@@agrajyadav2951 well that will complicate things. I think that requires non elementary functions because in a verticle circles, the time period involves elliptical integral
This video is great, but remember that a lot of us are physics students or aspiring physics students (maybe even some full blown physicists) and we're NOT afraid of some nitty gritty math, and indeed LOTS of it. For me, as a math major, the more the merrier!
I'm guessing you're an applied math major? As a pure math major, I disdain excessive calculation. The video was spectacular in my opinion. If I had to ask for something more though, I'd ask for him to go deep into this notion of a Legendre Transform.
This video and next are like perfect timing because i just started my self study of working through taylors classical mechanics. Who knew air resistance would make a projectile motion problem so complicated.....
Please do all the problems in all 3 formalisms. I'd love to see you tackle some very basic quantum mechanics using the standard Schrodinger Hamiltonian approach and another using the Feynman Path integral. Also are you versed in functional analysis? A detailed derivation of the Euler-Lagrange equations from the calculus of variations is a very instructive exercise (as is the informal derivation).
As a side note the action used to be called "Hamiltons principle function" and the principle of least action used to be known as Hamilton's principle. Originally Lagrange obtained the E-L equations from the lagrangian through the principle of virtual work (or maybe "Maupertuis's principle", not 100% sure) . It was Feynman who coined the term action and popularised it.
I have to say I didn't understand most of that but seeing the (KE +PE) part and also dH/dp made feel a bit better because at least I kinda know what they mean. Physics is weird, it seems more difficult, for me, than maths however just looking at you talking about it makes me want to learn. Yes, please to problems.
You forgot to mention that for nonconservative systems, the Hamiltonian does not correspond to T + U and that the more general way of attaining the Hamiltonian of a system is by taking the Legandre transform of the Lagrangian
It would be cool if you keep exploring this subject and eventually get to the more geometric and qualitative aspects of Hamiltonian systems, including the Liouville-Arnold theorem and KAM theorem
What I don't quite understand is how did physicists ,at the time when lagrangian mechanics was born, come up with the idea/hypothesis that matter follows path of stationary action? This idea is crucial since without this idea they wouldn't use lagrangian and Euler - Lagrange equation to derive correct EOM in harmony with Newton's 2nd law.
I found a problem in a University Book, wich is about a Pendulum, but the thing its that there's a nail after the object swings through angle=0, or swings to the other side, so the lenght would change for that part, it would be pretty cool solving that in some simple dummy form of Lagrange Equations, cuz you have to demonstrate that if you drop it under the height of the nail, it will get back to it (uses Potential and Kinetic energy)
This helped me understand the motivation behind using Lagrangian and Hamiltonian mechanics. I've been out of college for awhile, but I'm still interested in learning/reviewing things like this, and I get lost trying to understand the math without seeing the bigger picture sometimes. Thanks a lot
Yes. KE - PE. But why minus? Action is a quantity for which I cannot get a feel. So you can derive Newton's laws with action. Does that mean that Lagrange knew where he wanted to get before he started, and fiddled the books in order to get there?
Here's a question that has popped into my head only this year, 2021, way late in my life: so if between any charged particles, at every point in time, and every point in space, there exists an electromagnetic field, and if the photon is supposedly the "force carrier" of the EM field, then how often is a photon being exchanged between the charged particles? How often and how many? How can it even be a discrete finite number of photons being exchanged in a nonzero finite period of time, since there exist an UNCOUNTABLE number of points of time, and and UNCOUNTABLE number of points in space between the charged particles? Ok... well, treat light as a wave, rather than discrete photons. Got it. But, even with a wave formulation of light, you're still transferring momentum between the charges: are you not? Is this momentum exchange what causes the charged particles to move: either towards each other (if oppositely charged) or away from each other (if similarly charged)?
If it gives you a max, you sub in -L for it to correspond to a minimum. I've never heard of someone deliberately keeping the lagrangian in a form in which it leads to maximizing the action, except maybe in certain QFT phenomena like in Yang Mills.
I'll be taking my upper div CM class this fall, so go ham on those problems, man! Also, I was wondering if the 2 formalisms you talked about are approached by calculus of variations?
I knew it! I thought I was just confused (old white guys in wigs all look the same?), but that is definitely George Washington in the bottom picture there. They did live at the same time, roughly, but I kinda doubt they had *anything* to do with each other.
Is the hamiltonian (energy operator) in quantum mechanics requires the understanding of Legendre Transform or is it specific to classical mechanics only?
In the video, he said that the path it takes minimizes the action. This is not true. The path that satisfies the Euler-Lagrange equation is the critical point of the Action functional. More often than not, the path it takes is a saddle point of the functional, not the minimum.
Go on with them Poisson Brackets buddy! I'd prefer that you continue down the casual explanation route ... even if it gets a little more lecturey. If were gonna do problem videos it needs to be on that PGRE that were studying for!!!
hey dotson... do u think understand lagrangian using Legrendre Transform difficult to understand for an "kinda above average" highschool student and writing a short HS level report on it? Please I need some advice
EPIC Tube HD knowing what a legendre transform is doesn’t really tell you much about a lagrangian. What it tells you is how a lagrangian is related to a Hamiltonian.
For some reason, I can see you TAing the introductory physics mechanics class in graduate school and whipping out the Lagrangian and Hamiltonian on the unsuspecting students LOL
1337master multivariable calc is amazing. Its super cool what you can do with it, but takes alot of studying to get yhe conceptual understanding of it. I reccommend proffessor leonards videos on calc 3!
Do examples...essentially, one of every fundamental problem that these were invented for. That...would be a lot of work for you. I'd do it, but I just don't have the time.
