Total Solar Eclipse in April 2024

 

The total solar eclipse that will occur on Monday, April 8, 2024, will cross North America, going over Canada, the United States, and Mexico. When the moon moves between the sun and the Earth and briefly obscures the sun's face, this is known as a total solar eclipse. The term "path of totality" refers to the lunar shadow's trajectory across the surface of Earth. Thirty-one million Americans already reside inside the path of totality, making the celestial marvel accessible to them. The South Pacific Ocean will be the starting point of the total solar eclipse. The Pacific coast of Mexico will witness totality first in continental North America, weather permitting, at approximately 11:07 a.m. PDT.

Following its journey from Mexico, the eclipse will pass across Oklahoma, Arkansas, Missouri, Illinois, Kentucky, Indiana, Ohio, Pennsylvania, New York, Vermont, New Hampshire, and Maine before entering the United States in Texas. The complete solar eclipse will also be visible in a few isolated areas of Tennessee and Michigan. After crossing Southern Ontario into Canada, the eclipse will pass over Quebec, New Brunswick, Prince Edward Island, and Cape Breton. At 5:16 p.m. NDT, the eclipse will leave continental North America and cross the Atlantic coast of Newfoundland, Canada.

The total solar eclipse on April 8 is expected to be the largest mass travel event in U.S. history, drawing together eclipse professionals and amateur skywatchers alike.

Various things should be avoided during a solar eclipse. Following is a list of such things:

1) Do not look directly at the sun.

2) Avoid using optical devices.

3) Don't drive wearing eclipse glasses.

4) It is better to avoid travel.

5) Do not drive while trying to watch the eclipse.

6) Use proper eye protection.

7) Don't use optical devices without proper filters.

8) Supervise children while watching the event.

Prabir Rudra

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The Devil Comet is approaching us

Over the next few days, skywatchers will be able to see more than just the bright planet Mercury and April's total solar eclipse. Comet 12P/Pons-Brooks, sometimes known as the "devil comet," will be visible in the night sky for the next few days and might emerge during the April 8th total solar eclipse. Since Pons-Brooks only completes one orbit around the sun every 71 years, observing it is usually an opportunity that comes only once in a lifetime.

Pons-Brooks is a ball of ice and rock that is 10.5 miles wide. In comparison, it will be twice as large as the Mount Everest. It is currently traveling towards our sun with an extremely elliptical or stretched-out orbit. It features a frozen shell, or nucleus, encircling a core composed of solid ice, gas, and dust. A coma, or a cloud of frozen material, progressively seeps out of the comet's center and covers the nucleus. 

Pons-Brooks is cryovolcanic, unlike most other comets. It frequently erupts when the nucleus's fractures are opened by sun radiation. As a result, frozen cryomagma under extreme pressure erupts into space. The surrounding cloud of frozen particles enlarges and seems brighter than usual when this happens.

For the first time in 69 years, Pons-Brooks erupted violently in July 2023, leaving behind two separate gas and ice trails that resemble a pair of devil horns. It has persisted in erupting rather regularly.

When will it be visible?

Pons-Brooks may become visible to the unaided eye during the next few weeks as it passes through the inner solar system. It will stay that way until April 2 when it moves closer to the sun and disappears from view in the pitch-black night sky. The evenings of 30th and 31st March are considered to be the best time to see the comet as it will be nearest to the Earth at that time. On June 2, when it moves away from the sun, it will be at its closest to Earth. It will be roughly 139.4 million miles away and presents no known hazard to Earth. 

It is expected to become even more active in the upcoming weeks and be visible to the unaided eye with a maximum brightness magnitude of about 4.0, according to SETI Institute postdoctoral scholar Ariel Graykowski. The appearance appears brighter the lower the magnitude. It won't be very noticeable in the sky because "the limit for naked eye objects in dark, moonless skies is around 6 magnitudes," according to Graykowski.

Where should we look?

It is most seen in the early evening in the Northern Hemisphere toward the west-northwest horizon. Pons-Brooks is located low in the northwest sky, close to the Pisces constellation. It should look like a blazing ball of ice, trailed by the forks of its horns.

An hour or so after sunset, the comet should be visible to the unaided eye low in the west. As it approaches the sun, it will become somewhat brighter. You want to find a place where you can see the western horizon unhindered and where there are no city lights. Using binoculars might be a good idea as the comet might be difficult to find without them.

