In high school, we were all taught Newton's laws. Objects in motion tend to stay in motion, force equals mass times acceleration, and there is an equal and opposite response to every action. Isaac Newton created a universal theory of gravity based on the principles of motion, which applied equally well to apples falling from trees and planets moving in their orbits. However, Newton was unable to explain why his laws of motion were accurate and why they did not take another form. That discovery would be made by another legendary but lesser-known intellect.
Lagrange vs. Newton
We're accustomed to thinking about motion in terms of forces and accelerations, partly because it's a fairly natural way of viewing the world (for example, I push on something, and it moves), and partly because that's how Newton established his laws (and therefore how they're taught in school). However, studying forces and masses isn't the only approach to characterize our surroundings. Consider a ball being tossed in the air. That ball has a lot of relevant attributes, such as its location, velocity, acceleration, and mass. Some of these features may be quite helpful in forecasting the ball's future motion, while others may be less so. Newton found that the combination of mass, acceleration, and force was very powerful indeed, allowing him to formulate his famous Force = mass * acceleration equation as a fundamental law of the universe.
Joseph Louis Lagrange, a mathematician, physicist, and all-around genius, produced his own formulations some 150 years after Newton published his equations of motion. He discovered that by studying an object's kinetic and potential energy, he could deduce his own laws of motion. Lagrange discovered that the difference between an object's kinetic and potential energy revealed something important about the cosmos.
Stationary action
If I threw a ball at you, you'd probably have a good chance of catching it. You can do this because you've seen a lot of balls thrown at you throughout your life, and your brain has deduced that thrown objects follow a very consistent set of trajectories. Newton's breakthrough was his discovery of a universal rule of motion that could predict the trajectory of a thrown ball. But why should Newton's laws be considered correct? Why should a thrown ball have the same path? Why don't balls bounce backward or fly to Mars on their route to you? Why does the same thing happen every time? In other words, why do objects act the way they do instead of any other? The universe might have selected any behavior for thrown balls or other moving things. What causes Newton's laws to function?
Newton was unable to provide an answer, but Lagrange did. The key is the difference between the kinetic and potential energy of the moving item. If you see a ball in flight, for example, you may determine the difference at any point in time. At the end of the motion, you may put all of the differences together to produce a single value. For different historical reasons, this quantity is known as the action of the item in motion. You may envision many courses the ball could take when it is hurled at you. Different actions will be linked with each of these alternative pathways. And it turns out that the familiar way is the road with the least amount of motion, as anticipated by Newton's equations.
Creating laws of motion
Lagrange developed what we now refer to as the principle of least action. This one unifying concept underpins all physical laws, including Newton's laws. A simple formula is used to create a law of motion. To begin, record the kinetic and potential energies of the objects of interest. Then you calculate their difference. (In his honor, we now refer to this amount as "the Lagrangian.") Then, using a complicated mathematical approach known as the calculus of variations, you determine the equation that minimizes the action. What emerges is a brand-spankin' new physical rule.
Because it is such a strong and sophisticated (and universal) manner of approaching dynamics, all current physics is written in this language. Langrangians are the foundation of general relativity, electromagnetism, and even quantum field theory and the Standard Model, and scientists all around the world use Lagrange's principles to deduce laws of motion. These laws of motion regulate the movement of the planets in the solar system as well as the expansion of the cosmos itself. Lagrange's approach will always provide the answers you want, whether you use general relativity or the basic Newtonian form of gravity.
Comments