Interestingly, Newton was unable to explain why the laws of motion were true and why they had no other form. This discovery belongs to another legendary but lesser known genius.

Lagrange and Newton

We’re used to thinking about motion in terms of forces and accelerations – partly because it’s a very intuitive way of perceiving the world (e.g. I push something and it’s moving) and partly because it’s how Newton formulated his laws (and therefore we teach them in school).

But studying forces and masses is not the only way to describe the world around us. Think of a ball thrown into the air. This ball has many properties that can be useful to us, such as its position, velocity, acceleration and mass. Some of these features can be very useful for predicting future ball movement, and some not so much.

Newton discovered that the combination of mass, acceleration, and force was very powerful, and this allowed him to formulate his famous equation “force equals mass times acceleration” as the fundamental law of the universe.

About 150 years after Newton described the laws of motion, another mathematician, physicist, and all-around genius Joseph Louis Lagrange developed his own formulations. He discovered that he could also deduce his own laws of motion by looking at the kinetic and potential energy of an object.

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

fixed action

If I throw a ball at you, you’ll probably have a good chance of catching it. You can do this because you’ve seen so many balls thrown at you in your lifetime, and your brain quickly deciphers that thrown objects follow a fairly common trajectory. Newton’s insight lay in his ability to find a general law of motion that could predict the trajectory of this thrown ball.

But why do Newton’s laws have to be true? Why should a thrown ball follow a certain trajectory? Why don’t the balls bounce back or fly to Mars first as they fly towards you? Why does the same path repeat itself every time? In other words, why do objects behave this way and not otherwise? The universe can choose literally any behavior for thrown balls or other objects in motion. What makes Newton’s laws work?

Newton did not know the answer to this difficult question, but Lagrange did.

Why Newton’s Laws Work

The key point is the difference between the kinetic and potential energies of an object in motion. For example, if you watch a ball in flight, then at any moment you can calculate this difference. At the end of the move, you can add up all these differences to get a single number. This number, for various historical reasons, by the movement of a moving body.

You can imagine the different possible paths the ball could take when thrown at you. These different possible paths will have different actions associated with them. And the way we know – the path precisely predicted by Newton’s laws – turned out to be the path with the least possible number of actions.

Creating the laws of motion

Lagrange discovered what we today call the principle of least action. All physical laws, including Newton’s laws, are derived from this single unifying principle.

Follow a simple recipe to create a law of motion. First of all, the kinetic and potential energies of the objects we are interested in are recorded. Then their difference is taken. (We now call this quantity “Lagrange” after the scientist.) Then you apply a fancy mathematical technique called calculus of variation to find an expression that minimizes action. What we get as a result is a completely new law of physics.

All modern physics is written in this language as it is a very powerful and intelligent (and universal) way of approaching dynamics. general relativity, electromagnetism, and even quantum field theory and Standard Model – it all starts with Lagrangianism, and physicists all over the world apply Lagrangian rules to derive the laws of motion.

These laws of motion include the laws that govern the motion of the planets in the solar system and the expansion of the universe itself. Whether you’re using general relativity or the original Newtonian version of gravity, the Lagrange trick will always give you the answers you’re looking for.