Brachistochrone Problem

Problem Statement

Given two points A and B, find the path joining these points such that a bead sliding down under gravity (without friction) travels from A to B in the shortest time.

Mathematical Model

Physical Principles

When a bead slides along a path under gravity without friction:

• Energy conservation applies (potential energy converts to kinetic energy)
• The speed at any point depends on the vertical distance traveled
• The time required depends on the path shape

Speed at Any Point

Using conservation of energy, for a bead starting from rest at point A(0,0) and reaching point (x,y) where y is measured downward:

$\frac{1}{2}m{v}^2=mgy$

Solving for velocity:

$v=\sqrt{2gy}$

Time of Travel

For a path y(x), the infinitesimal time to travel along a segment of the curve is:

$dt=\frac{ds}{v}=\frac{\sqrt{1+(dy/dx{)}^2}dx}{\sqrt{2gy}}$

The total time of travel is therefore:

$T={\int }_A^{B}dt={\int }_A^B\frac{\sqrt{1+(dy/dx{)}^2}}{\sqrt{2gy}}dx$

Optimisation Setup

Design Function

• y(x) = the path shape (function, infinite-dimensional problem)

Objective Functional

We want to minimize the total travel time:

$L(y)={\int }_A^B\sqrt{\frac{1+(dy/dx{)}^2}{2gy}}dx$

Solution Using Variational Calculus

The Euler-Lagrange equation for this problem is:

$\frac{\partial f}{\partial y}-\frac{d}{dx}\left(\frac{\partial f}{\partial y^{\prime }}\right)=0$

Where $f(y,y^{\prime },x)=\sqrt{\frac{1+y^{\prime 2}}{2gy}}$ and $y^{\prime }=\frac{dy}{dx}$.

Applying the Euler-Lagrange Equation

Since $f$ does not explicitly depend on $x$, we can use a simpler form of the solution. The first integral of the Euler-Lagrange equation gives:

$f-y^{\prime }\frac{\partial f}{\partial y^{\prime }}=C$

where C is a constant of integration.

After substitution and algebraic manipulation, we get a differential equation whose solution is parametrized as:

$x=k(\theta -\sin \theta )$ $y=k(1-\cos \theta )$

where $k$ is a constant determined by the boundary conditions.

The Cycloid Solution

The parametric equations above describe a cycloid, which is the curve traced by a point on the circumference of a circle as it rolls along a straight line without slipping.

A cycloid has some remarkable properties:

• It is the brachistochrone (path of shortest time)
• It is also the tautochrone (path where the travel time is independent of starting position)
• It was later discovered to be the solution to several other optimization problems

Verification Through Boundary Conditions

To find the specific cycloid passing through points A and B:

1. Set the origin at point A
2. Determine parameter $k$ based on the coordinates of point B
3. Find the appropriate range of parameter $\theta$

Numerical Example

Suppose point A is at (0,0) and point B is at (1,1).

To find the cycloid passing through these points:

1. Use the parametric equations $x=k(\theta -\sin \theta )$ and $y=k(1-\cos \theta )$
2. Solve for $k$ and the value of $\theta$ at point B
3. The path between A and B is the cycloid segment for $\theta$ ranging from 0 to this value

Comparison with Other Paths

The brachistochrone solution can be compared with other paths:

• Straight line: Intuitively might seem fastest but isn’t
• Circular arc: Closer to optimal but still not as fast
• Cycloid: Optimal path, providing the shortest travel time

For a typical configuration with B directly below and to the right of A, the time savings can be significant:

• A bead on the cycloid path might reach B in 30% less time than on a straight path
• The deeper the destination point, the more pronounced the advantage

Physical Demonstration

The brachistochrone problem can be demonstrated physically with a set of tracks:

• One shaped as a straight line
• One shaped as a cycloid
• Perhaps additional tracks with other shapes

Beads released simultaneously will consistently reach the end of the cycloid track first, confirming the theoretical result.

Functional Optimisation Perspective

This problem is a classic example of functional optimization:

1. The “design variable” is a function y(x) rather than a finite set of parameters
2. The objective is a functional (a function of a function) rather than a standard function
3. The solution requires solving a differential equation (the Euler-Lagrange equation)
4. Boundary conditions (the fixed endpoints) constrain the solution

Generalizations

The brachistochrone problem can be generalized in several ways:

1. Different force fields (not just constant gravity)
2. Adding friction
3. Constraints on the path
4. Multiple particles with interaction
5. Alternative objectives (e.g., minimize energy)

Key Insights

1. The brachistochrone problem illustrates that intuition can be misleading in optimization
2. It demonstrates the power of variational calculus in solving optimization problems with infinite-dimensional design spaces
3. The cycloid solution has properties that make it important beyond just this problem
4. The principle of least time (Fermat’s principle) appears in many areas of physics, including optics
5. This problem exemplifies how optimization principles can reveal fundamental properties of natural phenomena