Equations that changed the course of history

Physics equations are forms of magic. They allow us to explain the past, such as why Halley’s comet visits every 76 years, and predict the future – as far as the ultimate fate of the Universe.

They place limits on the possible, as in the efficiency of an engine, and they reveal possibilities we could never have imagined, such as the energy inside an atom.

Occasionally over the past few centuries, a new equation endowed the next generation with a new magical tool, and so changed the course of history. Here are some of the most pivotal.

1. Newton’s second law of motion (1687)

What does it say?

Force equals mass times acceleration.

In other words …

It’s easier to push an empty shopping cart than a full one.

What did it teach us?

Together with Isaac Newton’s other two laws of motion (the first says you need a force to move something, the third says every action has an equal and opposite reaction), this equation forms the foundation of classical mechanics.

F=ma allowed physicists and engineers to calculate the value of a force. For instance, your weight (measured in newtons) is your mass (in kilograms) multiplied by acceleration due to gravity (on Earth, about 10 metres per second squared).

Saying you “weigh” 60 kilograms is incorrect in physics terms – your actual weight is about 600 newtons. This is the force pushing down on your bathroom scales.

But was it practical?

This equation was crucial to the arrival of the mechanical age. It’s used in almost every calculation which involves using force to cause movement.

It tells you how powerful an engine needs to be to power a car, how much lift an aircraft needs to take-off, how much thrust to lift a rocket, how far a cannonball flies.

2. Newton’s law of universal gravitation (1687)

What does it say?

Any two massive objects pull on one another across space. But the force decreases rapidly the further apart they are.

In other words …

We’re stuck to the Earth’s surface because our planet is comparatively big with lots more mass.

What did it teach us?

For centuries, the Universe had been divided into two realms – the earthly and the celestial. But Newton’s law of gravitation applied to everything. The same tug that causes an apple to fall from a tree keeps the Moon orbiting the Earth. Newton gave us the first direct connection between everyday life and the movement of the heavens.

But was it practical?

For a long time, the equation’s main use was to calculate the orbits of planets. The space-age of the 1950s and 60s saw it used in practice – to send satellites into orbit and astronauts to the Moon.

One failing, which Newton himself admitted, was that he did not know “why” gravity operated. It took nearly 230 years for Albert Einstein to come along and explain gravity as arising from the warping of spacetime by massive objects in his theory of general relativity.

Even so, general relativity is only used in extreme situations, such as when gravity is very strong, or when great precision is required, such as for GPS satellites. In most cases Newton’s 330-year-old equation is still good enough.

3. Second law of thermodynamics (1824)

What does it say?

Entropy (a measure of disorder) always increases.

In other words …

It’s no good crying over spilt milk. Disorder and mess are inevitable in the Universe.

What did it teach us?

While trying to analyse steam engine efficiency in the 19^th century, French physicist Sadi Carnot stumbled upon one of the most profound equations in all of science.

It tells us some processes are irreversible, and may even be responsible for the arrow of time. In one of its simplest forms, it says heat always travels from a warm object to a cold one.

It can also be applied to the grandest scales. Some have applied it to describe the ultimate fate of the Universe in the form of “heat death” where all the stars are burnt out and nothing’s left but waste heat.

Others have used it to wind back through time and describe the origin of the Universe in a moment of zero entropy (or perfect order) at the instant of the Big Bang.

But was it practical?

This law was important for developing technologies of the industrial revolution, from steam to internal combustion engines, to refrigerators and chemical engineering.

In real engines, some energy is always wasted – so the law also showed any efforts at perpetual motion were ultimately futile.

4. The Maxwell-Faraday equation (1831 and 1865)

What does it say?

You can create a changing electric field (left side of the equation) from a changing magnetic field (on the right) and vice versa.

In other words …

Electricity and magnetism are related!

What did it teach us?

In 1831, Michael Faraday discovered the connection between two natural forces, electricity and magnetism, when he found a changing magnetic field induced a current in a nearby wire.

Later, James Clark Maxwell generalised Faraday’s observation as one of his four fundamental equations of electromagnetism.

But was it practical?

This is the equation that powers the world. Most electric generators (whether in a wind turbine, coal-fired plant or a hydroelectric dam) work by converting mechanical energy (from steam or water) to rotate a magnet. By running this process in reverse, you get the electric motor.

More generally, Maxwell’s equations are still used in almost every application of electrical engineering, communications technology and optics.

5. Einstein’s mass-energy equivalence (1905)

What does it say?

Energy equals mass multiplied by the speed of light squared.

In other words …

Mass is really just a super-condensed form of energy.

What did it teach us?

Because of the size of the constant in the equation (the speed of light squared, an unimaginably huge number) a colossal amount of energy can be released through converting a tiny amount of mass.

But was it practical?

Einstein’s most famous equation hinted at the potential for the huge amounts of energy released in nuclear fission, when a large unstable nucleus breaks into two smaller ones. This is because the mass of the two smaller nuclei together is always less than the mass of the original big nucleus – and the missing mass is converted into energy.

The “Fat Man” atomic bomb dropped over Nagasaki in Japan on 9 August 1945 converted just one gram of mass to energy, but produced an explosion the equivalent around 20,000 tonnes of TNT.

Einstein himself had signed a letter to US president at the time Franklin Roosevelt recommending the atom bomb be developed – a decision he later regarded as the “one great mistake” of his life.

6. The Schrödinger wavefunction (1925)

What does it say?

It describes how the change of a particle’s wavefunction (represented by psi, the candlestick shaped symbol) can be calculated from its kinetic energy (movement) and its potential energy (the interactions on it).

In other words …

It’s the quantum version of F=ma.

What did it teach us?

When Erwin Schrödinger formulated his equation in 1925, it placed the new theory of quantum mechanics on firm footing by allowing physicists to calculate how quantum particles move and interact.

The equation looks a bit weird because it uses the mathematics of waves. (Subatomic particles are “wavy”, so their interaction is described as interference of waves, rather than like billiard balls.)

But was it practical?

In one of its simplest forms, it describes the structure of the atom, such as the arrangement of electrons around the nucleus, and all chemical bonding.

More generally it’s used for many calculations in quantum mechanics and is fundamental to much of modern technology from lasers to transistors, and the future development of quantum computers.

Related reading: New physics needed to explain the universe