The brightest planet Venus will become visible on one side of the sun as the sky grows darker. Jupiter, the second-brightest planet in the solar system, is located on the other side. Furthermore, you should be able to see Comet Pons-Brooks closer to Jupiter but between Jupiter and the sun if it is sufficiently bright.

Remember this is your only chance to see it because it won't be seen again until 2097. 

Prabir Rudra

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Big Bang: The beginning of the Universe

The Big Bang theory is the prevailing cosmological model for the observable universe's earliest known periods and its subsequent large-scale evolution. It proposes that the universe began as a hot, dense state approximately 13.8 billion years ago and has been expanding and cooling ever since.

Here's a brief overview:

1. Initial Singularity: The Big Bang theory proposes that the universe began as a hot, dense point known as a singularity. At this moment, all the matter, energy, space, and time were contained within this infinitesimally small point. As a result, certain qualities became infinite or undefined at the Big Bang, hence the name singularity.

2. Rapid Expansion: Approximately 13.8 billion years ago, the singularity underwent a rapid expansion known as cosmic inflation. This expansion caused the universe to cool down and expand, leading to the formation of subatomic particles and eventually atoms.

3. Formation of Elements: As the universe expanded and cooled, subatomic particles such as protons, neutrons, and electrons combined to form the first hydrogen and helium nuclei. This process, known as Big Bang nucleosynthesis, occurred within the first few minutes after the Big Bang.

4. Formation of Stars and Galaxies: Over millions to billions of years, gravity caused regions of higher density in the early universe to collapse, forming stars and galaxies. Within galaxies, stars formed from clouds of gas and dust, and many stars went on to form planetary systems.

5. Cosmic Microwave Background Radiation (CMB): About 380,000 years after the Big Bang, the universe cooled enough for neutral atoms to form. This event, known as recombination, led to the release of photons that continue to travel through the universe today as the cosmic microwave background radiation. The CMB provides a snapshot of the universe at that early stage.

6. Observable Universe: The expansion of the universe continues to this day. The observable universe, the part of the universe we can see, includes galaxies, clusters of galaxies, and cosmic structures, all moving away from each other due to the ongoing expansion.

The Big Bang theory is supported by a wealth of observational evidence, including the cosmic microwave background radiation, the observed distribution of galaxies, the abundance of light elements, and the large-scale structure of the universe. However, it's important to note that there are still unanswered questions and areas of active research within cosmology, such as the nature of dark matter and dark energy, and the conditions of the universe before the Big Bang. This is probably the reason why various competing theories like the steady state theory, cyclic universe model, multiverse hypothesis, quantum cosmology, etc. find their place in the literature.

Prabir Rudra

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Emmy Noether: Probably the best female Mathematician in the history

Consider that you are launching a ball into the air. The skill of juggling involves a dance between motion and rest, with the ball's speed decreasing as it rises and increasing as it descends. One of the fundamental ideas of physics is revealed in this dance: conservation laws. Earth, or juggling for that matter, is not the only place where conservation principles apply. These are general laws of physics that hold true in many branches of the subject. However, they aren't always clear-cut. Here's where the woman from Germany enters the scene. Her discovery revealed an incredibly straightforward relationship between these conservation rules and the symmetries of the world.

Emmy Noether, the woman who developed one of the most beautiful theorems in physics. Today (March 23) is her birthday.

Emmy Noether was a German mathematician who made significant contributions to abstract algebra and theoretical physics. She was born on March 23, 1882, in Erlangen, Germany, and died on April 14, 1935, in Bryn Mawr, Pennsylvania, United States.

She developed what is now known as Noether's theorem, which is fundamental in the study of symmetries in physics. This theorem relates symmetries and conservation laws, showing that for every continuous symmetry in a physical system, there is a corresponding conserved quantity.

Despite facing discrimination as a woman in academia during her time, Noether made significant contributions to mathematics and physics. She worked at the University of Göttingen, where she collaborated with prominent mathematicians and physicists such as David Hilbert and Albert Einstein. Her work laid the foundation for modern algebra and had a profound impact on theoretical physics, particularly in the development of quantum mechanics and general relativity.

Noether thought about the issue raised by Hilbert and Einstein, namely that the General Relativity Theory appeared to violate the law of conservation of energy. This led to the development of her well-known theorem, which fundamentally altered the path of physics.

