The problem of the brachistochrone, or of the tautochrone (because the brachistochrone is also the tautochrone), is an intriguing one, and important. This is because all objects falling on the brachistochrone from different heights, end up at the bottom of the brachistochrone simultaneously. Additionally if the brachistochrone is the path of least action (or of least time) then this will be the preferred path which all objects, accelerated by gravity, will follow in the universe.
Furthermore, if acceleration, in general, is equivalent to the acceleration of gravity, according to general relativity, then all accelerated objects (no matter what is the cause of the acceleration) will travel on brachistochrones. Therefore the geodesics which objects follow as they travel in spacetime will be brachistochrones.
In this document there will not be a mathematical description and analysis of the equations of the cycloid (the brachistochrone). Instead we will directly focus attention on a condition of synchronism (which I call principle of synchronicity). This condition relates the cycloid curve (the curve of the brachistochrone) to the linear path (the rim of the brachistochrone) which connects the two ends of the curve.
Thus instead of describing the associated distances with respect to the equation of the cycloid, we will treat them as the product of speed by time. Thus, on one hand, there will be the speed v of an object travelling on the brachistochrone at a time T (the time of the brachistochrone, or tautochrone), while, on the other hand, we will have the speed c of a photon travelling on the linear path (the rim of the brachistochrone) at a time equal to (or multiple of) its period. Such a connection will be described with examples later on.
The solution to the brachistochrone problem is not a straight line or some combination thereof, but a cycloid.
Here we may note some historical facts. According to Wikipedia (previous picture), the problem of the brachistochrone was brought forward by Johann Bernoulli in 1696, as follows:
“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?”
Bernoulli published his solution the following year, and he went on to show that it yields a cycloid.
Five other mathematicians responded to the problem with solutions: Isaac Newton, Jakob Bernoulli (Johann’s brother), Gottfried Leibniz, Ehrenfried Walther von Tschirnhaus and Guillaume de l’Hôpital.
Earlier, in 1638, Galileo had tried to solve a similar problem, but he drew the conclusion that the arc of a circle is faster than any number of its chords.
[https://en.wikipedia.org/wiki/Brachistochrone_curve]
The following picture illustrates the cycloid:
[https://tex.stackexchange.com/questions/196957/how-can-i-draw-this-cycloid-diagram-with-tikz]
The cycloid equations (in parametric form) are the following,
where R is the radius of the rolling (generating) circle which produces the cycloid (in the previous picture the same radius is called a).
Significantly enough, the brachistochrone is also the tautochrone:
Four balls slide down a cycloid curve from different positions, but they arrive at the bottom at the same time. The blue arrows show the points’ acceleration along the curve. On the top is the time-position diagram.
According to Wikipedia (previous picture), 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 of its starting point. The curve is a cycloid, and the time is equal to π times the square root of the radius (of the circle which generates the cycloid) over the acceleration of gravity. The tautochrone curve is the same as the brachistochrone curve for any given starting point.
The tautochrone problem was solved by Christiaan Huygens in 1659. He proved that the time of descent is equal to the time a body takes to fall vertically the same distance as the diameter of the circle that generates the cycloid, multiplied by π/2. In modern terms, this means that the time of descent is
where R is the radius of the circle which generates the cycloid, and g is the gravity of Earth.
[https://en.wikipedia.org/wiki/Tautochrone_curve]
Some geometric aspects of the brachistochrone are the following one:
In the previous image, the curved line S≡OC΄O΄ is the brachistochrone, generated by the circle of radius R≡CR, and it is equal to S=8R. The linear distance L≡OCO΄ is equal to L=2πR, while the vertical distance y≡CRC΄ is equal to y=2R.
Thus, we have
Also comparing the arc OC΄≡S/2 to the chord between the same points OC΄, which we may call Δr, we take
Furthermore, we have the following ratios,
Here Δx is the line OC (half the distance ΔL), and ΔS/2≡OC΄ is half the arc of the brachistochrone. These are some of the geometric proportions of the problem.
Notes:
For reasons which will become more obvious later on, it will be useful here to compare the brachistochrone to a sinusoidal wave, as well as to the perimeter of a cycle.
This is a related graph:
In the previous graph, the red line is a brachistochrone (in parametric form), with a generating circle of radius R=1/2. The green line corresponds to a sinewave (half its wavelength), with amplitude R=1. The orange line (also in parametric form) is half the perimeter of a circle with radius R=1.
The exact arc length of the sine curve for a full period is 7.64, thus 3.82 for half a wavelength.
[https://en.wikipedia.org/wiki/Sine#Arc_length]
Using the cyclic approximation of a wavelength (that a wavelength λ is equal to the perimeter of a circle whose radius R is equal to the amplitude of the wave), the wavelength will be
On the other hand, the length of the brachistochrone is 8R, where R is the radius of the generating circle. The corresponding brachistochrone in the previous graph has a generating circle of radius R=1/2, thus the length S of the brachistochrone will be half as much,
Thus the brachistochrone best approximates a sine wave (being about 5% bigger than the length of a sine wave), rather than the cyclic approximation (where the circle of reference has a perimeter about 15% shorter than the length of the sine wave).
