Delta's occurs in many places in math and engineering. In this case, we are referring to the variational operator. It bears no relation other uses like the Dirac delta function or the Kronecker delta.

How is it going? What do you think of modern physics, how confusing is it?

@@user-lc6jq1hi1r to me it is not that much, since usually you don't have to deal with the uncertainty as you do see in quantum mechanics, however, you will have to do some real world solutions in which will be harder for some, due to the scale, but to me the larger values just make more leeway for "rookie mistakes", I personally do not think of it as confusing, but others may say otherwise to me it is a shame that modern physics, and maybe biology is the reason why most people hate high school and college, and i think that we should dive more deeply into physics just to show that it is very important to our everyday lives, most of the reasons for this is because of the intense math involved, but as Richard Feynman has done it, he makes sure that physics in general is easier by the Feynman technique for integrals, differentials and more, if only more people knew about those subjects, maybe the world may be different today.

@@universum-theuniverseexpla6565 experimental physics mogs theoretical physics.

​@@universum-theuniverseexpla6565I have heard so many time about feyman technique...can you kindly briefly explain..I know there is a complete set of his lectures pdf available.

Stick with it!! This is more advanced material than before. Videos are a little bit mathematical at first because it is setting up a framework that we will use. As soon as you see a few examples of this theory being used in practice, it will fall into place for you. It's a little like using Lagrange's equations, which made our lives very easy once we started applying it to mechanics problems, but we've never derived Lagrange's equations...until now. These first few videos on this playlist explain the theory behind Lagrange's equations and other energy methods. Once you see some practical examples of this being used (which will come in the next few videos), then, you'll return to these videos and they will make more sense. I am very much trying to get you guys quickly to the point where we are solving practical problems, but I can't get there without laying out some theory first. The math behind this theory is actually very beautiful, but for most people, Variational Calculus takes a little while to get used to. It took the world top scientists 160 years to get to Hamilton's Principle from Newtonian Mechanics and I'm trying to get you there in a few videos.

took me some time but i get it now. Thank you for excellent explanation.

Thanks for you suggestion. Might take me a little while to get there, but I will definitely tackle a constraint problem soon.

🏀

Great you just cleared my confusion, thanks

1. It's not the derivative ∂y‘, but rather the variation 𝛿y' that we are integrating. The formula for integration by parts is: int u dv = uv - int v du. So, in our problem, dv = 𝛿y' therefore integrating gives us v = 𝛿y. Then we plug this into the formula. 2. This is how we integrate by parts. Just plugging into the formula above. Perhaps try to review this technique. 3. The reason we say that the entire expression must be equal to zero is as follows... - If the value of the integral, I, is to be a maximum or minimum, then the integral must not change its sign for all possible variations of 𝛿y. - However 𝛿y is arbitrary (ie can be positive or negative), therefore the part that multiplies 𝛿y equal to 0. - We conclude from this that 𝛿I = 0 is the necessary condition to find an extremal. Hope this makes sense.

Yes, I think that's correct.

I discuss linearity around the 10:45 mark.

It doesn't matter what causes it to be zero. The fact that both of these give us zero demonstrates that setting the first variation δI = 0 is the equivalent of setting dI/dε (@ ε=0) to 0. We are just showing that we will find the extremal using either method.

There is no restriction on ε as such, but we are interested in the behavior near extremals, and in the vicinity of extremals, values of ε --> 0, so really, the infinitesimal values are the ones of interest. And, "yes" to your last question. That's exactly the definition of an extremal - it's a stationary point so a little nudge by a variational amount (in terms of any variable) will have no change on the value of the function (because the slope is zero).

@@Freeball99 Thanks. Also, why are we interested in retaining the sign of the integral? Seems like merely a more complicated way of saying that the variation of the integral must be zero. Edit: Actually, isn't it the variation of the integral and not the integral itself that must retain its sign? For example Wikipedia states so.

Initially, we do treat y and its derivatives as independent parameters when taking the variation. The relationship between y and it’s derivatives is then accounted for when we integrate by parts and the combine terms to form the Euler-Lagrange equation. This enforces the relationship between y and its derivatives.

The variation of I at the extremal will be zero. Consequently, in the vicinity of the extremal, the value of I will not change and so its signs stays the same. This is similar to differential calculus where the derivative/slope is zero at the extremal and changes sign in the vicinity. So, the statement is correct that I (rather than the variation of I) should retain its sign.

Sorry for the delayed response, but I missed this until now. Yes, it's just the second term that is integrated by parts - since we need to remove the derivative on δy'.

@@Freeball99 thankyou and great work you are doing

In the region of a stationary point, a variational "nudge" away from this point should not change the value of the function. The idea is that if the value doesn't change, then its sign cannot change.

@@Freeball99 But isn't it true that what changes is just the variation of the functional not the value of the functional itself. I don't see why this variation cannot be changing signs. Could you please elaborate a bit? BTW, I am extremely grateful for these videos. I have been always shaky in the variational calculus part in my study of solid mechanics, and the mathematics of it seemed intimidating. Thanks a lot! Keep up the good work

In essence I just applied the formula for the Taylor series of a function but replaced the function with a functional. Then, instead of taking the derivative, I took the variation (bearing in mind that we only vary the dependent variables so the variation of x is zero).

@@Freeball99 Hello, also asked this as a seperate comment but seeing that this is discussed here too: What does it mean to vary Y'. Afaiunderstand, Y' is completely decided by Y. what is the meaning of the partial derivative of Y' then? Thanks.

The varied path (y_bar) is the sum of the optimal path (y) plus a variation to that path (𝛿y). So y_bar(x) = y(x)+ 𝛿y(x). The variation is modeled as 𝛿y(x) = ε·η(x) where η(x) is a shape function and ε is the magnitude. NOTE: ε (which is considered to be small) is NOT a function of x. The shape function η(x) is completely arbitrary EXCEPT that it must satisfy the geometric boundary conditions, namely that η(x1) = 0 and η(x2) = 0 (ie the varied path must start and end at the same locations as the actual path so the variation at these points is zero).

Not sure if I completely understand the question. Hopefully this covers it. The starting point is that we are dealing with path minimization type problems. In general, we are dealing with a functional that depends on an independent coordinate, x, and the dependent coordinate, y(x) and its derivatives (up to any order). We initially (in a previous video) treat y and its derivatives as independent coordinates, which allows us to derive the weak form of the governing equations. We then use the fact that y and its derivatives are not independent when we integrate by parts to arrive at the strong form.

The app is called "Paper" by WeTransfer. Running on an iPad Pro 13 inch and using an Apple Pencil. Quicktime is used to record the screen.

What the hill.. There is no God except Allah.

Solid Mechanics: A Variational Approach by Dym & Shames