Let to be a ball positioned at some point . Considering that only the gravitational force acts on the ball, what is the trajectory the ball should follow in order to reach a point in the shortest time? The solution of this problem is known as the brachistochrone curve.
Although similar, the brachistochrone curve has a different description than the tautochrone curve. The amusing fact is that they are, indeed, the same. The brilliant argument exposed next is due to Johann Bernoulli.
The principle of least time
Light has different speeds at different mediums. That’s why we perceive the phenomenon of refraction when observing objects underwater while ourselves stand out of it. Willebrord Snellius was the first to give a relation between the speed of light in different mediums and its angle of refraction. By denoting and constants related with the speed of light at mediums and (also called refraction indexes of mediums and ); the entry ray’s angle with the plane separating and by ; and the out ray’s angle, the relation is written as:
The tautochrone is the curve in which a ball, positioned anywhere on the curve, will take the same time to arrive at its lowest point.
In a physical model where uniform gravity is the only force acting on the ball, the tautochrone curve is a cycloid (the curve described by a fixed point in a rolling circle), as correctly proven by Huygens in 1673 [BOYDE] using pure geometric arguments. We are going to present two different ways to arrive at this result.
The expression “It is easy to see that…” was used many and many times by the scientist Pierre-Simon Laplace when he didn’t want to come into the details of its ideas. One of the main contributions of Laplace, the Laplace Transform, will be explained here, hopefully in an easy to see fashion.
Let . Its Laplace transform is defined as:
If is piecewise continuous and there exist real positive numbers such that , then the integral above is well defined for all values .
In the beginning of the last decade of the 17th century, Jacob Bernoulli proposed the problem of the catenary; in 1696, Johann Bernoulli proposed the problem of the brachistochrone; some years later, Daniel Bernoulli would suggest to Euler the right functional to solve the elastica problem. The Bernoulli’s family had a great influence in the development of the Calculus of Variations, which was formalized by Euler in his book of 1744. Here the catenary problem and its solution are described in detail.
Definition and first model.
In 1690, Jacob Bernoulli stated the following problem:
Assume you have a perfectly elastic wire (it can be deformed by the action of some force, but its shape is recovered immediately after the forces are ceased) made of a material with uniform density . The two extremities of the wire are attached at the top of two columns of the same height. Moreover, assume that the only forces acting on the wire are tension and gravity. What is the equation described by the wire?
A model that works perfectly fine in every single case. Unfortunately, models that get closer to this tend to be very complex and, as a consequence, unpractical. Sometimes the best thing is to focus on a particular case and use all the assumptions that come with it in your favor and then devise a model to solve this particular class. In the following, we are going to work through image restoration models. The concepts introduced here will be motivated using the problem of Image Denoising.
An example from politics
Try to answer the question: Who is going to be the next president of Brazil?
It could be very hard to answer, even if you are a Brazilian. Nonetheless, you can be more confident on your answer if additional pieces of information are given. For example, your guess will be different in each of the scenarios below.