The principle of least action

Here's a qualitative introduction to another way of looking at physics.

[kevinwaterman]

Around 9:30 when the narrator begins listing the reasons why this is worth knowing, the reason that kept popping into my head was (while closely related to reasons 1 and 2) is that it shows that the Newtonian methodology is not necessary. I think finding a separate way of solving these equations shows that while they are both extraordinarily accurate, they are both inventions, and that other inventions that solve these problems might still yet be more useful and powerful, and maybe even simpler than either. To me that is the most exciting thing about a first encounter with the principle of least action, is it can make a person realize that our descriptions are provincial, and the possibility of a great discovery, even in the realms we think to have mastered, seems to still be possible.

[Cosmalano]

Been reading quantum mechanics and path integrals but couldn't figure out what s was. Thanks man!

[joeboxter]

S, i believe, because historically this intergral was associated with length (ds) and that was denoted as ds, integrated S. And the idea of shortest path comes from Hygen's principle that light travels shortest distance. But its also L or H.

[Suav]

It might be useful to think about the picture in/convert it to three dimensions. Let the vertical direction still has the name "h", let there be another direction "s" denoting projection of a trajectory of the point mass on the "ground" and let "t" axis, for time, stays where it is. Then the interval \Delta{t} = t_2 - t_1 is a function of initial velocity only (we neglect wind, air resistance and all sort of other considerations).
Now take your fav CAS, or HTML5 Canvas and model the situation for different initial velocities. It is a useful exercise. The question can be rephrased. What parametre of the equation must have changed for the time interval to state constant and the curves vary like on the drawing?

[sergio]

Hello! Not using Newton's F = ma, allow me ask:
1 - Where does the { Action = Integral (K - U)dt} come from?
2 - Where does the {Lagrangian (K - U)} come from?
3 - Can I deduce that I must minimize the Action integral equation from minimizing the potential energy U?
4 - Can you elaborate?
Thanks!👋

[zoltankurti]

All right, I see great confusion in this comment section, thanks to the misleading title. In mechanics, this principle should be called stationary action, not least action, since it isn't that. The easiest example is a hatmonic oscillator, you can easily sgow that for given boundary conditions staying still has less action than the actual motion of the object. THIS IS NOT THE PRINCIPLE OF LEAST ACTION, BUT THE PRINCIPLE OF STATIONARY ACTION. And there isn't a deep phylosophical depth behind why this works. It just turns out to be mathematically equvivalent to newtons laws. It actually has a connection with quantum mechanics, you can basicakly derive the stationary action from quantum mechanics, but that is a pretty advanced topic.

[jimdogma]

Why does that graph remind me of the Jaws poster from the 70's?

[comprehensiveboy]

Principle of Least Action? Kinda sounds like my social life dude. :(

[andrejburcev]

Thanks for the informative video! Can you please clarify something. From the equation I can conclude that having a small kinetic energy and high potential energy does not really describe the behaviour of the equation. However it is the average DIFFERENCE between them that is important. So average difference should be as small as possible. I.e. both of the energies can be very large or very small, the minimum average difference is the one that wins. What am I missing? (obviously with the assumption that "Action" scalar is always positive)

[sinersaiyan]

How did they conclude that the action should be the integral of T - U, ?(or, in other words the Lagrangian) ? I don't quite understand where the T-U comes from and if there could be any other definition and we use this one as it is the simplest one. ( I know that it works as i've solved problems using this concept, but it is this theorical aspect in particular that I couldnt find the answer to.)

[MBailey]

Is there any relation between this and the Lifeguard's Calculation? I've come across both concepts separately, but they both seem to be ways of illustrating the same broader concept.

[mohamedbelebardi]

It must be called the principle of least energy because action is a force. can we arrive at this principle starting from the forces applied instead of the energies?

[simonjeffery]

Hi Physics Help. I really enjoyed this video and compliment you on your style in presenting the information. I can not believe that I never came across this Principal (not in high school or Engineering school) until I read that book "The Theoretical Minimum". As Jared points out above it ties nicely into Relativity and QM (I have not tackled String Theory yet). I am looking forward to viewing your other videos.

[bsingin]

Assuming the item were launched from h=0 in a consistent field of gravity, wouldn't this equation always = 0?

[PhysicsHelps]

Hey thanks for the feedback. This is just the first video of a playlist (still in progress right now). In the "Finding the path of least action" videos, you'll find the equivalence to Newton's 2nd law is addressed. You're right about quantum, but I'm waiting until I actually cover the Hamiltonian to allude to quantum mechanics since that's what's more commonly used in quantum.

[chancesire]

Physics University student here, love this video.

[Madvilllain]

Why would you want T to be small and U to be large? Wouldn't you want them both to be small if you're trying to minimize your differences... a small number subtracted by a large number is still a large difference?

Or by a minimum of "S"  do you mean negative values as well? ( I guess I'm thinking of absolute differences) 

[ultimatequantumguy]

Unless I am significantly misunderstanding, the way in which the narrator describes action at around 8:00 implies that it is impossible to throw a ball higher or lower than a fixed path. I'm assuming that the reason that this is not true is because we're setting the initial kinetic energy by throwing the ball at a fixed speed, and ergo kinetic energy. I still get the point, but it does feel like a slightly unclear description...

[comicrelief]

I'm not sure your depiction of why it works is right.

It should not be too hard to visualize.

[zphuo]

@8:00 is that mean that if start point and final point is fixed, then only middle path is possible?