Brachistochrone - Revision history

RobertLampel at 11:27, 4 September 2026

2026-09-04T11:27:52Z

← Older revision		Revision as of 11:27, 4 September 2026
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	}}		}}

	The '''brachistochrone problem''' (from Greek ''brachístos chrónos'', "shortest time") is one of the founding problems of the calculus of variations, and serves as a classical benchmark in optimal control~~. This description is taken from [[#OCPjl \| [1]]]~~.		The '''brachistochrone problem''' (from Greek ''brachístos chrónos'', "shortest time") is one of the founding problems of the calculus of variations, and serves as a classical benchmark in optimal control.

	First posed by Johann Bernoulli in 1696 as a public challenge, the problem asks for the shape of a frictionless curve connecting two fixed points in a vertical plane such that a particle released from rest at the upper point reaches the lower point in minimum time under gravity.		First posed by Johann Bernoulli in 1696 as a public challenge, the problem asks for the shape of a frictionless curve connecting two fixed points in a vertical plane such that a particle released from rest at the upper point reaches the lower point in minimum time under gravity.
	As an optimal control problem, the system state consists of the horizontal position, vertical position, and speed of the particle, while the control is the instantaneous angle of the velocity vector ~~with respect to~~ the ~~vertical~~.		As an optimal control problem, the system state consists of the horizontal position, the downward vertical position, and the speed of the particle, while the control is the instantaneous angle of the velocity vector measured below the horizontal.
	The terminal time is free and is the quantity to be minimised.		The terminal time is free and is the quantity to be minimised; it is handled by augmenting it as a global time-scaling variable and optimising over a normalised unit horizon.
	The analytical solution is a cycloid, obtainable through Pontryagin's minimum principle or the classical Euler–Lagrange equations.		The analytical solution is a cycloid, obtainable through Pontryagin's minimum principle or the classical Euler–Lagrange equations.

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	<math>		<math>
	\begin{array}{lll}		\begin{array}{lll}
	\displaystyle \min_{u,\, t_\mathrm{f}} && t_\mathrm{f} \\[6pt]		\displaystyle \min_{\theta,\, t_\mathrm{f}} && t_\mathrm{f} \\[6pt]
	\text{subject to} \\		\text{subject to} \\
	\quad \dot{x}_1(t) &=& ~~x_3~~(t)\,\~~sin~~(u(t)), \\		\quad \dot{x}(t) &=& v(t)\,\cos(\theta(t)), \\
	\quad \dot{x}_2(t) &=& ~~x_3~~(t)\,\~~cos~~(u(t)), \\		\quad \dot{y}(t) &=& v(t)\,\sin(\theta(t)), \\
	\quad \dot{x}_3(t) &=& g\,\~~cos~~(u(t)), \\		\quad \dot{v}(t) &=& g\,\sin(\theta(t)), \\
	\quad x(0) &=& (0,\; 0,\; 0)^T, \\		\quad (x(0),\, y(0),\, v(0)) &=& (0,\; 0,\; 0), \\
	\quad ~~x_1~~(t_\mathrm{f}) &=& x_~~{1,~~\mathrm{f}}, \\		\quad x(t_\mathrm{f}) &=& x_\mathrm{f}, \\
	\quad ~~x_2~~(t_\mathrm{f}) &=& ~~x_{2,~~\mathrm{f}}, \\		\quad y(t_\mathrm{f}) &=& y_\mathrm{f}, \\
	\quad u(t) &\in& ~~\left~~[~~-\tfrac{\pi}{2}~~,\, ~~\tfrac{~~\pi~~}{2}\right~~] \quad \forall\, t \in [0, t_\mathrm{f}],		\quad \theta(t) &\in& [0,\, \pi] \quad \forall\, t \in [0, t_\mathrm{f}],
	\end{array}		\end{array}
	</math>		</math>
	</p>		</p>

	where <math>~~x_1~~(t)</math> is the horizontal position, <math>~~x_2~~(t)</math> is the downward vertical position, <math>~~x_3~~(t)</math> is the speed of the particle~~, and~~ <math>u(t)</math> is the angle of the velocity vector ~~measured from~~ the downward ~~vertical~~.		where <math>x(t)</math> is the horizontal position (m), <math>y(t)</math> is the downward vertical position (m, positive downward), and <math>v(t)</math> is the speed of the particle (m/s).
	The speed dynamics ~~follow from conservation of energy under gravity:~~ <math>\dot{x}_3 = g\~~cos(u)~~</math>.		The control <math>\theta(t) \in [0, \pi]</math> is the angle of the velocity vector below the horizontal: <math>\theta = 0</math> corresponds to purely horizontal motion and <math>\theta = \pi/2</math> to straight downward motion.
	~~Note that~~ <math>~~x_3~~(0) = 0</math>, so the particle starts from rest.		The speed dynamics <math>\dot{v} = g\sin\theta</math> follow from the component of gravitational acceleration along the direction of motion.
			Since <math>v(0) = 0</math>, the particle starts from rest.
			The constraint <math>\theta \in [0, \pi]</math> restricts the particle to descent arcs (no upward motion).