It's likely true that Emmy Noether is the best female mathematician in history. Noether's theorem fundamentally changed our view of the cosmos. Her pioneering work in abstract algebra also revolutionized mathematics.

Contributions

Emmy Noether made several significant contributions to mathematics and theoretical physics during her career. Some of her most notable contributions include:

1. Noether's Theorem: This is arguably her most famous contribution. Noether's theorem states that for every continuous symmetry in a physical system, there is a corresponding conserved quantity. This theorem has profound implications in theoretical physics, particularly in the conservation laws of energy, momentum, and angular momentum.

2.  Abstract Algebra: Noether made significant contributions to abstract algebra, particularly in the development of ring theory and group theory. She introduced the concept of "ideals" in ring theory, which has become a fundamental concept in modern algebra. Her work provided important insights into the structure of algebraic systems.

3. Invariant Theory: Noether worked on invariant theory, a branch of mathematics concerned with properties that remain unchanged under a group of transformations. Her work in this area helped to clarify and extend the understanding of the relationships between symmetry and invariance.

4.  Algebraic Geometry: Noether made contributions to algebraic geometry, particularly in the study of algebraic curves and surfaces. She developed the theory of "Noetherian rings" and made significant advances in understanding the properties of algebraic varieties.

5.  Physics: While primarily a mathematician, Noether's work had a profound impact on theoretical physics. Her theorem has applications in various areas of physics, including classical mechanics, quantum mechanics, and general relativity. She collaborated with notable physicists such as Albert Einstein and David Hilbert, contributing to the development of their theories.

6.  Teaching and Mentorship: Noether was also known for her exceptional teaching skills and mentorship of students. Despite facing discrimination as a woman in academia, she mentored several prominent mathematicians and physicists, inspiring future generations of researchers.

Emmy Noether's contributions to mathematics and theoretical physics have had a lasting impact on these fields, and she is widely regarded as one of the most influential mathematicians of the 20th century. She has left a lasting legacy, with her work continuing to influence research in various fields of science and mathematics.

Prabir Rudra

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Abel Prize for 2024 goes to Michel Talagrand

 

The 2024 Abel Prize has been awarded to Michel Talagrand of the French National Center for Scientific Research (CNRS), Paris, France for his ground-breaking work in functional analysis and probability theory, which has excellent applications in statistics and mathematical physics.

Michel Talagrand is recognized for his contributions to stochastic processes and probability theory. Problems arising in the context of gambling or risk assessment served as the first impetus for the creation of probability theory. An overarching theme throughout Michel Talagrand's seminal discoveries is interacting with and comprehending the stochastic processes surrounding us. It is now evident that in the modern world, a deep comprehension of random events is crucial. For instance, our huge language models and weather forecasts are based on random methods.

Random events are constantly at work in the modern world and have far-reaching implications for everything from condensed matter physics to business logistics. Talagrand's work mostly focuses on comprehending and applying the "Gaussian distribution," which is often referred to as the "Normal distribution" or, because of its form, the "bell curve". The Gaussian distribution governs every aspect of our lives. Weight at birth, grades at school, and retirement ages of athletes are all seemingly random phenomena that nicely fit into the Gaussian distribution.

Specific areas of research

The prize has been awarded for the following three basic areas:

Suprema of stochastic processes: 

The greatest value that can be anticipated from a group of random values is known as the "supremum" of stochastic processes. A stochastic process generates a series of random values. Knowing the expected height of the biggest wave that will hit the beach the following year is helpful if the height of waves that smash onto a beach is determined by a stochastic process.

Concentration of measures: 

It may seem counterintuitive, but when a process relies on a variety of random sources, rather than becoming more complex, the various random components may be able to balance each other out and yield more predictable outcomes. For this, Talagrand has provided accurate quantitative estimations.

Spin glass: 

Leaving abstract probability theory behind, physicists were first surprised to learn that atoms may arrange themselves in a unique form of matter called a "spin glass". Talagrand finished the proof of Giorgio Parisi's work, which won him the Nobel Prize in 2021, by using his understanding of statistics and probability to establish bounds on the behavior of spin glass matter.

Talagrand is a strong problem solver in addition to being an outstanding mathematician. He has significantly advanced our knowledge of random processes, particularly Gaussian processes. His contributions have changed probability theory in several ways. Professor Helge Holden, chair of the Abel Prize Committee, adds, "His proof of the renowned Parisi formula for the free energy of spin glasses is an amazing accomplishment."

