Brachistochrone: Difference between revisions

Revision as of 21:28, 24 February 2021

Introduction

To find the shape of the curve which the time is shortest possible. . .

We use WxMaxima to do the calculus part.

Theory

Variational Calculus and Euler--Lagrange Equation

The time from 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 P_a} to 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 P_b} is 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} \frac 1 v ds } 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 ds= \sqrt{1+y'{^2}}dx} is the Pythagorean distance measure and 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} 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=f(y,y')} 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

Since 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} does not depend on 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 x} , we may use the simplified E--L formula 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-y' \frac{\partial f}{\partial y'}= } Constant. 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'} = \frac{\partial }{\partial y'} \sqrt{\frac{1+y'^{2}}{2gy}} = \frac1{\sqrt{2gy}} \frac{\partial }{\partial y'} \sqrt{1+y'^{2}} = \frac{2y'}{\sqrt{2gy}} \frac1 {2\sqrt{1+y'^{2}}} } .

Thus, we have

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 - y' \frac{\partial f}{\partial y'} = \sqrt{ \frac{1+y'{^2}}{2gy} } - \frac{y'{^2}}{\sqrt{2gy} \sqrt{1+y'^{2}}} = C } So we have 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{1+y'{^2}}{\sqrt{2gy(1+y'{^2})}} - \frac{y'{^2}}{\sqrt{2gy} \sqrt{1+y'^{2}}} = \frac{1}{\sqrt{2gy(1+y'{^2})}} = C } and multiplying this with the denominator and rearring, we have 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 y\left( 1 + y'^{2}\right) = \frac1{2gC^2} = k^2 } by redefining the constant. The standard solution to this equation is given by

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 \begin{align} x &= \frac12 k^2( \theta - \sin\theta) \\ y &= \frac12 k^2( 1 - \cos\theta) \end{align} }

and is the equation of a cycloid.

The same

energy : 1/2*m*v^2 = m*g*y;
v_sol : solve( energy, v);
v_sol : v_sol[2];
EL_f : rhs( sqrt(1+'diff(y,t)^2)/v_sol );
doof_dooyp : diff( EL_f, 'diff(y,t));
EL: EL_f - 'diff(y,t)*doof_dooyp = C;
EL_dy : solve(EL, y);
ode2(EL_dy[1]^2,y,t);

but the ode2 solver cannot handle the nonlinear differential equation.

Rolling Ball: Angular momentum

The rotational energy is 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 E_\text{rot} = \frac12 I \omega^2} and by applying non-slipping condition 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 = \omega r} 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 E_\text{rot} = \frac {v^2}{2r^2} I} .

energy : 1/2*m*v^2 + 1/2*I*v^2/r^2= m*g*y;
. . .

gives

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 y= \[\frac{m\, {{r}^{2}}+I}{2 {{C}^{2}} g m\, {{r}^{2}}\, {{\left( \frac{d}{d t} y\right) }^{2}}+2 {{C}^{2}} g m\, {{r}^{2}}}\] }

Friction

The normal force follows the path, and thus is given by 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 \vec T = \frac{dx}{ds}\vec x + \frac{dy}{ds} \vec y } , but The friction depends on the normal force of the path. The normal force is perpendicular to the previous, thus we have

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 \vec N = - \frac{dy}{ds}\vec x + \frac{dx}{ds} \vec y }

The conservation of energy does not apply here, but we have Newton's Second Law, 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 \vec F = m \frac{d \vec v}{dt}} . We need the components along the curve 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 s} . Thus we have

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 \begin{align} \vec F &= \vec G - \vec F_\mu \\ &= mg \frac{dy}{ds} - \mu mg \frac{dx}{ds} \end{align} }

Clearly, for the left hand side of NII we have 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 m\frac{dv}{dt} = mv \frac{dv}{ds} = m\frac12 \frac {d v^2}{ds}} , and by including the differential part only, we have

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 \begin{align} \frac12 v^2 &= g( y - \mu x ) \\ v&= \sqrt{2g(y-\mu x)} \end{align} }

and $f$ for the Euler--Lagrange equation is $f={\sqrt {\frac {1+y'^{2}}{2g(y-\mu x)}}}$

Euler--Lagrange

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}{dt}\frac{\partial f}{\partial y'} =0 }

${\begin{aligned}{\frac {\partial f}{\partial y}}&={\sqrt {\frac {1+y'^{2}}{2g}}}{\frac {\partial }{\partial y}}(y-\mu x)^{-1/2}\\&={\sqrt {\frac {1+y'^{2}}{2g}}}{\frac {-1}{2(y-\mu x)^{3/2}}}\\&=-{\frac {1}{2(y-\mu x)}}{\sqrt {\frac {1+y'^{2}}{2g(y-\mu x)}}}\\\end{aligned}}$

