Deriving the Second Variation | Calculus of Variations

Derivation of the Second Variation of Variational Calculus. This is basically the analog to the second derivative in ordinary calculus, in that it allows you to determine the nature of your function for a particular functional (e.g. whether it's a minimum like a straight line minimizing distance on a plane, maximum etc.).

As you'll see in this video: the derivation is more involved than my derivation for the regular Euler-Lagrange equation. Unlike Euler-Lagrange, the second variation is also much harder to apply as it's not a simple matter of solving a differential equation.

Questions/requests? Let me know in the comments!

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[joshematics]

These videos are so helpful, it would be awesome to see you start an electrodynamics series

[annelashbrook]

Thank you so much for this Calculus of Variations series! I have taken notes on every video and appreciate how you call back on concepts from single variable calculus along the way. I learn so much quicker watching your videos than reading through a textbook.

[sonicd]

Impatiently waiting for the follow-up video showing an example of the second variation! lol~

[eulefranz]

Oh sick! I never knew there was something like this. This is interesting, especially that eta, eta' is still left. I have to think about that

[Funkaar_Studio]

I'm watching your videos since my school days, thank you so much for making these videos❣️

[DitDitDitDahDahDahDitDitDit]

Sal has the great skill of pacing his listeners, and keeping his videos from sounding like read lectures. That’s hard to teach. 😅

[lizhenji]

Hi, thanks a lot to your lectures, I was confused about this for a very long time. I saw several papers using the Lagrange method to solve the minimization or maximization problem, but they only calculate for first derivatives and do not show whether the optimum is a maximum or minimum(like the information bottleneck paper and deterministic annealing paper for fuzzy C-means). Your lectures helps me a lot! Thanks!!!

[erfanmohagheghian]

Regarding the second derivative test in ordinary calculus: the test is inconclusive if the second derivative is zero. It does not say that it is a saddle point. It could be a local min (like x^4), a local max (like -x^4) or a saddle point (like x^3)

[steveshaver]

Hi! Can you provide a sample calculation where you use this second variation to validate a real extremal as a max or min? Can this second variation equation be used to find a “saddle” extremal function?

[cyrusIIIII]

Where are the rest of videos for examples on calculus of variations?

[tayebtchikou]

Thank you very much for this very helpful playlist and I wish from you to explain more topics in Calculus of Variations like finding the extremal y of the functional I that depends on the higher derivatives of y using Euler-Poisson Equation and Dirichlet's principle and the eigenvalue problems and its applications on Quantum Mechaaics.

[davidchallener]

The second derivative=0 does NOT imply a saddle point. Consider f(x)=x^4

[OmarAhmed-icfw]

Why did we assume that all functions eta(x) must have a continuous second derivative w.r.t x? I cannot get the point in this assumption.

[封錦童]

This video is awesome. I have a question. What is the meaning if Q = 0 ?

[chasethescientistsaturre]

The only thing I'm confused is this: What is Q=0 I need answers pls.

[Peter_]

Is there any way to use the Euler-Lagrange equations as an alternative to Newton's Second Law when friction is included?
For example, a simple situation would be a falling point mass in a vacuum, where the Lagrangian would become L = 0.5⋅m⋅ẋ² - m⋅g⋅x, and then you would get ẍ = -g from that if you plugged L into the Euler-Lagrange equation, just as expected.
But that situation - and all other situations that I encountered of this kind in college - always assumed that things happened in a vacuum, and I was wondering how this stuff works if you also take kinetic friction into account?
Do you just treat that as an added term for kinetic energy, or how does that work?