This episode is supported
by The Great Courses Plus. 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. Thanks to the Great Courses
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Physics and Our Universe by Richard Wolfson. It gives an overview of all of
the core material of college intro physics courses. Help support pull the series,
and start your free trial by clicking on the link below
or going to thegreatcoursesp lus.com/spacetime. 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.
