I have watched videos on the brachistochrone problem and how to find the quickest path a particle can take between two points. However they never gave an exact function for the path. I thought of using the least action principle but it seems as if it just gives you an idea of how to solve it and doesn't give the exact answer. So what is the exact function for the brachistochrone between points $(x_1, y_1)$ and $(x_2, y_2)$?
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4$\begingroup$ See en.wikipedia.org/wiki/Cycloid $\endgroup$– PM 2RingCommented Oct 22, 2022 at 20:22
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4$\begingroup$ Wikipedia as well, en.wikipedia.org/wiki/Brachistochrone_curve. Use Beltrami's identity on the integral $\endgroup$– basicsCommented Oct 22, 2022 at 20:27
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1$\begingroup$ Related : What is the position as a function of time for a mass falling down a cycloid curve?. $\endgroup$– FrobeniusCommented Oct 23, 2022 at 2:30
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2 Answers
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It's a cycloid.
The parametric equations that describe it are $$x = r(\theta-\sin\theta), \quad y=-r(1-\cos\theta)$$ where $\theta$ is a parameter, and the curve is traced out from the origin.
If you substitute the $y$ equation into the $x$, you can get a relation of the form
$$x=\arccos\bigg( 1+\dfrac yr\bigg)-\dfrac{\sqrt{r^2-(r+y)^2}}r$$
which is as close as you can get to a an "exact function," though note that the above relation is restricted for $0\leq x\leq \pi$.
answered Oct 22, 2022 at 20:56
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$\begingroup$ What's $r$ here? Is it the distance between the two points? $\endgroup$ Commented Oct 24, 2022 at 16:35
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$\begingroup$ the radius of the circle tracing out the cycloid curve. $r$ can be determined using your final position (x2, y2), assuming you set (x1,y1) to zero so it starts rolling from the origin @KamalSaleh $\endgroup$ Commented Oct 24, 2022 at 17:26
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This is more a comment than an answer, I post it here because comment space is to small for this comment.
While it is the case that the brachistochrone problem is usually approached with calculus of variations, the principle of least action is not applicable.
(I recognize that it could be that you use 'least action principle' and 'calculus of variations' as equivalent expressions. Anyway: 'calculus of variations' has a much wider scope than 'principle of least action'. Calculus of variations is used both in statics and dynamics; principle of least action is exclusive to dynamics.)
The thing is: the brachistochrone problem is closer related to the category of statics problems than the category of dynamics problems.
What you are looking to solve for is a static shape.
Another way of seeing the static nature: when Johann Bernoulli figured out the shape of the brachistochrone he did so by recognizing an analogy with finding the trajectory of light through a medium with a gradient in optical density.
Light moves so fast that it can be treated as moving from start point to end point instantaneously. In that sense the trajectory of light is a static shape.
Further reading:
Article by Paul Rojaz, 2014
The straight line, the catenary, the
brachistochrone, the circle, and Fermat
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$\begingroup$ So the trajectory of the light can be found using snell's law. That's a genius move by Johann Bernoulli. $\endgroup$ Commented Oct 24, 2022 at 16:35