Brachistochrone: Difference between revisions
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t = \int_{P_a}^{P_b} \frac 1 v ds | t = \int_{P_a}^{P_b} \frac 1 v ds | ||
</math> | </math> | ||
where <math>ds= \sqrt{1+y'2}dx</math> is the Pythagorean distance measure and <math>v</math> is determined from the the law of conservation of energy | where <math>ds= \sqrt{1+y'^2}dx</math> is the Pythagorean distance measure and <math>v</math> is determined from the the law of conservation of energy | ||
<math> | <math> | ||
\frac12 mv^2 = mgy | \frac12 mv^2 = mgy | ||
Revision as of 18:30, 16 February 2021
Introduction
To find the shape of the curve which the time is shortest possible. . .
Theory
The time from to is where is the Pythagorean distance measure and is determined from the the law of conservation of energy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac12 mv^2 = mgy } . giving Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v = \sqrt{2gy}} . Plugging these in, we get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t = \int_{P_a}^{P_b} \sqrt{\frac{1+y'2}{2gy}}dx = \int_{P_a}^{P_b} f dx} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is the function subject to variational consideration.
According to the Euler--Lagrange differential equation the stationary value is to be found, if E-L equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial f}{\partial y} - \frac{d}{d x}\frac{\partial f}{\partial y'} = 0} is satisfied.
No Friction
We get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial f}{\partial y'}}
Because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial f}{\partial x}=0}
,
Friction
Rolling Ball
References
https://mathworld.wolfram.com/BrachistochroneProblem.html
https://physicscourses.colorado.edu/phys3210/phys3210_sp20/lecture/lec04-lagrangian-mechanics/
