What is Energy - Script? (Patreon)

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 it from breakfast, you can lose it by mowing the lawn. Energy seems near tangible to us – we imagine it as an 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 is 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 Leibnitz 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 always conserved, assuming no friction and perfect bounciness. Leibnitz called this early incarnation of energy vis viva – the living force.

Fun side note – Leibnitz was famously a rival of Isaac Newton. Besides the whole co-inventing calculus thing, Leibnitz’s vis viva was seen as 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, Émilie du Châtelet, 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 this 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 Châtelet’s gravitational potential energy – mass times gravitational acceleration times height - is just a statement about how much kinetic energy – m-v^2 - an object like this ball would gain were it to fall from a given height under constant acceleration. But that’s just Newtonian mechanics – and as a funtime 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 falls. 

OK, let’s now 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. 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 as it gained speed on its 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 potential energy in the right way.

This may sound arbitrary or trivial, but it’s not. The concept of energy is incredibly powerful. 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 any journey the ball takes through the field is retraced, then 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’ll 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 balls 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. 

In the case of air resistance, some kinetic energy 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 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 that 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 also there are an impossibly large number of ways those particles can move from one spot to the next. 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 the internal energy due to fluid pressure.

And the concept of energy and its conservation led to new types of mechanics that supplanted Newtonian mechanics. For example Lagrangian mechanics, which in its simplest form follows the difference between kinetic and potential energy. It reproduces the same equations of motion as Newtonian mechanics but without having to keep track of innumerable fiddly force vectors.

Then there’s Hamiltonian mechanics, which traces the evolution of the total energy of a 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 of a room full 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, Lagrangean mechanics makes a quantum comeback here. It's inherently consistent with special relativity, unlike Hamiltonian mechanics. 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 behaviour of the physical universe, it’s also a hint. A hint of something more fundamental. 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 of 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’ll 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 its 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. Actually, conservation of energy is generally invalid in the context of Einstein’s theory of general relativity due to the potential time-evolution of space. Hey, every good physics lesson should end with the prof saying that everything he just told you is wrong. Well, not wrong, just more interesting. Energy is not fundamental, but it’s a clue to the deeper, truly fundamental properties of spacetime.