Incidentally the definition and measurement of a ‘wavelength’ is rather ambiguous, as the wavelength itself is an elusive entity. If we take a photograph of a sea wave, we can measure the wavelength as the distance between two peaks, for example. But when the wave is something we cannot see, what we receive is the ‘beat’ per unit of time (the frequency f) of the oscillation, or we measure the time it takes for one beat to occur (the period T). This is the most realistic and tangible aspect of an invisible wave (such as waves in spacetime). Thus, assuming that the speed of the wave is constant (the speed of light c in our case), then the wavelength will just be λ=c/f=cT. This length represents an oscillation, and also a reference unit of distance (if we use light to measure distances).
Furthermore, if acceleration, in general, is equivalent to the acceleration of gravity, according to general relativity, then all accelerated objects (no matter what is the cause of the acceleration) will travel on brachistochrones. Therefore the geodesics which objects follow as they travel in spacetime will be brachistochrones.
In this document there will not be a mathematical description and analysis of the equations of the cycloid (the brachistochrone). Instead we will directly focus attention on a condition of synchronism (which I call principle of synchronicity). This condition relates the cycloid curve (the curve of the brachistochrone) to the linear path (the rim of the brachistochrone) which connects the two ends of the curve.
Thus instead of describing the associated distances with respect to the equation of the cycloid, we will treat them as the product of speed by time. Thus, on one hand, there will be the speed v of an object travelling on the brachistochrone at a time T (the time of the brachistochrone, or tautochrone), while, on the other hand, we will have the speed c of a photon travelling on the linear path (the rim of the brachistochrone) at a time equal to (or multiple of) its period. Such a connection will be described with examples later on.
Here we may note some historical facts. According to Wikipedia (previous picture), the problem of the brachistochrone was brought forward by Johann Bernoulli in 1696, as follows:
“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?”
Bernoulli published his solution the following year, and he went on to show that it yields a cycloid.
Five other mathematicians responded to the problem with solutions: Isaac Newton, Jakob Bernoulli (Johann’s brother), Gottfried Leibniz, Ehrenfried Walther von Tschirnhaus and Guillaume de l’Hôpital.
Earlier, in 1638, Galileo had tried to solve a similar problem, but he drew the conclusion that the arc of a circle is faster than any number of its chords.
[https://en.wikipedia.org/wiki/Brachistochrone_curve]
The following picture illustrates the cycloid:
The cycloid equations (in parametric form) are the following,
Significantly enough, the brachistochrone is also the tautochrone:
According to Wikipedia (previous picture), 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 of its starting point. The curve is a cycloid, and the time is equal to π times the square root of the radius (of the circle which generates the cycloid) over the acceleration of gravity. The tautochrone curve is the same as the brachistochrone curve for any given starting point.
The tautochrone problem was solved by Christiaan Huygens in 1659. He proved that the time of descent is equal to the time a body takes to fall vertically the same distance as the diameter of the circle that generates the cycloid, multiplied by π/2. In modern terms, this means that the time of descent is
[https://en.wikipedia.org/wiki/Tautochrone_curve]
Some geometric aspects of the brachistochrone are the following one:
In the previous image, the curved line S≡OC΄O΄ is the brachistochrone, generated by the circle of radius R≡CR, and it is equal to S=8R. The linear distance L≡OCO΄ is equal to L=2πR, while the vertical distance y≡CRC΄ is equal to y=2R.
Thus, we have
Notes:
For reasons which will become more obvious later on, it will be useful here to compare the brachistochrone to a sinusoidal wave, as well as to the perimeter of a cycle.
This is a related graph:
In the previous graph, the red line is a brachistochrone (in parametric form), with a generating circle of radius R=1/2. The green line corresponds to a sinewave (half its wavelength), with amplitude R=1. The orange line (also in parametric form) is half the perimeter of a circle with radius R=1.
The exact arc length of the sine curve for a full period is 7.64, thus 3.82 for half a wavelength.
[https://en.wikipedia.org/wiki/Sine#Arc_length]
Using the cyclic approximation of a wavelength (that a wavelength λ is equal to the perimeter of a circle whose radius R is equal to the amplitude of the wave), the wavelength will be
Incidentally the definition and measurement of a ‘wavelength’ is rather ambiguous, as the wavelength itself is an elusive entity. If we take a photograph of a sea wave, we can measure the wavelength as the distance between two peaks, for example. But when the wave is something we cannot see, what we receive is the ‘beat’ per unit of time (the frequency f) of the oscillation, or we measure the time it takes for one beat to occur (the period T). This is the most realistic and tangible aspect of an invisible wave (such as waves in spacetime). Thus, assuming that the speed of the wave is constant (the speed of light c in our case), then the wavelength will just be λ=c/f=cT. This length represents an oscillation, and also a reference unit of distance (if we use light to measure distances).
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