Así NEWTON Humilló a los MEJORES Matemáticos del Mundo (Royal Society)

The Legendary Challenge of Johan Bernoulli

Introduction to the Mathematical Challenge

• The legend recounts that in 1696, Johan Bernoulli, a prominent figure in mathematics, presented two legendary problems to the scientific community and the Royal Society.

• He allowed six months for solutions, but no satisfactory answers emerged from notable mathematicians like L'Hôpital.

The Arrival of Newton

• After an additional six months without resolution, a significant character was introduced: Isaac Newton. The speaker hints at an anecdote involving "the claw of a lion" and how Newton outperformed all scientists in one night.

• Viewers are encouraged to engage with the content by liking the video and considering membership for access to university-level math courses.

The Nature of Bernoulli's Problems

• Bernoulli's challenge revolved around determining which curve allows objects to descend most rapidly under gravity—a problem he had empirically studied but could not solve mathematically.

• His findings were referenced in popular culture (e.g., Spider-Man 2), highlighting their significance despite lacking mathematical formulation.

The Competitive Spirit Among Mathematicians

• Bernoulli issued his challenge to renowned mathematicians including L'Hôpital, Hooke, Halley, and others who were eager to prove their prowess rather than merely win a prize.

• Despite their efforts over a year, only L'Hôpital submitted a solution that he himself deemed unsatisfactory due to its lack of elegance.

Newton's Involvement Through Halley

• After a year without substantial progress on the problems, L'Hôpital suggested sending them to Newton as a potential last resort—believing even he might struggle with them.

• Halley agreed to deliver the problems personally; however, upon receiving them, Newton initially dismissed them arrogantly.

A Turning Point: Newton’s Response

• Despite his initial disinterest in reading the letter from Bernoulli, it is implied that something changed overnight leading him towards solving these challenges.

The Genius of Newton and the Brachistochrone Problem

The Discovery of Newton's Solution

• Bernoulli received an anonymous letter containing a solution to two mathematical problems, which he recognized as elegant and masterful.

• The solution was attributed to Isaac Newton, who resolved the issues in just one night, showcasing his extraordinary intellect compared to other mathematicians of the time.

• When asked how he knew it was Newton's work, Bernoulli referenced "the claws of the lion," indicating that Newton had decisively outperformed his contemporaries without claiming credit for himself.

Newton's Impact on Mathematics

• Newton not only provided a solution but also subtly challenged those who doubted him by stating he did not appreciate being mocked by foreigners regarding mathematics.

• This incident led to significant developments in mathematics, particularly in the field known as calculus of variations, which emerged after Newton’s findings were expanded upon by mathematicians like Euler.

Educational Opportunities

• The speaker invites viewers to join their channel for access to various mathematics courses ranging from precalculus to advanced topics such as tensor calculus and group theory.

• Currently, 2,300 members are benefiting from these educational resources, including videos and exercises designed for comprehensive learning.

Understanding the Brachistochrone Problem

• The problem involves determining the curve traced by a point under gravity between two vertical points (A and B), aiming for minimal travel time.

• Contrary to Galileo’s assumption about inclined planes, this curve is known as the brachistochrone.

Mathematical Foundations

• To solve this problem, energy conservation principles are applied: potential energy equals kinetic energy. This leads to equations involving mass (m), gravitational acceleration (g), and height (y).

• The length of arc from point A to point B is expressed mathematically using derivatives and integrals related to y with respect to x.

Calculus of Variations Approach

• Two key equations are established: one relating changes over time with respect to velocity along the arc length; another transforms into a functional form suitable for variational methods.

Understanding the Brachiostochrone Problem

Derivation of the Equation

• The discussion begins with establishing that certain variables equal zero, leading to a simplified equation involving partial derivatives of F with respect to y and y prime.

• The speaker applies associativity in calculus, resulting in an expression where the derivative of F concerning x minus another term equals zero.

• This leads to the conclusion that for a solution curve, F minus y prime times its partial derivative must equal a constant, indicating that all derivatives yielding zero are constants.

Characteristics of the Solution Curve

• The derived solution indicates that the relationship involves constants and roots, ultimately defining the brachiostochrone as a cycloid.

• The cycloid is parameterized by specific equations for x and y, demonstrating how it relates to gravitational forces and motion.

Application of Cycloid in Physics

• A second problem is introduced regarding finding a curve such that lines drawn from given points maintain a constant sum; this connects back to Newton's discovery related to cycloids.

• The elegant representation of dy/dx is discussed, linking it back to properties of circles and emphasizing similarities between both problems.

Conclusion on Mathematical Concepts

• The speaker reflects on the complexity and beauty of variational calculus while encouraging viewers to engage further with these mathematical concepts through membership options.