How did Newton solve the fastest descent problem?

Newton received Bernoulli's challenge and, after finding it too enticing to ignore, solved it overnight. He submitted his solution anonymously to 'Philosophical Transactions,' leading Bernoulli to recognize it as Newton's work.

What is Fermat's principle of least time?

Fermat's principle of least time states that light travels between two points along the path that requires the least amount of time. This principle explains phenomena like reflection and refraction, and was pivotal in deriving Snell's Law.

What is the principle of least action?

Proposed by Maupertuis, the principle of least action suggests that nature minimizes a quantity called 'action' (mass times velocity times distance) on the path taken between two points. It generalizes Fermat's principle of least time.

What is the Lagrangian and Hamilton's principle?

The Lagrangian (L) is typically defined as the kinetic energy minus the potential energy (T-V). Hamilton's principle states that the path a system takes is the one for which the integral of the Lagrangian over time (the action) is stationary (usually a minimum).

Is the principle of least action equivalent to Newton's laws?

Yes, for classical mechanics, the principle of least action, when formulated using the Euler-Lagrange equations, yields Newton's second law (F=ma). However, the principle of least action is more general and applies to a wider range of physics.

Why is the principle of least action important beyond classical mechanics?

The principle of least action has proven to be fundamental across physics, appearing in areas like electromagnetism and even playing a role in the development of quantum theory, as hinted by its connection to the UV catastrophe.

Key Moments

The Closest We’ve Come to a Theory of Everything

Q: How did Euler and Lagrange contribute to the principle of least action?

Euler rigorously developed Maupertuis's principle using calculus of variations, adding conditions like energy conservation. Lagrange later provided a general proof and developed the Lagrangian formulation (kinetic minus potential energy), which became standard.

Veritasium

Education3 min read33 min video

Oct 29, 2024|16,367,205 views|367,406|13,964

veritasium science physics Veritasium math quantum mechanics Johann Bernoulli Isaac Newton principle of least action problem of fastest descent Snell’s Law Pierre Louis Maupertuis

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Key Moments

TL;DR

Physics unified by the principle of least action, minimizing 'action' across different phenomena.

Key Insights

The problem of fastest descent led to the brachistochrone curve (cycloid), illustrating optimization.

Fermat's principle of least time, explaining light refraction, is a precursor to broader optimization principles.

Maupertuis proposed the principle of least action (mass x velocity x distance) to unify physics.

Euler and Lagrange provided mathematical rigor and generalized the principle of least action.

Hamilton's principle (integral of kinetic minus potential energy over time) is the modern form of least action.

The principle of least action, expressed through the Euler-Lagrange equations, unifies classical mechanics, optics, and beyond.

THE PROBLEM OF FASTEST DESCENT AND THE BRACHISTOCHRONE

The exploration begins with a classical physics problem: finding the fastest path for a mass to slide from point A to point B. Common intuition suggests a straight line, but a slightly curved ramp, the brachistochrone (or cycloid), accelerates the mass more effectively, reducing total travel time. This problem, famously posed and solved by great mathematicians, highlights that nature doesn't always follow the most direct path but optimizes for time.

FERMAT'S PRINCIPLE OF LEAST TIME AND LIGHT REFRACTION

Ancient philosophers pondered how light travels, concluding it takes the shortest path in a uniform medium. However, when light passes between different media, it bends (refracts). Pierre de Fermat proposed that light minimizes travel time, not distance. This principle, elegant and profound, explains Snell's law of refraction and demonstrates nature's tendency towards optimization, a concept that would later inspire broader theories.

MAUPERTUIS' PRINCIPLE OF LEAST ACTION

Building upon Fermat's ideas, Pierre Louis Moreau de Maupertuis proposed a more universal principle: the principle of least action. He posited that nature minimizes a quantity called 'action,' defined as mass times velocity times distance. This revolutionary idea suggested that a single rule could govern diverse physical phenomena, from the motion of particles to the behavior of light, though it was initially met with skepticism.