	== Parameters ==		== Parameters ==
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	! Symbol !! Value !! Description		! Symbol !! Value !! Description
	\|-		\|-
	\| align=center \| <math>g</math> \|\| align=right \| 9.81 \|\| Gravitational acceleration (m s<sup>−2</sup>)		\| align=center \| <math>g</math> \|\| align=right \| 9.81 \|\| Gravitational acceleration (m s<sup>−2</sup>)
	\|-		\|-
	\| align=center \| <math>x_~~{1,~~\mathrm{f}}</math> \|\| align=right \| 1 \|\| Target horizontal position (m)		\| align=center \| <math>x_\mathrm{f}</math> \|\| align=right \| 1.0 \|\| Target horizontal position (m)
	\|-		\|-
	\| align=center \| <math>~~x_{2,~~\mathrm{f}}</math> \|\| align=right \| 1 \|\| Target downward vertical position (m)		\| align=center \| <math>y_\mathrm{f}</math> \|\| align=right \| 0.5 \|\| Target downward vertical position (m)
	\|}		\|}

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	<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">		<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
	Image: Brachistochrone.png \| Optimal trajectory and control for the brachistochrone problem (cycloid solution).		Image:Brachistochrone.png\| Optimal trajectory and control for the brachistochrone problem (cycloid solution).
	</gallery>		</gallery>

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	The analytical solution is a cycloid: the locus traced by a point on the rim of a circle rolling along a flat surface.		The analytical solution is a cycloid: the locus traced by a point on the rim of a circle rolling along a flat surface.
	It can be parametrised as		It can be parametrised as
	<math>~~x_1~~(\~~theta~~) = r(\~~theta~~ - \sin\~~theta~~)</math>, <math>~~x_2~~(\~~theta~~) = r(1 - \cos\~~theta~~)</math>,		<math>x(\phi) = r(\phi - \sin\phi)</math>, <math>y(\phi) = r(1 - \cos\phi)</math>,
	where the radius <math>r</math> is determined by the boundary conditions.		where the radius <math>r</math> is determined by the boundary conditions.
	The brachistochrone problem is a standard example in textbooks on optimal control and the calculus of variations, and is used as a test case for direct transcription methods due to its free terminal time and the mild singularity at <math>t = 0</math> where the speed vanishes.		The brachistochrone problem is a standard example in textbooks on optimal control and the calculus of variations, and is used as a test case for direct transcription methods due to its free terminal time and the mild singularity at <math>t = 0</math> where the speed vanishes.
	~~This~~ formulation and further discussion can be found in [[#~~OCPjl~~\|[1~~]]] and [[#Bryson\|[2~~]]].		A detailed formulation and further discussion can be found in [[#Bryson\|[1]]].

	== References ==		== References ==
	<span id="~~OCPjl~~">[1]</span> Caillau, J.-B., Cots, O., Gergaud, J., & Martinon, P. OptimalControlProblems.jl: a collection of optimal control problems with ODE's in Julia. https://github.com/control-toolbox/OptimalControlProblems.jl<br>		<span id="Bryson">[1]</span> Bryson, A. E., & Ho, Y.-C. (1975). <i>Applied Optimal Control: Optimization, Estimation, and Control</i>. Taylor & Francis.<br>
	~~<span id="Bryson">[2~~]</span> Bryson, A. E., & Ho, Y.-C. (1975). <i>Applied Optimal Control: Optimization, Estimation, and Control</i>. Taylor & Francis.<br>


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	[[Category:ODE model]]		[[Category:ODE model]]
	[[Category:Free terminal time]]		[[Category:Free terminal time]]
	~~[[Category:Classical mechanics]]~~

RobertLampel: /* References */

2026-09-03T11:49:20Z

References

← Older revision		Revision as of 11:49, 3 September 2026
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			[[Category:MIOCP]]
	[[Category:ODE model]]		[[Category:ODE model]]
	[[Category:Free terminal time]]		[[Category:Free terminal time]]
	[[Category:Classical mechanics]]		[[Category:Classical mechanics]]

RobertLampel: /* Reference Solutions */

2026-09-03T11:42:35Z

Reference Solutions

← Older revision		Revision as of 11:42, 3 September 2026
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	<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">		<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
	Image:Brachistochrone.png \| Optimal trajectory and control for the brachistochrone problem (cycloid solution).		Image: Brachistochrone.png \| Optimal trajectory and control for the brachistochrone problem (cycloid solution).
	</gallery>		</gallery>

RobertLampel: Created page with "{{Dimensions |nd = 1 |nx = 3 |nw = 1 }} The '''brachistochrone problem''' (from Greek ''brachístos chrónos'', "shortest time") is one of the founding problems of the calculus of variations, and serves as a classical benchmark in optimal control. This description is taken from [1]. First posed by Johann Bernoulli in 1696 as a public challenge, the problem asks for the shape of a frictionless curve connecting two fixed points in a vert..."

