i )=∮_a^b▒√((1+(y^2))/(2g(y-ux)) dx) ………………………(2)
Since the computation of the equations is a bit messy, therefore, only an outline of the key steps is shown below,
Applying the Euler Lagrange equation to the above equation gives.
d/dx (Fy^i )-Fy=0
Where F is the integrand in equation (2) used to give the second order differential equation,
(1+(y^i )^2 )(1+μy^i )+2(y-μx) y^ii=0
By substituting two equations and doing a partial fraction integration, the above equation can be reduced to
(1+(y^i )^2)/(1+μ〖 y〗^i )^2 =C/(y-μ x)………… (3)
The constant C in the equation is non-negative
The substitution y^i=cot〖(θ⁄2)〗 into (3), from the classical problem lead, will be used to attain a parametric solution for the optimal curve. The parametrization for the cycloid can be denoted as;
〖(θ)=ρ(θ-cosθ ) and x〗_c (θ)=ρ(θ-sinθ )
The new curve, which is the fastest, for the problem of “frictional” Brachistochrone can be expressed in the following form,
x(θ)=x_c (θ)+μ ρ (1-cosθ )
y(θ)=y_c (θ)+μ ρ (1-sinθ )
The parameterizations in the above equations and cycloid remain valid in the of 0 and ϴf whereby ϴf and ρ are determined in order for the curves to pass through the ending point (a, b). Figure 6 below demonstrates the relationship between the new curve and cycloids. It is important to note that there are similar repetitive patterns. These patterns have vertical tangents tat are even multiples of πρ. However so, the minima of the curves do not occur at the same place. There is a sloping line that indicates whereby the new curves stops as opposed to the cycloid that makes back to the X-axis.
Figure 6
The sloping line is usually called the “line of repose.” The line as minimum gradient µ. This the is the point at which the bead will start to roll. The above derived solution is invalidated when the condition y<µx applies. This restricts the allowable end points; the restrictions are consistent with the physical feasibility of the curve. The bead cannot be able to make it back to its original point due to the loss of energy due to friction. Figure 7 below demonstrates the relationship between a frictional Brachistochrone with a classic Brachistochrone, a cycloid. There descent time is compared to a frictionless cycloid.
Figure 7
When comparing speed for all valid ending points, the new fastest curve always lies below the cycloid throughout its length. If the ending point is placed at the line of repose, distinctive results between the two curves can be obtained. The fastest curve will lie above the cycloid at all its length if friction is factored in the curve. This is subject to test treatment. Figure 8 below shows the fastest curve that lies above the cycloid with a frictional coefficient µ of 0.1. The ending points of the curves however must satisfy the strict inequality, y<µx. This is because it usually takes the bead an infinite time to reach the line of repose.
Considering our first intuition that the normal component of acceleration hinders curvature, figure 8 below shows the coefficients of friction and how they affect the Brachistochrone curve. The effect of the coefficients is that they tend to push the curve to be more of a straight line. As the curve tends to be pushed to be more of a straight line, descent time reduces.
Figure 8
Reflection and application.
There are several applications of the Brachistochrone curve. The first application of the Brachistochrone curve is in the aircraft routing and crew pairing. This application has enhanced aircraft operations through maximizing time by establishing optimal routes. The other use is surfing. The curve helps in determining the shortest path while ensuring that you enjoy every view one that a person wants to see. The Brachistochrone curve is a significant breakthrough in surfing. The curves application extends to the engineering world. For the electrical engineers who want to make a most effective circuit board, he can use the curve to establish the optimal distance he or she requires. A traveling salesperson traveling through different points can find the curve essential in his daily business. The salesperson can use the curve to establish the optimal path that he needs to follow. This will save him time and cost hence enhancing his business. The Brachistochrone curve is essential in classification and pattern recognition. Our eyes use the curve to classify and recognize patterns. The curve is applicable both for 3D and 2D color classification and identification. This curve as well helps computers to read written handwriting. This curve has revolutionized the computer word.
Fundamental truths are not established by lazy scientists strolling around, trying to figure out something to discover. The truths are found through the application of interesting and challenging tasks that create new versions of knowledge. Mathematics is vast, and most discoveries come from niche fields with very little exposure to even the general mathematical public. Some mathematicians such as Hilbert believe that it is sensible to pursue a specific single field wholeheartedly rather than having a divided interest and attention. He, in his namesake 23 problems, listed the 23rd problem to be “Further development of the methods of the calculus of variations8.” However, it is important to acknowledge that discoveries in mathematics will be furthered by a continuous appetite for curiosity by students of this field to keep working and researching rather than a proverbial carrot in the form of recognition of fame. This investigation has reminded us that the nature of mathematics requires that to improve one mathematical skill significantly, one should not be entirely dependent on what a textbook indicates or suggests. The reason is that some of the recent mathematical discoveries done on these textbooks constitute a far-from-exhaustive list, rife with personal bias. Rather, one should define problems and contemplate them from a different perspective elsewhere. One can discover lots of numerous mathematical association of American books, articles, physics applications and lecture notes by just doing a simple exploration like browsing through the internet. These discoveries can also be achieved through other non-conventional sources of knowledge such as brilliant deductive skills of colleagues or a professors’ perseverance and determination to develop and deliver an outstanding project. During this investigation, I interacted with the calculus of variations which is usually taught in the college level. I also understood more about parametric equations and partial derivatives as well as the branch between the world of geometry and analytic calculus.
Most importantly, I realized that I had my shortfalls, especially in the traditional calculus. I had problems remembering the quotient rule, the product rule, and the chain rule. This resulted to a week full of frustrations, head-scratching, and puzzlement at my notes. However, I learned that when I want to practice and train my skills in the future, I will indeed have to turn to mathematics textbooks that can be trusted but when I am willing to establish a fact for myself. The most important basic principle is to first Work on having a solid foundation in math basics. Then practice and practice to create a positive feedback loop that makes you feel good about yourself and like math more. For instance, one day I will have to find out why it is said that the cycloid has an interesting property that it satisfies the tautochrone curve. Mathematics is sort of like the biggest sandbox puzzle game in existence—you are welcome to try and put things together however you wish, with the constraint that it should be sensible, and that you should try to create something new, exciting, and unexpected. People make new mathematical “discoveries” every day. Well, I suppose it depends on what you mean by the term “discoveries.” If proving results and building theories counts, then we are on the same page
References
Benson, Donald C. “An Elementary Solution of the Brachistochrone Problem.” The American Mathematical Monthly 76.8 (1969): 890-94
Haws, LaDawn, and Terry Kiser. “Exploring the Brachistochrone Problem.” The American Mathematical Monthly, vol. 102, no. 4, 1995, pp. 328–336.
Barra, Mario. “The Cycloid.” Educational Studies in Mathematics 6.1 (1975): 93-98
Freire, Alex. The Brachistochrone Problem. Knoxville, Tennessee: University of Tennessee – Department of Mathematics, n.d. PDF
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