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A Brachistochrone curve, 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 ignoring friction. Image File history File links The cycloid as brachistochrone. ...
It has been suggested that gravitation be merged into this article or section. ...
Friction is the force that opposes the relative motion or tendency of such motion of two surfaces in contact. ...
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 and also passes through B. This is the brachistochrone curve. The brachistochrone thus does not depend on the body's mass or on the strength of the gravitational constant. Cycloid (red) generated by a rolling circle A cycloid is the curve defined by a fixed point on a wheel as it rolls, or, more precisely, the locus of a point on the rim of a circle rolling along a straight line. ...
The problem can be solved with the tools from the calculus of variations. Calculus of variations is a field of mathematics which deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. ...
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. 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: Fermats principle assures that the angles given by Snells law always reflect lights quickest path between P and Q. Fermats principle in optics states: This principle was first stated by Pierre de Fermat. ...
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ...
 , where h represents the altitude difference between the current position and the starting point. 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: Snells law is the simple formula used to calculate the refraction of light when travelling between two media of differing refractive index. ...
 , where θ represents the angle of the trajectory with respect to the vertical. 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. This article is about the mathematical concept of tangent. For other meanings, 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:
 . At any given point on the trajectory we have:  . Inserting this expression in the previous formula, and rearranging the terms, we have:  . Which is the differential equation of the opposite of a cycloid generated by a circle of diameter D. In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...
Cycloid (red) generated by a rolling circle A cycloid is the curve defined by a fixed point on a wheel as it rolls, or, more precisely, the locus of a point on the rim of a circle rolling along a straight line. ...
History Galileo incorrectly stated in 1638 in his Two New Sciences that this curve was an arc of a circle. Johann Bernoulli solved the problem (by reference to the previously analysed tautochrone curve) before posing it to readers of Acta Eruditorum in June 1696. Five mathematicians responded with solutions: Isaac Newton, Jakob Bernoulli (Johann's brother), Gottfried Leibniz and Guillaume de l'Hôpital. Four of the solutions (excluding l'Hôpital's) were published in the May 1697 edition of the same publication. Galileo Galilei Galileo Galilei (Pisa, February 15, 1564 – Arcetri, January 8, 1642), was an Italian physicist, astronomer, and philosopher who is closely associated with the scientific revolution. ...
Events March 29 - Swedish colonists establish first settlement in Delaware, called New Sweden. ...
The Discourses and Mathematical Demonstrations Relating to Two New Sciences (1638) was Galileos final book and a sort of scientific testament covering much of his work in physics over the preceding thirty years. ...
A circle, in Euclidean geometry, is the set of all points at a fixed distance, called the radius, from a fixed point, the centre. ...
Johann Bernoulli Johann Bernoulli (Basel, July 27, 1667 - January 1, 1748) was a Swiss mathematician. ...
A tautochrone curve is the curve for which the time taken by a particle sliding down it under uniform gravity to its lowest point is independent of its starting point. ...
The year 1696 had the earliest equinoxes and solstices for 400 years in the Gregorian calendar, because this year is a leap year and the Gregorian calendar would have behaved like the Julian calendar since March 1500 had it have been in use that long. ...
Sir Isaac Newton, PRS, (4 January [O.S. 25 December 1642] 1643 – 31 March [O.S. 20 March] 1727) was an English physicist, mathematician, astronomer, alchemist, inventor and natural philosopher who is generally regarded as one of the most influential scientists in history. ...
Jakob Bernoulli. ...
Gottfried Wilhelm Leibniz (also von Leibni(t)z) (July 1 (June 21 Old Style) 1646, Leipzig – November 14, 1716, Hanover) was a German polymath, deemed a genius in his day and since. ...
Guillaume François Antoine, Marquis de lHôpital (1661 – February 2, 1704) was a French mathematician. ...
Events September 20 - The Treaty of Ryswick December 2 – St Pauls Cathedral opened in London Peter the Great travels in Europe officially incognito as artilleryman Pjotr Mikhailov Use of palanquins increases in Europe Christopher Polhem starts Swedens first technical school. ...
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. Joseph-Louis de Lagrange did further work that resulted in modern infinitesimal calculus. It has been suggested that Leonhard Euler/EB1911 biography be merged into this article or section. ...
1766 was a common year starting on Wednesday (see link for calendar). ...
Calculus of variations is a field of mathematics which deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. ...
Joseph Louis Lagrange (January 25, 1736 – April 10, 1813) was an Italian mathematician and astronomer who later lived in France and Prussia. ...
Infinitesimal calculus is an area of mathematics pioneered by Gottfried Leibniz based on the concept of infinitesimals, as opposed to the calculus of Isaac Newton, which is based upon the concept of the limit. ...
Another rivalry, between Newton and Leibniz, also contributed to this development. Each claimed to have solved the brachistochrone problem before the other, and they continued to quarrel over their subsequent work on the calculus.
Etymology In Greek, brachistos means "shortest" and chronos means "time". Note that there is no "y" in the Greek root, so "brachystochrone", while rather common, is a spelling mistake.
See also Calculus of variations is a field of mathematics which deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. ...
A tautochrone curve is the curve for which the time taken by a particle sliding down it under uniform gravity to its lowest point is independent of its starting point. ...
External links - Iagsoft's "Brachistochrone Construction" (Java applet)
- [1] (in French but has an excellent animated illustration)
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