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Topic: Brachistochrone

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Brachistochrone problem

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	Brachistochrone curve - Wikipedia, the free encyclopedia
		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.
		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).
		Hence, the brachistochrone curve is tangent to the vertical at the origin.
	en.wikipedia.org /wiki/Brachistochrone (623 words)

	Brachistochrone curve (Site not responding. Last check: 2007-10-08)
		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 translates along it under the action of constant gravity.
		Galileo incorrectly stated in 1638 in his Discourse on two new sciences that this curve was an arc of a circle.
		The brachistochrone curve was proved to be a cycloid.
	www.uncover.us /en/wikipedia/b/br/brachistochrone_curve.html (232 words)

	The Brachistochrone (Site not responding. Last check: 2007-10-08)
		Exercise 4: Approximate the shape of the brachistochrone from the point (0 meters, 2 meters) to the point (2 meters, 2 meters) by representing the path of minimum time as four line segments, but with no specifications on either the x- or y-coordinates of the endpoints of the line segments.
		Exercise 6: Approximate the shape of the brachistochrone from the point (0 meters, 2 meters) to the vertical line x = 2 (to a point with x-coordinate of 2 and an unknown y-coordinate) by representing the path of minimum time as four line segments that span equal horizontal intervals.
		Exercise 7: Approximate the shape of the brachistochrone from the point (0 meters, 2 meters) to the vertical line x = 2 (to a point with x-coordinate of 2 and an unknown y-coordinate) by representing the path of minimum time as eight line segments that span equal horizontal intervals.
	www.mathcs.emory.edu /~fox/NewCCS/ModuleII/MIIP9E.html (308 words)

	Brachistochrone problem
		The brachistochrone problem was posed by Johann Bernoulli in Acta Eruditorum in June 1696.
		Now Huygens had shown in 1659, prompted by Pascal's challenge about the cycloid, that the cycloid is the solution to the tautochrone problem, namely that of finding the curve for which the time taken by a particle sliding down the curve under uniform gravity to its lowest point is independent of its starting point.
		The first problem of this type [calculus of variations] which mathematicians solved was that of the brachistochrone, or the curve of fastest descent, which Johann Bernoulli proposed towards the end of the last century.
	www-gap.dcs.st-and.ac.uk /~history/HistTopics/Brachistochrone.html (1928 words)

	The Brachistochrone
		Example 1: Let us approximate the shape of the brachistochrone from the point (0 meters, 2 meters) to the point (2 meters, 1 meter) by representing the path of minimum time as four line segments that span equal horizontal intervals.
		Example 2: Let us now approximate the shape of the brachistochrone from the point (0 meters, 2 meters) to the point (2 meters, 1 meter) by representing the path of minimum time as four line segments that span equal vertical intervals.
		Example 3: Let us approximate the shape of the brachistochrone from the point (0 meters, 2 meters) to the point (2 meters, 1 meter) by representing the path of minimum time as four line segments, but with no specifications on either the x- or y-coordinates of the endpoints of the line segments.
	www.mathcs.emory.edu /~fox/NewCCS/ModuleII/ModIIP9.html (2011 words)

	AMCA: Brachistochrones Under Central Forces by Garry J. Tee (Site not responding. Last check: 2007-10-08)
		Johann Bernoulli 1st proved in 1696 that the brachistochrone (curve of quickest descent) under uniform gravity is the cycloid.
		The bounded and unbounded brachistochrones are separated by a critical brachistochrone, which is bounded in radius.
		For inverse square forces the brachistochrones are constructed in terms of elliptic integrals — except that (for repulsion) the critical brachistochrone is constructed in terms of elementary functions.
	at.yorku.ca /c/a/d/l/23.htm (186 words)

	Brachistochrone (Site not responding. Last check: 2007-10-08)
		The center track is a brachistochrone, the curve along which a particle moving under the influence of gravity alone will traverse the distance between the endpoints in minimum time.
		Another unusual property of the brachistochrone is that the time required to reach the bottom of the track is independent of the starting point on the track.
		This can be shown by releasing two balls from different positions on the brachistochrone simultaneously and observing that they reach the bottom together.
	physicsnt.clemson.edu /physdemo/1/demos/Dbrachi2.htm (126 words)

	Atomic Rocket: Mission Table
		Impulse trajectory I-3 is near the transition between delta V levels for high impulse trajectories and low brachistochrone trajectories (it is a hyperbolic solar escape orbit plus 30 km/s).
		Impulse trajectory I-2 is in-between I-1 and I-3 (it is equivalent to an elliptical orbit from Mercury to Pluto).
		Brachistochrone trajectories are labeled by their level of constant acceleration: 0.01 g, 0.10 g, and 1.0 g.
	www.projectrho.com /rocket/rocket3o.html (449 words)

	The Brachistochrone
		The brachistochrone problem is a seventeenth century exercise in the calculus of variations.
		The brachistochrone is a cycloid, but that cycloid is not the only curve satisfying the equation.
		The brachistochrone cannot go down from there, because its derivative would be undefined, and also it would have downward concavity.
	whistleralley.com /brachistochrone/brachistochrone.htm (2469 words)