${\begin{aligned}{\frac {\partial f}{\partial y'}}&={\frac {1}{\sqrt {2g(y-\mu x)}}}{\frac {\partial }{\partial y}}{\sqrt {1+y'^{2}}}\\&={\frac {y'}{2{\sqrt {2g(y-\mu x)}}{\sqrt {1+y'^{2}}}}}\\\end{aligned}}$

${\begin{aligned}{\frac {d}{dt}}{\frac {\partial f}{\partial y'}}&={\frac {-\left(2y''{\sqrt {2g(y-\mu x)}}{\frac {y'}{\sqrt {1+y'^{2}}}}+{\sqrt {2}}g(y'-\mu x'){\frac {\sqrt {1+y'^{2}}}{\sqrt {g(y-\mu x)}}}\right)y'+2{\sqrt {2}}y''{\sqrt {y-\mu x}}{\sqrt {1+y'^{2}}}}{\left(2{\sqrt {2g(y-\mu x)}}{\sqrt {1+y'^{2}}}\right)^{2}}}\end{aligned}}$

Plugging these into EL we have

${\begin{aligned}-{\frac {1}{2(y-\mu x)}}{\sqrt {\frac {1+y'^{2}}{2g(y-\mu x)}}}&={\frac {-\left(2y''{\sqrt {2g(y-\mu x)}}{\frac {y'}{\sqrt {1+y'^{2}}}}+{\sqrt {2}}g(y'-\mu x'){\frac {\sqrt {1+y'^{2}}}{\sqrt {g(y-\mu x)}}}\right)y'+2{\sqrt {2}}y''{\sqrt {y-\mu x}}{\sqrt {1+y'^{2}}}}{\left(2{\sqrt {2g(y-\mu x)}}{\sqrt {1+y'^{2}}}\right)^{2}}}\\\\-4g{\frac {(1+y'^{2})^{3/2}}{\sqrt {2g(y-\mu x)}}}&=-\left(2y''{\sqrt {2g(y-\mu x)}}{\frac {y'}{\sqrt {1+y'^{2}}}}+{\sqrt {2}}g(y'-\mu x'){\frac {\sqrt {1+y'^{2}}}{\sqrt {g(y-\mu x)}}}\right)y'+2{\sqrt {2}}y''{\sqrt {y-\mu x}}{\sqrt {1+y'^{2}}}\\\end{aligned}}$

Rolling Ball with radius

References

https://mathworld.wolfram.com/BrachistochroneProblem.html

https://physicscourses.colorado.edu/phys3210/phys3210_sp20/lecture/lec04-lagrangian-mechanics/

http://hades.mech.northwestern.edu/images/e/e6/Legeza-MechofSolids2010.pdf

https://www.tau.ac.il/~flaxer/edu/course/computerappl/exercise/Brachistochrone%20Curve.pdf

https://mate.uprh.edu/~urmaa/reports/brach.pdf

https://medium.com/cantors-paradise/the-famous-problem-of-the-brachistochrone-8b955d24bdf7

https://wiki.math.ntnu.no/_media/tma4180/2015v/calcvar.pdf

@@ Line 85: / Line 85: @@
 === Rolling Ball: Angular momentum ===
-The rotational energy is <math>E_\text{rot} = \frac12 I \omega^2</math> and by applying non-slipping condition <math>v = \omega r</math> we get <math>E_\text{rot} = \frac {v^2}{2r^2} I</math>. The conservation energy states
+The rotational energy is <math>E_\text{rot} = \frac12 I \omega^2</math> and by applying non-slipping condition <math>v = \omega r</math> we get <math>E_\text{rot} = \frac {v^2}{2r^2} I</math>.
+<syntaxhighlight>
+energy : 1/2*m*v^2 + 1/2*I*v^2/r^2= m*g*y;
+. . .
+</syntaxhighlight>
+gives
 <math>
-\begin{align}
+y=
-E_\text{kin} &= E_\text{rot} \\
+\[\frac{m\, {{r}^{2}}+I}{2 {{C}^{2}} g m\, {{r}^{2}}\, {{\left( \frac{d}{d t} y\right) }^{2}}+2 {{C}^{2}} g m\, {{r}^{2}}}\]
-\frac12 mv^2 + \frac{v^2}{2r^2}I &= mgy \\
-v^2 &= \frac{mgy}{m + I/(2r^2)} \\
-&= \frac{2mgr^2}{2m + I}
-\end{align}
 </math>

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Revision as of 21:28, 24 February 2021

Contents

Introduction

Theory

Variational Calculus and Euler--Lagrange Equation

No Friction

Rolling Ball: Angular momentum

Friction

Euler--Lagrange

Rolling Ball with radius

References

Navigation

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Wiki tools

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Anonymous

Search

Brachistochrone: Difference between revisions

Revision as of 21:28, 24 February 2021

Introduction

Theory

Variational Calculus and Euler--Lagrange Equation

No Friction

Rolling Ball: Angular momentum

Friction

Euler--Lagrange

Rolling Ball with radius

References

Navigation

Wiki tools

Page tools

Categories