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A Brachistochrone curve, (Greek - "brachistos" shortest, "chronos" time), or curve of fastest descent, is the curve between two points that is covered in the least time by a body that starts at the first point with zero speed and passes down along the curve to the second point, under the action of constant gravity and assuming no friction. Gravitation is a natural Phenomenon by which objects with Mass attract one another Friction is the Force resisting the relative motion of two Surfaces in contact or a surface in contact with a fluid (e

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The brachistochrone is the cycloid

Given two points A and B, with A not lower than B, there is just one upside down cycloid that passes through A with infinite slope, passes also through B and does not have maximum points between A and B. A cycloid is the curve defined by the path of a point on the edge of circular wheel as the wheel rolls along a straight line This particular inverted cycloid is a brachistochrone curve. The curve does not depend on the body's mass or on the strength of the gravitational constant.

The problem can be solved with the tools from the calculus of variations. Calculus of variations is a field of Mathematics that deals with functionals, as opposed to ordinary Calculus which deals with functions.

Note that if the body is given an initial velocity at A, or if friction is taken into account, the curve that minimizes time will differ from the one described above.

Proof

According to Fermat’s principle: The actual path between two points taken by a beam of light is the one which is traversed in the least time. In Optics, Fermat's principle or the principle of least time is the idea that the path taken between two points by a ray of light is the path that can be Hence, the brachistochrone curve is simply the trajectory of a beam of light in a medium where the speed of light increases following a constant vertical acceleration (that of gravity g). The conservation law can be used to express the velocity of a body in a constant gravitational field as:

v=\sqrt{2gh},

where h represents the altitude difference between the current position and the starting point. In Physics, a conservation law states that a particular measurable property of an isolated Physical system does not change as the system evolves It should be noted that the velocity does not depend on the horizontal displacement.

According to Snell's law, a beam of light throughout its trajectory must obey the equation:

\frac{\sin{\theta}}{v}=K,

for some constant K, where θ represents the angle of the trajectory with respect to the vertical. In Optics and Physics, Snell's law (also known as Descartes' law or the law of refraction) is a formula used to describe the relationship Inserting the velocity expressed above, we can draw immediately two conclusions:

1- At the onset, when the particle velocity is nil, the angle must be nil. Hence, the brachistochrone curve is tangent to the vertical at the origin. For the tangent function see Trigonometric functions. For other uses see Tangent (disambiguation.

2- The velocity reaches a maximum value when the trajectory becomes horizontal.

For simplification purposes, we assume that the particle (or the beam) departs from the point of coordinates (0,0) and that the maximum velocity is reached at altitude –D. Snell’s law then takes the expression:

\frac{\sin{\theta}}{\sqrt{-2gy}}=\frac{1}{\sqrt{2gD}}.

At any given point on the trajectory we have:

\sin{\theta}=\frac{dx}{\sqrt{dx^2+dy^2}}.

Inserting this expression in the previous formula, and rearranging the terms, we have:

\begin{pmatrix}\frac{dy}{dx}\end{pmatrix}^2=-\left(\frac{D+y}{y}\right).

Which is the differential equation of the opposite of a cycloid generated by a circle of diameter D. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the A cycloid is the curve defined by the path of a point on the edge of circular wheel as the wheel rolls along a straight line

History

Galileo incorrectly stated in 1638 in his Two New Sciences that this curve was an arc of a circle. Galileo Galilei (15 February 1564 &ndash 8 January 1642 was a Tuscan ( Italian) Physicist, Mathematician, Astronomer, and Philosopher The Discourses and Mathematical Demonstrations Relating to Two New Sciences ( Discorsi e dimostrazioni matematiche intorno a due nuove scienze, 1638 was Galileo's Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the Johann Bernoulli solved the problem (by reference to the previously analysed tautochrone curve) before posing it to readers of Acta Eruditorum in June 1696. Johann Bernoulli ( Basel, 27 July 1667 - 1 January 1748 was a Swiss Mathematician. A tautochrone or isochrone curve is the curve for which the time taken by an object sliding without friction in uniform Gravity to its lowest point is independent Acta Eruditorum ( Latin for reports acts of the scholars) was the first Scientific journal of the German lands, published from Five mathematicians responded with solutions: Isaac Newton, Jakob Bernoulli (Johann's brother), Gottfried Leibniz, Ehrenfried Walther von Tschirnhaus and Guillaume de l'Hôpital. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements For other family members named Jacob see Bernoulli family. Jacob Bernoulli (also known as James or Jacques) ( Basel Ehrenfried Walther von Tschirnhaus (or Tschirnhausen) ( April 10, 1651 &ndash October 11, 1708) was a German Mathematician Guillaume François Antoine Marquis de l'Hôpital (1661 &ndash February 2, 1704) was a French Mathematician. Four of the solutions (excluding l'Hôpital's) were published in the May 1697 edition of the same publication.

In an attempt to outdo his brother, Jakob Bernoulli created a harder version of the brachistochrone problem. In solving it, he developed new methods that were refined by Leonhard Euler into what the latter called (in 1766) the calculus of variations. Year 1766 ( MDCCLXVI) was a Common year starting on Wednesday (link will display the full calendar of the Gregorian calendar (or a Calculus of variations is a field of Mathematics that deals with functionals, as opposed to ordinary Calculus which deals with functions. Joseph-Louis de Lagrange did further work that resulted in modern infinitesimal calculus. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives

Etymology

In Greek, brachistos means "shortest" and chronos means "time". Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly

See also

External links

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