A physicist sees a guy standing on the edge of a rooftop and shouts, don't do it.

You have so much potential.

We all know what it feels like to be energetic to have energy it's this something that allows us to move, be active, get out of bed in the morning.

And you can have more or less of the stuff.

You can get from breakfast.

You can lose it by mowing the lawn.

Energy seems near tangible to us.

We imagine it as this ephemeral substance or a mystical influence.

But that intuitive sense has inspired us to discover the most powerful and useful concept in all of physics.

In physics, energy is not a substance nor is it mystical energy, it's a number, a quantity.

And the quantity itself isn't even particularly fundamental.

Instead, it's a mathematical relationship between other more fundamental quantities.

It was 17th century polymath Gottfried Leibniz who first figured out the mathematical form of what we call kinetic energy, the energy of motion.

He realized that the sum of mass times velocity squared for a system of particles bouncing around on a flat surface is conserved.

It adds up to the same number, even though the speeds of individual particles changes, at least assuming there's no friction and perfect bounciness.

Leibniz called this early incarnation of energy vis viva, the living force.

Fun side note, Leibniz was famously arrival of Isaac Newton.

Besides the whole co-inventing calculus thing, Leibniz's vis viva seen as a competition to Newton's idea of conservation of momentum.

Newtonian mechanics had only recently revolutionized physics.

And few accepted that our man Isaac could ever be wrong.

Besides, this vis viva thing wasn't conserved in the event of friction, while momentum was.

It took another genius, Emilie du Chatelet to show that vis viva, or energy, as Thomas Young eventually named it, is conserved.

It can never be destroyed, only changed in form.

When she introduced the idea of gravitational potential energy, she put the laws of conservation of energy and momentum on equal footing.

The brilliant experiments of James Prescott Joule and others extended the idea to include heat energy.

Energy is always conserved but only if you account for all types of energy.

The law of conservation of energy is an incredibly powerful tool.

But it's not exactly surprising.

For example, du Chatelet's gravitational potential energy, mass times the gravitational acceleration times height, is just a statement about how much kinetic energy, 1/2 mv squared an object, like this ball, would gain were it to fall from a given height and a constant acceleration.

But that's just Newtonian mechanics.

And as a fun time exercise for the student, see if you can show that the square of the change in velocity of a falling ball is proportional to the distance it fell.

OK. Now let's say that the ball hits the ground and bounces up again with perfect elasticity and no air resistance.

It starts moving up with the same speed and kinetic energy it landed with.

Now, as long as the downward gravitational acceleration doesn't change over time, the ball should lose speed on its way up at the same rate it gained speed on the way down.

So it should reach the same height that it was dropped from.

Gravitational potential energy gets converted to kinetic energy in the fall and then back to exactly the same amount of potential energy in the rise.

Sure, energy is conserved but only if we define kinetic and potential energy in the right way.

This may sound arbitrary or trivial, but it's not.

The concept of energy is incredibly powerful.

And the key is this reversibility in the conversion between kinetic and potential energy.

The reversibility seems simple for a falling ball.

But even a complex path through a gravitational field can be broken down into little, perfectly reversible steps.

That's true even if the gravitational acceleration changes from one point in space to the next.

The key is that the field doesn't change over time.

Then, if the ball follows some path through the field and then retraces its path, the conversion between kinetic and potential energy will happen in reverse.

And actually, it doesn't even matter if it takes one path out and a different path back.

As long as the ball ends up back where it started, it will always have the same combination of kinetic and potential energy as when it left.

And actually, we can be even more general.

If an object travels between two different points in a gravitational field, it will always experience the same conversion between potential and kinetic energy, no matter what path it takes.

This is a feature of what we call a conservative force.

Every path taken between two points within a conservative force field takes the same amount of work, the same shift between kinetic and potential energy.

You trade with perfect efficiency between motion and the potential for motion.

Energy is the currency of that trade.

Of course, anything that saps energy from the ball as it moves will mess with this transaction.

It may strike other bulls and grant them some of its kinetic energy.

Any impacts may remove energy if they aren't perfectly elastic.

It may encounter energy sapping effects, like friction or air resistance, so-called dissipative or non-conservative forces.

