# The Brachistochrone Problem

This problem was formulated by Johann Bernoulli, in Acta Eruditorum, June 1696

## Contents

## Problem description

"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."

In this example, we solve the problem numerically for A = (0,0) and B = (10,-3), and an initial speed of zero.

The mechanical system is modelled as follows:

where (x,y) is the coordinates of the point, v is the velocity, and theta is the angle between the direction of movement and the vertical.

```
% Copyright (c) 2007-2008 by Tomlab Optimization Inc.
```

## Problem setup

toms t toms tf p = tomPhase('p', t, 0, tf, 20); setPhase(p); tomStates x y v tomControls theta % Initial guess % Note: The guess for tf must appear in the list before expression involving t. x0 = {tf == 10 icollocate({ v == t x == v*t/2 y == -1 }) collocate(theta==0)}; % Box constraints cbox = {0.1 <= tf <= 100 0 <= icollocate(v) 0 <= collocate(theta) <= pi}; % Boundary constraints cbnd = {initial({x == 0; y == 0; v == 0}) final({x == 10; y == -3})}; % ODEs and path constraints g = 9.81; ceq = collocate({ dot(x) == v.*sin(theta) dot(y) == -v.*cos(theta) dot(v) == g*cos(theta)}); % Objective objective = tf;

## Solve the problem

```
options = struct;
options.name = 'Brachistochrone';
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);
x = subs(collocate(x),solution);
y = subs(collocate(y),solution);
v = subs(collocate(v),solution);
theta = subs(collocate(theta),solution);
t = subs(collocate(t),solution);
```

Problem type appears to be: lpcon ===== * * * =================================================================== * * * TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2010-02-05 ===================================================================================== Problem: --- 1: Brachistochrone f_k 1.878940328102787900 sum(|constr|) 0.000000473296672960 f(x_k) + sum(|constr|) 1.878940801399460800 f(x_0) 10.000000000000000000 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 45 ConJacEv 45 Iter 30 MinorIter 194 CPU time: 0.125000 sec. Elapsed time: 0.125000 sec.

## Plot the result

To obtain the brachistochrone curve, we plot y versus x.

```
subplot(3,1,1)
plot(x, y);
subplot(3,1,2)
plot(t, v);
% We can also plot theta vs. t.
subplot(3,1,3)
plot(t, theta)
```