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Hey Crazies.
In physics, we talk about Newton’s laws a lot.
Like a lot a lot.
But I wouldn’t actually want to use Newtonian mechanics to model something like a double pendulum.
It’s just too complicated.
So today, we’ll be learning about a powerful alternative:
Lagrangian Mechanics!
Mechanics? You mean like the people that fix stuff?
What do they have to do with Newton’s laws?
Actually a lot, but that’s not what we’re talking about.
Mechanics is a branch of physics dealing specifically with motion.
It’s the set of mathematical tools we use to predict the path objects will take
and there are a lot ways we can do that.
Newtonian mechanics, Lagrangian mechanics, Hamiltonian mechanics,
and even quantum mechanics.
Can’t we just pick one and forget the rest?
I wouldn’t recommend that.
They all have their own realms of usefulness.
For example, if you’ve got a very simple scenario,
Newtonian mechanics is definitely the way to go.
Newton’s laws of motion are all about forces.
Don’t have any forces?
Newton’s first law says steady motion in a straight line.
Is there a downward gravity?
Newton’s second law says you get an acceleration.
That’s just free fall or maybe a projectile.
As long as you can keep track of all the forces acting on something
and you know some initial conditions, you can predict the motion.
Cause and effect is built right into the laws.
Is there a ball on a ramp?
Gravity pulls it down and friction makes it roll.
Do you have a simple pendulum?
The tension makes it swing in an arc.
Unfortunately, not all scenarios are that simple.
Consider the double pendulum.
This thing is a beast.
It’s a beast!
There are two objects, each with their own forces.
The top one has three forces and the bottom one has two, for a total of five.
Three of those five are changing in time.
It’s the quintessential example for chaos theory.
Slightly different initial conditions can result in wildly different paths.
Enter Lagrangian Mechanics.
Named after Giuseppe Luigi Lagrangia
I mean, Joseph-Louis Lagrange who published them in 1788.
In Newton’s laws, if you know where this pendulum is and what it’s doing,
then the forces tell you what happens next;
then what happens after that, then what happens after that, and so on.
A cause leads to an effect.
But Lagrange thought to himself:
What if we could consider the entire path at once?
Hmm.
If we want to do that, we’ll need to look for some kind of pattern.
There must be something that all paths share.
As the expression goes though, nobody lives in a vacuum.
Lagrange was not the first person think of this.
To the timeline!
Lagrange might have been developing his mechanics in the late 1700s,
but Fermat was already dabbling with this in 1662
and he built his ideas on the works of ibn al-Haytham from 1021.
Nobody lives in a vacuum!
Anyway, Fermat’s principle states that, out of all the paths available,
light always takes the path that requires the least amount of time.
But, Newton’s laws got published only a couple decades later
and it was clear that matter doesn’t move the same way light does.
So, Lagrange had a problem:
If matter doesn’t follow the path of least time, what path does it follow?
He needed to find a pattern that all paths share.
Pattern!
Forces aren’t going to help us with that.
We’re going to need energy and work.
In simplest terms, energy is the amount of stuff that could happen
and work is the amount of stuff that does happen.
Mind you, we didn’t actually know any of that at the time.
As far as Lagrange was concerned, it was just a useful quantity without a direction,
something we’d call a scalar quantity.
We wouldn’t find meaning in it or even call it energy until the 1800s.
How is this supposed to help us with Lagrangian mechanics?
It’s the basis of the whole thing.
These days, we’re aware of a lot of different energies,
but they can be organized into two major categories:
Kinetic Energy and Potential Energy.
Kinetic energy is associated with motion and potential energy with location.
They’re essentially the scalar versions of the vectors velocity and position.
If energy is the amount of stuff that could happen
and stuff happening usually involves some kind of path,
then it stands to reason that energy and paths are related to each other.
Lagrange took both major categories of energy, kinetic and potential,
and combined them into a single quantity we now call the Lagrangian.
If we map out all possible Lagrangians for an object across space,
we can add the values along a possible path and get an idea of its efficiency.
We call this efficiency the "action" and it’s a really big deal.
The action is that pattern we’re looking for.
Pattern!
There are a lot of paths available, but it only takes one of them:
The path with the smallest variation in the action.
Jargon Alert. Jargon Alert. Jargon Alert.
What? Ok, fair, I have been using a bunch of 10 dollar words.
Let me explain.
Say you’ve got a graph like this.
How much it’s changing is given by its slope.
The place where the slope is zero, there’s no change
and there is very little variation around that place.
Yes, it’s a math thing.
A change of zero can occur at one of three places:
a minimum, a maximum, and a saddle point.
We call all these places “stationary points.”
Knowing that, we can write the principle of stationary action,
which tells us objects will take the path with the smallest variation.
Paths around the real one show very little difference in their total action,
so that’s the path the object takes.
So, wait, does it check all the different paths to see which one to take?
No, no, no. Don’t be ridiculous.
Just because we can use this model to predict where the object will go,
that doesn’t mean the object can.
It just does whatever it’s going to do.
We’re the ones coming up with rules and patterns.
It’s just a mathematical tool for making predictions.
Remember, while Lagrangian mechanics might consider the path as a whole,
Newtonian mechanics doesn’t.
It’s still valid and has causality built into it.
But, if Lagrangian mechanics works so well, why don’t we use that all the time?
Do you always use the most powerful tool for the job?
I mean, would you use a sledgehammer to open a walnut?
No, of course not!
Anyway, there are better ways to visualize this stuff.
Newtonian mechanics is done in three-dimensional position-space.
All three axes represent normal everyday position measured in meters or miles
or whatever your distance unit of choice is.
In Lagrangian mechanics, we use something called generalized coordinates,
which is a terrible name because they tend to be very specific to the scenario.
I prefer to call them degrees of freedom.
For example, this simple pendulum only has one degree of freedom:
the angle of the string.
You could very easily show the motion on a single axis.
The double pendulum has two degree of freedom:
the angle of each of the two strings.
Its motion could be tracked on a two-dimensional grid.
Each point on this grid represents a possible configuration of the pendulum,
so we call this a configuration space.
It’s not a real space like the 3D space we live in.
It’s just an abstract representation.
You could even include a speed or momentum axis to get an idea of motion.
Here’s the simple pendulum in something we call phase space.
Yes, you’re technically doing Hamiltonian mechanics at that point,
but there really isn’t much difference.
So what’s Lagrangian mechanics?
It’s just another way of looking at a physical system.
In normal Newtonian mechanics, we start by looking at a single moment,
then step forward or backward according to the rules of cause and effect.
In Lagrangian mechanics, we look at the path as a whole between two events.
According to the principle of stationary action,
the path taken by the object is the one with very little variation in energy over time.
If your system is really simple or complex forces like friction are involved,
Newtonian mechanics is probably going to be your best option.
If your system is in the sweet spot in-between,
maybe try out Lagrangian mechanics.
It’s especially useful in quantum mechanics,
where the concept of a force almost completely loses meaning.
Lagrangian mechanics is just a tool.
Whether or not you use it is up to you.
So what do you think of Lagrangian mechanics?
Fun? Weird? Both?
Let us know in the comments.
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The featured comment comes from dilophi who said:
The delayed choice experiment is still mysterious after this video.
That’s OK,
but it’s only mysterious because none of us actually understand it.
Just remember that it’s not mysterious because of magic.
Anyway, thanks for watching.