2026-09-03T11:41:56Z

Created page with "{{Dimensions |nd = 1 |nx = 3 |nw = 1 }} The '''brachistochrone problem''' (from Greek ''brachístos chrónos'', "shortest time") is one of the founding problems of the calculus of variations, and serves as a classical benchmark in optimal control. This description is taken from [1]. First posed by Johann Bernoulli in 1696 as a public challenge, the problem asks for the shape of a frictionless curve connecting two fixed points in a vert..."

New page

{{Dimensions
|nd = 1
|nx = 3
|nw = 1
}}

The '''brachistochrone problem''' (from Greek ''brachístos chrónos'', "shortest time") is one of the founding problems of the calculus of variations, and serves as a classical benchmark in optimal control. This description is taken from [[#OCPjl | [1]]].

First posed by Johann Bernoulli in 1696 as a public challenge, the problem asks for the shape of a frictionless curve connecting two fixed points in a vertical plane such that a particle released from rest at the upper point reaches the lower point in minimum time under gravity.
As an optimal control problem, the system state consists of the horizontal position, vertical position, and speed of the particle, while the control is the instantaneous angle of the velocity vector with respect to the vertical.
The terminal time is free and is the quantity to be minimised.
The analytical solution is a cycloid, obtainable through Pontryagin's minimum principle or the classical Euler–Lagrange equations.

== Mathematical formulation ==

<math>
\begin{array}{lll}
\displaystyle \min_{u,\, t_\mathrm{f}} && t_\mathrm{f} \\[6pt]
\text{subject to} \\
\quad \dot{x}_1(t) &=& x_3(t)\,\sin(u(t)), \\
\quad \dot{x}_2(t) &=& x_3(t)\,\cos(u(t)), \\
\quad \dot{x}_3(t) &=& g\,\cos(u(t)), \\
\quad x(0) &=& (0,\; 0,\; 0)^T, \\
\quad x_1(t_\mathrm{f}) &=& x_{1,\mathrm{f}}, \\
\quad x_2(t_\mathrm{f}) &=& x_{2,\mathrm{f}}, \\
\quad u(t) &\in& \left[-\tfrac{\pi}{2},\, \tfrac{\pi}{2}\right] \quad \forall\, t \in [0, t_\mathrm{f}],
\end{array}
</math>


where <math>x_1(t)</math> is the horizontal position, <math>x_2(t)</math> is the downward vertical position, <math>x_3(t)</math> is the speed of the particle, and <math>u(t)</math> is the angle of the velocity vector measured from the downward vertical.
The speed dynamics follow from conservation of energy under gravity: <math>\dot{x}_3 = g\cos(u)</math>.
Note that <math>x_3(0) = 0</math>, so the particle starts from rest.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>g</math> || align=right | 9.81 || Gravitational acceleration (m s−2)
|-
| align=center | <math>x_{1,\mathrm{f}}</math> || align=right | 1 || Target horizontal position (m)
|-
| align=center | <math>x_{2,\mathrm{f}}</math> || align=right | 1 || Target downward vertical position (m)
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Brachistochrone.png | Optimal trajectory and control for the brachistochrone problem (cycloid solution).
</gallery>

== Miscellaneous and Further Reading ==
The analytical solution is a cycloid: the locus traced by a point on the rim of a circle rolling along a flat surface.
It can be parametrised as
<math>x_1(\theta) = r(\theta - \sin\theta)</math>, <math>x_2(\theta) = r(1 - \cos\theta)</math>,
where the radius <math>r</math> is determined by the boundary conditions.
The brachistochrone problem is a standard example in textbooks on optimal control and the calculus of variations, and is used as a test case for direct transcription methods due to its free terminal time and the mild singularity at <math>t = 0</math> where the speed vanishes.
This formulation and further discussion can be found in [[#OCPjl|[1]]] and [[#Bryson|[2]]].

== References ==
[1] Caillau, J.-B., Cots, O., Gergaud, J., & Martinon, P. OptimalControlProblems.jl: a collection of optimal control problems with ODE's in Julia. https://github.com/control-toolbox/OptimalControlProblems.jl 
[2] Bryson, A. E., & Ho, Y.-C. (1975). Applied Optimal Control: Optimization, Estimation, and Control. Taylor & Francis. 

[[Category:ODE model]]
[[Category:Free terminal time]]
[[Category:Classical mechanics]]