Life

Michel Talagrand was born in France in 1952 and graduated from the University of Paris VI with a Ph.D. in mathematics in 1977. He studied for a while at Ohio State University in the United States. He is married and has two sons. He is a member of the French Academy of Sciences and has won various honors and awards. One of the amazing things that he runs on his website is that he challenges mathematicians to solve riddles under the banner "Become rich with my prizes."

Prabir Rudra

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The explosion of Corona Borealis: Once in a lifetime sight in the sky

According to NASA, astronomers anticipate that a "new star" will emerge in the night sky at any moment between now and September, 2024. This celestial spectacle is expected to be once in a lifetime.

The Milky Way's Corona Borealis, or Northern Crown constellation, which is situated between the Boötes and Hercules constellations, is where the anticipated brightening event, known as a nova, will occur.

A nova is the quick, brief explosion from a collapsing star known as a white dwarf, whereas a supernova is the catastrophic death of a large star.

The "Blaze Star," or T Coronae Borealis, is a binary system in the Corona Borealis that consists of an aged red giant star and a dead white dwarf star. When stars run out of hydrogen for nuclear fusion and start to fade, red giants are created. NASA estimates that in roughly 5 or 6 billion years, our sun will turn into a red giant, sputtering and growing as it releases layers of material and probably vaporizing the inner planets of the solar system. In this context, Earth's future is yet unknown.

An explosive event occurs in T Coronae Borealis approximately every 79 years.

Because of their close proximity, the stars in the orbital pair interact furiously with one another. As the red giant heats up, it becomes more unstable, shedding its outer layers that fall as matter onto the white dwarf star.

According to the space agency, the interchange of stuff leads the white dwarf's atmosphere to steadily heat up until it undergoes a "runaway thermonuclear reaction," which results in a nova.

How to keep an eye on the event?

Astronomers are closely monitoring T Coronae Borealis once more after it last had a spectacular outburst in 1946.

In an email, NASA Meteoroid Environments Office lead William J. Cooke stated that "most novae happen unexpectedly, without warning." Nonetheless, T Coronae Borealis is among the galaxy's ten recurrent novae. From the last eruption in 1946, we know that the star will darken for a little more than a year before brightening up quickly. T Coronae Borealis started to dim in March of last year, and between now and September, some researchers anticipate that it may go nova. We are unable to predict with greater accuracy than the few months it will take for this to occur, given the current state of knowledge.

It is anticipated that the star system, which is 3,000 light-years away from Earth and usually too faint to be seen with the unaided eye, would become as bright as Polaris, or the North Star.

When the nova reaches its maximum brightness, it will appear as if a new star has emerged. It can be seen with binoculars for a little over a week and without any equipment for a few days, after which it will fade and be lost from view for another 80 years or more.

The nova will be visible from the Northern Hemisphere, appearing in a little arc between the constellations Hercules and Boötes.

The Neil Gehrels Swift Observatory, located in space, will be used by astronomers to analyze the celestial event through ultraviolet and X-ray light, while the Hubble Space Telescope will be used to observe the nova.

Cooke stated, "Repeated novae like T Coronae Borealis provide us insights into the thermonuclear runaway that occurs on the surface of the white dwarf when the star goes nova and help us understand the mass transfer between the stars in these systems."

Updates regarding the eruption and its appearance will be available from the NASAUniverse account on X, formerly known as Twitter.

Cooke remarked that the brightness of Nova Cygni, which he saw last in 1975, was comparable to that of T Coronae Borealis. It is not anticipated that Nova Cygni will encounter another explosion.

"On August 29, I was outside as a teenage astronomy geek who was about to start college," Cooke recalled. "Looking up at the sky, I saw that there was a star that shouldn't have been there in the Cygnus constellation. We discovered that we were staring at a nova after I persuaded some pals who thought I was insane to look at it! It was a really memorable event that confirmed my professional decision to pursue astronomy. I used to make jokes about how undergraduate physics had to blow out a star for me to have to endure it.

Prabir Rudra

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Wormhole: The missing link between us and our future home

 

The term "wormhole" usually describes a theoretical spacetime tunnel-like structure that may link two different points in space or time, possibly enabling instantaneous or faster-than-light travel between far-off places. The idea comes from theoretical physics, specifically from the perspective of general relativity.

