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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.
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In our recent Space Time journal club,
we talked about the discovery of the amazing Chronos, the planet
eating star.
You guys had plenty to say.
Sebry asked whether it's possible for different parts
of a star-forming nebula to have different ratios of elements?
And if so, couldn't binary pairs have different ratios
or metallicities?
Well, the answer is, yes.
These star-forming clouds can vary in metallicity
across their vast widths, which are often
hundreds of light years.
But a binary pair will form very close together,
from the same knot of collapsing gas.
Over these smaller scales, metallicity
isn't expected to vary very much.
Richard Brockman points out that 15 Earth masses
of terrestrial material is a lot a planet for a star to consume,
at least compared to our solar system,
which only has little over two Earth
masses in the solar system.
In fact, it may be that our solar system
is unusual in this case.
In other star systems, we frequently
see one or more super-Earths with several Earth masses each.
The Trappist 1 system has seven planets all close to
or larger than the Earth.
And that's around a tiny red dwarf star.
So what happened to our solar system?
Well, solar system formation models
do indicate a scenario in which our sun may have swallowed
some early terrestrial planets.
The grand tack hypothesis suggests
that Jupiter may have migrated to Mars' orbit
before moving back out again.
That could have sent a new generation of planets
spiraling into the sun.
The remaining protoplanetary debris
was only just enough to form a few planets that we see today.
If only the sun had a binary partner,
we could actually test this by comparing metallicities.
Sebastian Elytron points out the ridiculousness
of astronomers calling all elements heavier
than helium a metal, because technically, lithium is grunge,
not metal.
Bravo, Sebastian.
I have nothing to add to that.