	TPS Archives
		The study suggests that, indeed, the shortest line between the gates may not be the fastest.
		The mathematical solution to the concept of brachistochrone is well known in mathematical circles.
		Today, virtually every text on classical dynamics or variational calculus deals with the brachistochrone problem in theory, but application of this theory to sports is rare.
	www.psia.org /psia_2002/education/TPSArticles/coaching/tpsfall96cycloids.asp (1111 words)

	brachistochrone -- Encyclopædia Britannica
		The calculus of variations evolved from attempts to solve this problem and the brachistochrone (q.v.) problem.
		Isoperimetrics was made the subject of an investigation by 17th- and 18th-century Swiss mathematicians, the...
		Includes information on the brachistochrone problem and his contribution to finding a solution.
	www.britannica.com /eb/article-9016094?tocId=9016094 (409 words)

	parnovsky
		On this basis a differential equation of a brachistochrone is built and solved in the next section of this article.
		(6), (8) define the brachistochrone equation in a parametrical representation.
		At the start of falling the brachistochrone inclination is close to vertical and N is small.
	info.ifpan.edu.pl /firststep/aw-works/fsV/parnovsky/parnovsky.html (1846 words)

	Brachistochrone curve -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		Brachistochrone curve -- Facts, Info, and Encyclopedia article
		Given two points A and B, with A not lower than B, there is just one upside down (A line generated by a point on a circle rolling along a straight line) cycloid that passes through A with infinite slope and also passes through B.
		The problem can be solved with the tools from the (The calculus of maxima and minima of definite integrals) calculus of variations.
	www.absoluteastronomy.com /encyclopedia/b/br/brachistochrone_curve.htm (386 words)

	Brachistochrone with Coulomb Friction
		Brachistochrone with Coulomb Friction: SIAM Journal on Control and Optimization Vol.
		This paper formulates and solves in closed form the problem of finding the minimum-time path of a particle between two points in a uniform gravitational field when motion of the particle is resisted by a force proportional to the normal force exerted on the particle by the path.
		The problem solution involves the reformulation of the classical brachistochrone of Bernoulli in terms of a singular control problem in which the time derivative of the heading angle of the particle is the control parameter.
	epubs.siam.org /sam-bin/dbq/article/28795 (166 words)

	Brachistochrone Revisited (Site not responding. Last check: 2007-10-08)
		The branch of applied mathematics known as the "Calculus of Variations" has its origins in the brachistochrone problem posed in 1696 by Johann Bernoulli.
		The problem of the path taken by a "frictionless" bead accelerated from rest by gravity that would minimize the time of transit is reviewed.
		This illustrates using the asymptotic and Taylor series to explicitly construct the Brachistochrone Solution.
	my.execpc.com /~aplehnen/brach/Brachist.html (444 words)

	PIRA 1D15.00 VELOCITY, POSITION, AND ACCELERATION (Site not responding. Last check: 2007-10-08)
		Balls released at any height on the brachistochrones reach the middle at the same time.
		Two balls released on opposite sides of a cycloid always meet in the middle regardless of handicap.
		Use the brachistochrone and tautochrone properties of a cycloid to make an actual slide track in amusement parks.
	www.physics.ncsu.edu:8380 /pira/1mech/1D15.html (351 words)

	The Brachistochrone
		The brachistochrone and cycloid have a very rich math and physics literature.
		The brachistochrone is for "the shrewdest mathematicians of all the world.
		Johnson and others suggest the Euler-Lagrange formula and boundary conditions applied to the brachistochrone will define a differential equation whose solution is similar to finding critical points (usually a maximum or minimum) in ordinary calculus.
	curvebank.calstatela.edu /brach/brach.htm (662 words)

	The Brachistochrone Revisited
		One of the origins of the branch of applied mathematics known as the "Calculus of Variations" is the brachistochrone problem posed in 1696 by Johann Bernoulli.
		The first trajectories considered consist of piecewise linear segments and their ability to approximate the solution of the brachistochrone is presented.
		Thus, the solution of the Brachistochrone problem is a cycloid with the interpretation that the positive constant k is the diameter of the "generating circle".
	matcmadison.edu /alehnen/brach/brachchistochrone.html (5057 words)

	Brachistochrone Revisited (Site not responding. Last check: 2007-10-08)
		This illustrates using the asymptotic and power series to explicitly construct the Brachistochrone Solution.
		A plot of the parameter Theta Max versus r = a/L.
		This also illustrates using the asymptotic and power series to construct the Brachistochrone Solution.
	my.execpc.com /BB/72/aplehnen/Brachist.html (298 words)

	Travel by Brachistochrone
		The Brachistochrone Transit Company claims to be able to beat this time, making the trip in 2.9 minutes, at an average speed of 614 mph.
		In another article (Curves), I show that the brachistochrone is also the tautochrone.
		That is, the time taken to reach the bottom from any point on the curve is the same.
	www.du.edu /~etuttle/math/brach.htm (1609 words)

