What Makes Newton’s Laws Work? Here is the simple trick.

Paul M. Sutter is an astrophysicist at SUNY Stony Brook and the Flatiron Institute, host of “ask an astronaut and “space radio” and author of “how to die in space.”

We’ve all learned Newton’s Laws in high school: moving objects tend to stay in motion, force equals mass multiplied by acceleration, and for every action there is an equal and opposite reaction. Of these laws of motion, Isaac Newton discovered a universal theory of gravity this applied as much to apples falling from trees as to planets moving in their orbits.

But Newton could not explain why his laws of motion were correct and why they had no other form. This discovery would come from another legendary genius, but less famous.

Related: Astronauts become billiard balls to demonstrate Newton’s third law (video)

Lagrange versus Newton

We’re used to thinking of motion in terms of forces and accelerations – partly because it’s a very intuitive way of seeing the world (e.g. I press on something and it moves) and partly because that’s how Newton formulated his laws (and hence how we are taught them in school).

But looking at forces and masses is not the only way to describe the world around us. Think of a ball thrown in the air. This ball has a lot of properties that we might find useful – things like its position, speed, acceleration, and mass. Some of these properties can be very useful in predicting the future motion of the ball, and others less so.

Newton discovered that the combination of mass, acceleration and force was indeed very powerful, which enabled him to formulate his famous Force = mass * acceleration the equation as the fundamental law of the universe.

About 150 years after Newton developed his laws of motion, another versatile mathematician, physicist and genius, Joseph Louis Lagrange, developed his own formulations. He discovered that by examining the kinetic and potential energies of an object, he could also deduce his own laws of motion.

Specifically, Lagrange discovered that the difference between an object’s kinetic energy and potential energy reveals something deeply deep in the universe.

Stationary action

If I threw a ball at you, you’d probably have a good chance of catching it. You can do this because in your lifetime you’ve seen a lot of balls thrown at you and your brain has deciphered that thrown objects follow a fairly common set of trajectories. Newton’s insight was his ability to find a general law of motion that could predict the trajectory of that thrown ball.

But why should Newton’s laws be correct? Why should a thrown ball follow the familiar path? Why don’t the bullets first jump backwards or shoot towards March on the way to you? Why does the same path happen every time? In other words, why do objects behave the way they do, rather than in another way? The universe could have chosen literally any behavior for thrown balls or any other moving object. What Makes Newton’s Laws Work?

Newton didn’t have the answer, but Lagrange did.

The key is the difference between the kinetic and potential energies of the moving object. If you are watching a ball in flight, for example, then at every moment of time, you can calculate this difference. At the end of the motion, you can add up all of these differences and get a single number. This number is called, for various historical reasons, the stock of the moving object.

You can imagine different possible paths the ball might take when thrown at you. These different possible paths will be associated with different actions. And it turns out that the familiar path – the path exactly predicted by Newton’s laws – is the path with the least possible action.

Creation of the laws of motion

Lagrange discovered what is now called the principle of least action. All physical laws, including Newton’s laws, stem from this single unifying principle.

To develop a law of motion, you follow a simple recipe. First, you write down the kinetic and potential energies of the objects of interest. Then you take their difference. (We now call this quantity “the Lagrangian” in his honor.) Then you apply a sophisticated mathematical technique called the calculus of variations to find the expression that minimizes the action. What emerges is a whole new law of physics.

All modern physics is written in this language, because it is a powerful and intelligent (and universal) way of approaching dynamics. General relativity, electromagnetismand even quantum field theory and the standard model all start as Langrangians, and physicists all over the world apply Lagrange’s rules to derive laws of motion.

These laws of motion include those which govern the motion of the planets in the solar system and the expansion of the universe himself. Whether you use general relativity or the original Newtonian version of gravity, Lagrange’s trick will always give you the answers you seek.

Learn more by listening to the “Ask a Spaceman” podcast, available on iTunes (opens in a new tab) and askespaceman.com. Ask your own question on Twitter using #AskASpaceman or following Paul @PaulMattSutter and facebook.com/PaulMattSutter.

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Maria D. Ervin