Wormholes are frequently discussed as solutions to Einstein's field equations in astrophysics and cosmology. These equations control the gravitational effects of matter and energy on the curvature of spacetime. Although the equations of general relativity make wormholes theoretically viable, they have not been detected or verified to exist in reality, hence they are still considered speculative.

Scientists, authors, and filmmakers have all been captivated by wormholes, which are often depicted in science fiction as a way to travel through space or journey back in time. Wormhole stability, traversability, and the exotic stuff needed to stabilize them are only a few of the many unresolved concerns in the nascent scientific field of wormhole research.

More recently, the word "wormhole" has been used metaphorically to refer to systems or procedures that allow for unusually fast data transfer, financial transactions, or communication between distant sites or systems in domains like computer science and finance. A "wormhole" in the context of blockchain technology, for instance, might be a bridge that permits token transfers between several blockchain networks.

Given the challenges and uncertainties like existence, stability, need for exotic matter, traversability, time dilation, and paradoxes, interstellar travel via wormholes remains highly speculative at this point. While it's an intriguing concept that has captured the imagination of scientists and science fiction enthusiasts alike, practical implementation would require significant advancements in our understanding of fundamental physics and the development of technologies far beyond our current capabilities. But if this turns out to be a success then humanity can seriously think in terms of colonizing other habitable planets in the future, when the resources of our planet get completely depleted. 

It is said that the future of humanity lies in space and wormholes might just be the missing link between us and our destination (future home). But we have a long way to go!!

Prabir Rudra

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Pi Day 2024

Pi Day is an annual celebration observed on March 14th (3/14) around the world. It's a day dedicated to the mathematical constant π (pi), which is approximately equal to 3.14159. Here are some key aspects and traditions associated with Pi Day:

•  Origin: 

Pi Day was first celebrated in 1988 by Larry Shaw, a physicist at the Exploratorium, a museum of science, art, and human perception in San Francisco. Shaw organized various activities and celebrations to mark the day, including a circular parade and eating pie.

• Significance of Pi: 

Pi is a mathematical constant that represents the ratio of a circle's circumference to its diameter. It's an irrational number, meaning its decimal representation never ends or repeats. Pi has fascinated mathematicians, scientists, and enthusiasts for centuries due to its unique properties.

• Activities and Events: 

Pi Day is often celebrated in schools, universities, museums, and math clubs through various activities and events. These may include pi recitation competitions, pi-themed quizzes, math art contests, and hands-on activities related to circles and geometry.

• Pie Eating: 

One of the most popular traditions associated with Pi Day is indulging in pies, both savory and sweet. Whether it's apple pie, pumpkin pie, pizza pie, or any other kind, enjoying a slice of pie on Pi Day has become a widespread tradition.

• Educational Outreach: 

Pi Day provides an opportunity for educators to engage students in learning about mathematics and its applications in a fun and interactive way. Teachers often use the day to introduce concepts related to circles, geometry, and the history of mathematics.

• Pi Memorization: 

Some enthusiasts challenge themselves to memorize and recite as many digits of pi as possible. While memorizing pi to hundreds or even thousands of digits is impressive, it's important to note that the practical applications of pi typically require only a few decimal places.

• Online Celebrations: 

With the rise of social media and online communities, Pi Day celebrations have expanded to the digital realm. People share pi-related memes, jokes, artwork, and trivia on platforms like Twitter, Instagram, and Reddit, fostering a sense of camaraderie among math enthusiasts worldwide.

Overall, Pi Day is a fun and inclusive celebration of mathematics that encourages people of all ages to appreciate the beauty and ubiquity of numbers in our lives. Whether you're a seasoned mathematician or just someone who enjoys a good slice of pie, there's something for everyone to enjoy on Pi Day. Wish everyone a very happy Pi Day 2024!!

Prabir Rudra

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Classical Mechanics: A journey from Newton to Lagrange and Hamilton

Classical mechanics, as understood today, is largely attributed to the foundational contributions of Sir Isaac Newton, building upon the work of earlier thinkers like Galileo Galilei. Here is a brief overview of the contributions of Newton and Galileo to classical mechanics:

 

Galileo Galilei (1564-1642):

• Experimental Approach:

Galileo is often considered the father of modern physics due to his emphasis on experimental methods. He conducted a series of experiments on inclined planes and falling bodies to study the fundamental principles of motion.