	Washington University in St. Louis Magazine (Site not responding. Last check: 2007-10-08)
		Some years ago, Carl Bender's father was putting Bender's son, Michael, to bed by recounting the story of a well-known physics problem: that of the brachistochrone, a quickest path that a bead on a wire can take to get from one point to another.
		The curved shape of the brachistochrone was well-known; it had been worked out 300 years before.
		Quantum mechanics—the study of the behavior of very small particles—is much newer than the brachistochrone problem, but still, its basic principles were worked out early in the 20th century.
	magazine.wustl.edu /Spring05/CarlBender.htm (1497 words)

	millennium
		The minimum time path between two points in a constant one-g gravitational field (at different elevations) is a cycloid also known as the brachistochrone (shortest time from the Greek) when concave up.
		The cyloid is the locus of a point on a circle (or rim of a wheel) as the circle (or wheel) rolls along a smooth surface and the point on the wheel returns back to its origin (one revolution through 2
		If the horizontal displacement were increased to 160 feet, the advantage of the cycloid would be more pronounced: flat board 4.64 sec, cycloid 3.91 sec and circular arc 4.54 sec.
	www.navworld.com /navcerebrations/millennium/millennium.htm (717 words)

	Exploring the Brachistochrone Problem -- from Mathematica Information Center
		In the light of the attention given to a national crisis in mathematics education, concerned mathematics instructors are always looking for innovative ways to present and reinforce ideas.
		Computer technology can help educators compete for students' attention and at the same time enhance the learning process by 1) bringing an added dimension--visualization--to the presentation of mathematical concepts, 2) giving students greater flexibility to explore and discover ideas on their own, 3) making more advanced topics accessible to a wider range of classes.
		These learning aspects will be discussed in the context of some Mathematica packages for exploring the classic Brachistochrone problem and interesting variations.
	library.wolfram.com /infocenter/Articles/1073 (143 words)

	Brachistochrone problem references (Site not responding. Last check: 2007-10-08)
		T Koetsier, The story of the creation of the calculus of variations : the contributions of Jakob Bernoulli, Johann Bernoulli and Leonhard Euler (Dutch), in 1985 holiday course : calculus of variations (Amsterdam, 1985), 1-25.
		J Peiffer, Le problème de la brachystochrone à travers les relations de Jean I Bernoulli avec L'Hôpital et Varignon, in Der Ausbau des Calculus durch Leibniz und die Brüder Bernoulli, Basel, 1987, Studia Leibnitiana Sonderheft 17 (Wiesbaden, 1989).
		B Singh and R Kumar, Brachistochrone problem in nonuniform gravity, Indian J. Pure Appl.
	www-gap.dcs.st-and.ac.uk /~history/HistTopics/References/Brachistochrone.html (229 words)

	brachistochrone - OneLook Dictionary Search
		Tip: Click on the first link on a line below to go directly to a page where "brachistochrone" is defined.
		BRACHISTOCHRONE : 1911 edition of the Encyclopedia Britannica [home, info]
		Phrases that include brachistochrone: brachistochrone problem, brachistochrone curve
	www.onelook.com /?w=brachistochrone (86 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Asymptotic Expansions for Small r The small r limit of the Brachistochrone problem is of some interest since in many textbooks the solutions, or at least the graphs of the solutions, usually have EMBED Equation.COEE2 .
		O'Connor and E. Robertson, “The Brachistochrone Problem” From The MacTutor History of Mathematics Archives: HYPERLINK "http://turnbull.mcs.st-and.ac.uk/~history/HistTopics/Brachistochrone.html" \l "s1" http://turnbull.mcs.st-and.ac.uk/~history/HistTopics/Brachistochrone.html - s1 9.
		J.Zeng, "A Note on the Brachistochrone Problem", The College Mathematics Journal 27, 206-208, (1996).
	matcmadison.edu /alehnen/brach/brach.doc (2948 words)

	Projects (Site not responding. Last check: 2007-10-08)
		By Fermat's Principle, we can treat this curve as the trajectory of light which passes through an optically nonhomogeneous medium.
		In short, the light trajectory is a brachistochrone.
		So, the brachistochrone appears to be a cycloid.
	www.math.wpi.edu /Course_Materials/MA1024A97/projects/project2.html (734 words)

	REFERENCES
		Giulio Venezian, Terrestrial Brachistochrone, AJP 34, 701-704 (1966).
		Edge, The Brachistochrone - or, the longer way round may be the quickest way home, TPT 23, 372-373 (1985).
		Figueroa, G. Gutierrez, and C. Fehr, Demonstrating the Brachistochrone and Tautochrone, TPT 35, 494-498 (1997).
	www.physics.umd.edu /lecdem/services/refs/refsc.htm (5493 words)

	Research Experience for Undergraduates (Site not responding. Last check: 2007-10-08)
		The brachistochrone for a material point with arbitrary initial velocity.
		The brachistochrone of a point of variable mass with constant relative rates of losing and gaining particles.
		On a brachistochrone in a field of constant force.
	math.fullerton.edu /mathews/n2003/catenary/CatenaryBib/Links/CatenaryBib_lnk_3.html (537 words)

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