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The Brachistochrone Problem: A J Preston, October 2004

1.Imagine a particle traveling down a smooth surface from A to B with no air resistance, inclined at an angle q to the horizontal. Let’s suppose the particle starts from rest and travels a distance L.

Since u = 0, s = L, and a = g sin q and and

then

2.Now consider the quickest route on a curve joining A to B shown below. Would it be quicker to take route ACB or route ADB ?

If we simplify the model and take 2 straight line sections for each of the two routes we have the diagram shown below:

Clearly, the acceleration along the slope is g sin q along AC and DB and g cos q along AD and CB.

If we then draw speed-time graphs with gradients g sin q and g cos q and use the fact that the areas under the two sections are both equal to L , then we have:

3.Clearly then, the quickest route is ADB. | Back to contents

From part (1),

To find the speed at D,we can use

to obtain

Then, time between D and B is given by

and so applying

we have

The total time for the journey is then given by

4.We can consider the question “What shape of curve will ensure the particle travels from A to B in the shortest possible time. For example, consider traveling along the arc of a circle: | Back to contents

And by the conservation of energy from A to E:

Gain in Kinetic Energy = Loss in Gravitational Potential Energy

Hence,

Hence,

5.In order to find the best possible curve, we need to talk about Partial differentiation. | Back to contents

Consider, for example the function

If we treat the x as constant then since f is a function now of p we can write:

And in the same way we can find:

So that

6. We can now introduce an area of Mathematics called: The calculus of variations. | Back to contents

In this topic, we want to find the function y(x) such that:

is a minimum given the particular boundary conditions:

where

7. It can be shown (with an extensive theoretical treatment) that finding y(x) to minimise the integral in (6) is equivalent to solving the famous Euler-Lagrange equation : | Back to contents

for

(7i)

Note that if

i.e f is not explicitly a function of x , we can show that:

(7ii)

 

8. Example 1 | Back to contents

Using the Euler-Lagrange equations, show that the formula y(x) satisfying

making

stationary is

Show further, that the minimum value of the integral in this case is

9. Example 2: Proof that the shortest distance between 2 points is a straight line. | Back to contents

It can be shown that if we have a cartesian equation y = y(x) , then the length of an arc s is given by:

Hence, using the Euler-Lagrange equation show that

is stationary when

where

are constants.

[i.e. This proves that the shortest distance between 2 points is a straight line].

 

10. The Brachistochrone problem | Back to contents

We now have all the machinery to turn our attention to this wonderful problem.

In the diagram below, s is the length of arc from A to E.

Now, by the conservation of energy we have

or

(10i)

Now the time taken to get from A to B is

since

Where s measures the distance along the curve.

Now we know that the length of an arc is given by

And so

and substituting for v from (10i) therefore gives

The problem is thus to find a function y(x) which minimises

where

Now since x is explicitly absent from F we can use (7ii) which said that

But here,

Hence, substituting in (7ii) we have:

where

is a constant.

Therefore

Hence,

where

is a constant.

Hence,

and so

Now we obviously require the curve to point downwards at ,

and so

From which

So,

where

is a constant

Now, we can integrate this by a change of variables,

(10ii)

And so

We therefore have

Hence

And do

However, the boundary conditions stated that when

and

Hence,

11. The solution is therefore best given parametrically by:

and from (10ii)

The value of the constant

can be found from the fact that at

The curve traced out is that of a cycloid. This is the curve traced out by a point on a circle, rolling on a fixed line:

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