EULER AND LAGRANGE: RIGOR AND GENERALIZATION

Mathematicians Leonhard Euler and Joseph-Louis Lagrange provided the crucial mathematical framework for Maupertuis' principle. Euler refined it by replacing sums with integrals and introduced necessary conditions like energy conservation. Lagrange later provided a general proof and reformulated the principle using kinetic and potential energies (T-V), leading to the Euler-Lagrange equations, which became a powerful tool for solving complex mechanics problems.

HAMILTON'S PRINCIPLE AND THE MODERN FORMULATION

William Rowan Hamilton further advanced the principle, reformulating it as the principle of stationary action. This modern version integrates the Lagrangian (T-V, kinetic minus potential energy) over time. Hamilton's principle requires specified start and end points and times, and it applies to paths where the 'action' is stationary (often a minimum, but not always), providing a unified approach to describing motion.

THE EULER-LAGRANGE EQUATIONS: A UNIVERSAL TOOL

The Euler-Lagrange equations, derived from the principle of least action, offer a systematic way to find the equations of motion. Instead of using forces directly, one defines the system's kinetic and potential energies and plugs them into these equations. This method is particularly powerful for complex systems, allowing for easier calculation in various coordinate systems and enabling the unification of mechanics, optics, and even leading to quantum theory.

IMPLICATIONS FOR THE THEORY OF EVERYTHING

The principle of least action, in its various forms, has become a cornerstone of modern physics. It elegantly unifies seemingly disparate physical laws under a single optimization principle. From classical mechanics to electromagnetism and quantum theory, this concept provides a fundamental rule that underlies the behavior of the universe, suggesting that nature operates with remarkable efficiency and economy, minimizing 'action' across all scales.

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Common Questions

The problem of fastest descent, also known as the brachistochrone problem, asks for the shape of a ramp that will allow a mass to travel from point A to point B in the shortest possible time. It's not necessarily a straight line. The solution is a cycloid.

Topics

Principle Of Least Action Principle Of Least Time Calculus Of Variations Lagrangian Mechanics Newton's Laws Euler-Lagrange Equations Hamilton's Principle Brachistochrone Problem Cycloid Fundamental Physics

Mentioned in this video

personLeonhard Euler

A mathematician who defended Maupertuis's principle of least action, rigorously developing it using calculus of variations and introducing necessary conditions.

conceptNewton's Second Law

Shown to be equivalent to the principle of least action when applied to mechanics, representing forces.

conceptUV catastrophe

A problem in atomic physics that was partially solved using the concept of action, hinting at its importance in quantum theory.

personWilliam Rowan Hamilton

Reformulated the principle of least action in terms of an integral over time (Hamilton's principle), introducing the concept of the Hamiltonian.

conceptcycloid

The curve that represents the solution to the brachistochrone problem, being the fastest path for a mass to descend.

personPierre-Louis Moreau de Maupertuis

Proposed the principle of least action, suggesting nature minimizes 'action' (mass x velocity x distance), a generalization of Fermat's principle.

personSamuel König

A friend of Maupertuis who criticized his principle of least action, accusing him of plagiarism and poor physics.

personHero of Alexandria

An ancient philosopher who contemplated how light travels, realizing it follows the shortest path in a single medium.

conceptHamilton's principle

The modern formulation of the principle of least action, using an integral over time and requiring specified start and end times.

bookPhilosophical Transactions

A journal where Newton submitted his solution to the fastest descent problem anonymously.

personJoseph-Louis Lagrange

Provided a general proof for the principle of least action using Euler's methods and developed the Lagrangian formulation.

softwareEuler-Lagrange equation

The differential equation derived from the principle of least action that describes the equations of motion.

personPierre de Fermat

Proposed the principle of least time, suggesting light travels between two points in the shortest possible time, which led to Snell's Law.

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