Lagrangian and Hamiltonian Mechanics in Under 20 Minutes: Physics Mini Lesson

When you take your first physics class, you learn all about F = ma---i.e. Isaac Newton's approach to classical mechanics. But there's a lot more to mechanics than F = ma, and modern physicists actually more commonly use two other formulations of mechanics: the Lagrangian and Hamiltonian formalisms. In this video, I'll teach you the basics of both. They're not only powerful approaches to classical mechanics, they're also fundamental to the way we think about quantum mechanics!

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About physics mini lessons:
In these intermediate-level physics lessons, I'll try to give you a self-contained introduction to some fascinating physics topics. If you're just getting started on your physics journey, you might not understand every single detail in every video---that's totally fine! What I'm really hoping is that you'll be inspired to go off and keep learning more on your own.

About me:

[iyziejane]

If the Lagrangian and Hamiltonian formulations look pretty similar, to the point of almost being different notations, this is because Hamilton invented the term "Lagrangian" and codified Lagrangian mechanics as we know it, and it was Hamilton's obsession with notation that led him to make the equations look as symmetrical as possible with the P's and Q's, which paid off 100 years later with quantum mechanics

[sanori-cs]

1:49 Newtonian formulation
5:44 Lagrangian formulation (L = K - U)
10:59 Hamiltonian formulation (H = K + U)

[davidgustavsson]

I wish I had learned this before quantum mechanics. We essentially had a half semester course racing from "what is an operator" through "what's a Hilbert space" to "this is the Schrödinger equation, good luck!". It hasn't even occurred to me to try using Hamiltonian mechanics in classical physics.

[givemeyourfish]

I went to graduate school for engineering and that was the best explanation of the Lagrangian/Hamiltonian I have ever listened to.

[mintakan]

Future topic suggestion. Noether's theorem. Symmetry. Why is this so important for physics and math?

[douglasstrother]

As a Physics Freshman, I recall reading the terms "Lagrangian and Hamiltonian Mechanics" in the course description for the Upper Division Classical Mechanics couse and thinking "What does that even mean?".
I figured that I'll learn that when I get there. I got there about 40 years ago!

[NovaWarrior]

Absolutely awesome. I finally found somewhere that got past the H=KE+PE of Hamiltonian mechanics AND actually explained the point. Thank you.

[tedsheridan]

Very clear and well presented. I briefly learned Lagrangian and Hamiltonian formulations 20 years ago in Dynamics and promptly forgot them. Now I'm teaching myself more physics and they keep popping up. Thank you!

[Bayners]

What a clear summary, with well thought out supporting materials. You cut to the essence but leave pointers for people to find the details. Great work!

[dtcarrick]

As a physics teacher I can safely say this is amazing! Succinct and encouraging for a student. Well Done.

[obetancourtra]

Thanks for this. I've worked with a considerable amount of lagrangians and hamiltonians in my macroeconomics class to determine optimal paths of investment or consumption. It's always interesting to see where our mathematical tools come from.

[Snwar]

the Lagrangian was the most beautiful thing when meeting it in the early courses of studying physics. the way you can just throw away all the complicated geometric/vektor assesments you have in newtons method and just use the energies is so efficient

[gideonk]

Fantastic explanation!
Regarding the 2 different types of curves in phase-space after 17:00, I presume the internal ones, which touch the horizontal axis (dp/dt = 0) are where the pendulum swings back and forth (momentarily zero velocity when changing directions). The 2 external curves are where the pendulum swings/rotates around the pivot point: one is clockwise rotation and the other is counter-clockwise rotation.

[Ligatmarping]

Nice work! Im a math guy who started studying a little physics after many years; I like it a lot.Greetings from Argentina.

[abdullahkarolia]

I've been confused for a whole semester on Lagrangian mechanics and this actually made it very clear, I might actually pass now, thanks!

[JeffSchneiderMusic]

Me thinks this is going to be a great YouTube channel!

[justchecking]

Finally you have enabled me to understand these three formulations of mechanics that I first learned in graduate school in 1968. I have no need of them now as a retired scientist but thank you!

[anshumantripathi]

This channel will soon reach million subs.

[PunmasterSTP]

This really blew my mind, and once again I'm so glad that educational material exists on YouTube. Thank you for spreading your knowledge; it was mechanawesome! 👍

[oak]

these videos hit different and get more appreciation post graduation, forgot what got me into physics in the first place but your videos bring me back in