But ultimately, all fundamental forces are conservative, as long as you consider all of the particles involved.

For example, the molecules causing air resistance are just tiny particles.

They exchange kinetic energy with perfect efficiency with the particles comprising the ball.

If we account for every particle and field involved, then the transaction between kinetic and potential energy is a zero sum game.

The energy ledger is always balanced.

Energy calculations are about balancing the books and accounting for all of the places energy can be stored.

In the case of air resistance, the kinetic energy transfer to the air particles ends up as heat.

With the falling ball, we're actually including the entire ball-Earth system when we add in gravitational potential energy, because that energy is stored in the Earth's gravitational field.

Sometimes, we even need to account for the potential energy in the forces that bind subatomic particles together, the energy of mass, which we talk about in earlier episodes.

Tracking the shift between different forms of energy allows us to predict the behavior of the universe in ways that would otherwise be impossible.

Just adding conservation of energy to Newton's mechanics gives an extra constraint that allows us to solve problems we couldn't otherwise.

But it allows us to go much further.

The universe is complicated.

Newtonian mechanics is great at describing the motion of simple systems of a few rigid objects.

But try to describe the behavior of the countless particles in, say, a stream of water or a universe, and it's pretty hopeless.

Such systems contain an impossibly large number of particles.

But there are also an impossibly large number of ways those particles can move from one spot to another.

Energy doesn't care what the individual particles are doing.

Instead, the concept of energy allows us to write down equations describing the evolution of the entire system.

For example, Bernoulli's equation predicts the flow of fluids by demanding the conservation of the kinetic and potential energy of the fluid and also of the internal energy due to fluid pressure.

It ignores the individual particles in the fluid.

And the concept of energy and its conservation has led to new types of mechanics that have supplanted Newtonian mechanics, for example Lagrange mechanics, which, in its simplest form, follows the evolving difference between kinetic and potential energy.

It produces the same equations of motion as Newtonian mechanics but without having to keep track of those innumerable fiddly force vectors.

Then there's Hamiltonian mechanics, which traces the evolution of the total energy of the system.

Hamilton's equation describes the motion of individual particles but can also describe the evolution of extremely complex systems, for example, the combined behavior of many celestial objects acting on each other, giving us the virial theorem, or a roomful of air in statistical mechanics.

The concept of energy is so versatile that Hamilton's approach was even adapted to quantum mechanics.

The quantum Hamiltonian operator describes the total energy of a quantum system and allows us to describe anything from the motion of a single particle in Schrodinger's equation to complex interactions of particles and fields in quantum field theories.

Actually, Lagrangian mechanics makes a quantum comeback here.

The way it uses energy is inherently consistent with special relativity, unlike Hamiltonian mechanics.

And that's important for describing fast-moving things.

Lagrangian mechanics is the inspiration behind Feynman's path integral approach to quantum mechanics.

And the Lagrangian quantum field theory is the basis for high-energy particle physics.

So what is energy?

Well, besides being a powerful accounting tool for describing the behavior of the physical universe, it's also a hint, a hint of something more fundamental.

See, the law of conservation of energy arises because of symmetry, in particular time translational symmetry.

Energy is conserved if the physics of a system, for example, the nature of a force field, stays the same over time.

In fact, for every symmetry in our universe, there exists a conserved quantity.

For example, the law of conservation of momentum is due to spatial translation symmetry.

Physics works the same whether you're here or a kilometer that way.

This relationship between conservation laws and symmetries was discovered by mathematician Emmy Noether and Noether's theorem is something we will come back to.

But for now, let's think about one implication.

What if the universe as a whole is not time symmetric, for example, in the case of an expanding universe?

Our universe looks fundamentally different from one moment to the next, at least on cosmic scales, where it's expansion becomes significant.

And in fact, energy is not conserved on those scales.

This leads to effects like dark energy and the accelerating expansion of the universe.

And actually, conservation of energy is generally invalid in the context of Einstein's general theory of relativity due to the potential time evolution of space.

Hey, every good physics lesson should end with the prof saying that everything they just told you is wrong.

Well, not wrong, just more interesting.

Energy is not fundamental.

It's a clue to the deeper truly fundamental properties of spacetime.