Finding the path of least action (part 2)
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This is part 2 of the derivation of the Euler-Lagrange equation for a special coordinate system.
@happyhedgehog6450 6 жыл бұрын
Even though there were some algebraic errors I found this video really helpful in understanding the concepts.
@oldiexian 8 жыл бұрын
Little math detail: Explanation for Delt S instead of dS is that it is a very very small change, not a differential. We only designate the change of a quantit, x, that is not path function (with an exact differential), with dx.
@jocider5698 6 жыл бұрын
Shouldnt the potential energy terms be subtracted in the integral expression based on the definition of action?
@Craptola1 8 жыл бұрын
I understand why this works for paths that are close to each other. But why is the difference function necessarily small, surely there are paths between those two points different enough that you can't ignore the eta squared terms.
@PhysicsHelps 8 жыл бұрын
+Edgy Fuckwad Good question. In principle, there could be multiple, quite different paths that satisfy this. The whole thing is analogous to finding the minima and maxima (extrema) of a function by taking the derivative, setting it to zero, and solving for the input. Lots of functions have many extrema. I don't have a proof for you, but my guess at why we're only worried about one extremum in these physics problems is that everything in a real Lagrangian is raised to power 2 at most, meaning there's only one extreme path. Just like how parabolas only have one min/max.
@M_1024 Ай бұрын
@@PhysicsHelps The other local extremes are also valid. For example for light there is an analogus "priniciple of shortest path". If we have light moving between point A and point B then it will move in a straight line (the shortest path) or bounce of a mirror (local minimum) or if the Universe "wraps around" at the end, it could also go around and come back (the maximum).
@venkatasaiprasannachigiche1704 6 жыл бұрын
You are explanation is good. Please use some bright colors for illustration as you are using a dark background.
@shrodindsy8992 6 жыл бұрын
please ,why the Taylor expansion of the potentiel ?thank you , great work keep on please :)
@sicktoaster 10 жыл бұрын
When you said that "+" was a mistake with the potential energy was it also a mistake when you put "+" after it and before the "mdx-bar/dt..." part of the equation?
@bartkwezelstaart9306 9 жыл бұрын
sicktoaster He corrects it in the next video
@jeffreyhowarth7850 2 жыл бұрын
At 14:13 is that the Laplace equation?
@richardaversa7128 6 жыл бұрын
I'll be you could have done all this in one video if you used dot notation for the time derivatives 😙
@user-qn5hv5uw2l 3 жыл бұрын
Ты крут
@joeboxter3635 5 ай бұрын
@3:00 when did the derivitive of the square become the square of the derivative? What you are claiming is d/dt (x^2) = (dx/dt)^2 really? I thought its 2x dx/dt, by the composite function therom. So you are saying 2x dx/dt = (dx/dt)^2 for all functions? REALLY?!!!
@HT-rq5pi 8 жыл бұрын
using different colours was a very bad choice, should've just used subscripts.
@GabrielSavageMusic 7 жыл бұрын
studies have shown that there is a large group of people that find the color coordination a much more obvious and thus intuitive way to show the relationship that is unfolding or being derived in these often 'pattern matched' steps that occur in problems. However, the main negative drawback is that anyone who does not require the use of these extra guided steps ends up having to endure through every step that is taken to change colors :P
@user-bc9wj2kk3x 5 жыл бұрын
veeery non rigorous. that argument "if eta is small, its derivative squared is veeery small" is simply false -- just look at xsin(1/x), it's arbitrarily close to 0 when x->0, but its derivative is arbitrarily large in any neighbourhood of 0. i get that purpose of such videos is to give intuitive explanation blablabla, but actually its way better to just state the euler lagrange equation without proof rather than give such a horrible one
@joeboxter3635 5 ай бұрын
Not to mention the derivative of the square is the square of the derivative!!! @3:00
@rejkomfort2090 9 жыл бұрын
Make a concept before making a video. Colored variables and confusion between paths are so unscientific. Not to mention basic algebra: is it + or - or ...?!
@swizzbeats1212 8 жыл бұрын
+Rej Komfort He corrects the arithmetic mistake in with an annotation, and if you followed the other videos it's obvious what he's doing...so maybe do some research before you want to criticize other peoples work.
@rejkomfort2090 8 жыл бұрын
Isaac Newton That's why people make concepts before doing any job!
@swizzbeats1212 8 жыл бұрын
People also make mistakes, and realize them after! C'mon man!
@rejkomfort2090 8 жыл бұрын
Isaac Newton But this is a recorder and edited video not a live show. Make concept, read it several times and then do a video. No mistakes!
@swizzbeats1212 8 жыл бұрын
That may be true, but sometimes you just don't notice the mistake and need someone else to look at it to point it out!
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