•  Law of Inertia:

Galileo formulated the principle of inertia, which states that an object at rest will remain at rest, and an object in motion will continue in motion with a constant velocity unless acted upon by an external force. This laid the groundwork for Newton's first law of motion.

• Uniform Acceleration:

Galileo made significant progress in understanding uniformly accelerated motion. He developed mathematical descriptions of falling bodies, showing that the distance traveled is proportional to the square of the time elapsed.

 

Isaac Newton (1642-1727):

• Three Laws of Motion:

Newton formulated three laws of motion, which are fundamental principles of classical mechanics.

First Law (Law of Inertia): An object at rest stays at rest, and an object in motion remains in motion with a constant velocity unless acted upon by a net external force.

Second Law: The force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma).

Third Law: For every action, there is an equal and opposite reaction.

• Law of Universal Gravitation:

Newton proposed the law of universal gravitation, which states that every mass attracts every other mass in the universe with a force proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

• Mathematical Formulation:

Newton's work was not only conceptual but also highly mathematical. He introduced calculus to describe motion and developed differential equations to express the relationship between force, mass, and acceleration.

• Unified Theory:

 Newton's laws of motion and the law of universal gravitation provided a unified framework for understanding a wide range of phenomena, from the motion of celestial bodies to the behavior of objects on Earth. This marked a significant departure from earlier fragmented approaches.

Together, the contributions of Galileo and Newton laid the foundation for classical mechanics, providing a systematic and mathematical framework for understanding the motion of objects and the forces acting upon them. Their work is essential for understanding the physical world and continues to be a fundamental part of the study of physics.

Development into an advanced form of mechanics

There was a lull following Newton, and it took until the end of the eighteenth century for classical mechanics to advance further. The development of classical mechanics from Newton to Lagrange and Hamilton represents a significant evolution in the understanding of physical systems. Here is a more detailed overview of this progression:

 

• Newtonian Mechanics (17th century):

Isaac Newton (1642-1727):

Newton formulated the three laws of motion and the law of universal gravitation, establishing a comprehensive framework for classical mechanics. His work laid the groundwork for understanding the motion of objects and the force interactions between them. Newtonian mechanics was basically a force-based mechanics where all motions were considered to be directly associated with forces.

• Principle of Least Action (18th century):

Pierre Louis Maupertuis (1698-1759):

    Maupertuis introduced the principle of least action, suggesting that natural processes follow paths that minimize or maximize a certain quantity called action. The principle states that "the actual path taken by a physical system between two points in its configuration space is the one for which the action, defined as the integral of the Lagrangian over time, is stationary—either a minimum, maximum or a saddle point".

• Analytical Mechanics and Euler-Lagrange Equations (18th century):

Leonhard Euler (1707-1783):

 Euler made significant contributions to mechanics, formulating the Euler-Lagrange equations. These equations provided a mathematical framework for expressing the equations of motion using the principle of least action.

• Lagrangian Mechanics (18th century):

Joseph-Louis Lagrange (1736-1813):

Lagrange further developed the analytical approach to mechanics, introducing the Lagrangian formulation. In his work "Mécanique Analytique" (1788), Lagrange employed generalized coordinates and the principle of least action to derive the equations of motion.

 

• Hamiltonian Mechanics (19th century):

William Rowan Hamilton (1805-1865):

Hamilton built upon Lagrange's work and introduced Hamiltonian mechanics. He reformulated the principle of least action in terms of the Hamiltonian function (H= Kinetic Energy + Potential Energy) leading to Hamilton's principle. In 1833, he published "On a General Method in Dynamics," which introduced the Hamiltonian formulation of mechanics. This approach uses generalized coordinates and momenta to describe the dynamics of a system, and it is equivalent to the Lagrangian formulation. Hamiltonian mechanics provides an alternative perspective, particularly useful in certain physical problems and in the context of quantum mechanics.

In summary, the development of classical mechanics from Newton to Lagrange and Hamilton involved a transition from empirical laws to more abstract and general mathematical formulations. The most striking feature of this advanced form of classical mechanics is that it is no longer a force-based mechanics like its Newtonian counterpart, but it is now based on energy. Euler, Lagrange, and Hamilton each contributed significantly to the analytical and variational approaches that are foundational to classical mechanics today. The Lagrangian and Hamiltonian formulations provide powerful tools for solving complex problems and have applications in various branches of physics.

Prabir Rudra

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