Let's Learn Physics: All About Oscillators
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- čas přidán 17. 03. 2021
- Harmonic oscillators are incredibly fundamental to almost all areas of physics. But how do we deal with systems that have more than one harmonic oscillators which all talk to each other? In this stream, we will discuss how to solve systems of coupled harmonic oscillators using the method of normal mode decomposition. We will then move on to deriving the wave equation by taking the limit of infinitely many coupled harmonic oscillators. Thinking about the underlying symmetries will give us our first hints as to why we might need a theory beyond Galilean relativity.
I did make a small error in this video: at 31:40, the second solution should be A2=-m1/m2 A1 and B2=-m1/m2 B1, not A1=-m1/m2 A2 and B1=-m1/m2. Hopefully this doesn't cause too much confusion!
This is my favorite physics channel. So underrated
Thank you for the video!
Awesome Video
Do the Normal Modes form a preferred basis in the space of functions(is this a Hilbert Space already?) or are they just the ones we're comfortable with mathematically because sinusoidals are easy? Can i change the basis for fourier analysis? Or are these physical observables again?
I'm currently trying to get familiar with tensors and when i see a preferred basis I get nervous...
Yes, this was a good one :)
Hi Narf, great questions, as usual. I will preface this with a disclaimer saying that my knowledge of functional analysis is fairly weak. However, I am confident that at least the case of normal modes on a finite string span a Hilbert space (specifically, I believe it is called L^2[a,b] where a and b are the endpoints of the string). Extending this to the case of the infinite string and the coupled oscillators is probably a bit mathematically touchy since in these cases, the range which you are covering, (-infinity, infinity) in either space (string) or time (coupled oscillators) is not closed. My guess is that these still span Hilbert spaces, but I am not super confident in that statement.
I also don't believe there is a preferred basis in this case. As an incredibly simple example, the basis functions used in the video could be sines and cosines, but they could also be in terms of sines *or* cosines exclusively (where the extra degree of freedom shows up as a phase shift in the argument of the sine or cosine), complex exponentials or even polynomials (just Taylor expand your sines/cosines) and each solution in terms of these separate bases is equally valid (i.e. when all is said and done, they solve our differential equation and satisfy any set of boundary conditions). Of course, that is not a comprehensive proof, but I would be very surprised if there was a preferred basis for this sort of analysis.
Sorry I can't be more mathematically rigorous with this discussion, but I have been trying to beef up my understanding of this area of math anyway recently, so if I come across a more satisfactory answer, I will certainly let you know!
@@zapphysics I discussed this a bit with someone who is more mathematically savvy than I(aka my dad) and we came up with this.
The Sin/cos basis is convenient for the case of oscillators but ultimately arbitrary. The Fourier Transform is a decomposition of the element in the vector space into its coefficient representation in that particular basis. The Taylor Expansion and Laplace Transform are examples that use different basis. So there should be a Jacobian to transform the basis between these?
The "simplest" basis would be that of delta functions, in which case the coefficients are just the function values.
The dimensionality of the Hilbert Space of "all functions" is always infinite, but depending on if you use the continuous or discrete transforms it can be countably or uncountably infinite.
We also talked about how you'd construct the dual space/basis in each case but that's more off topic. (Also we discussed this in German so if the math terms are wrong I hope you still understand what i mean.)
By the way. Do you know the channel Eigenchris? They make very mathematical videos on tensors and geometry and recently started on Relativity. I think there would be great collab potential between you two as I feel your styles would complement each other well. Maybe about the geometry of QFT. Just a thought :)
As always, thank you for your answer!
@@narfwhals7843 I think your dad's explanation is excellent. I am certainly convinced!
Also, I have not heard of Eigenchris. I will have to look into them, they sound interesting. Thanks for the recommendation!
I think there's an error on the correction at 40:50 .. Shouldn't the second term be -m2/m1 A1 ^(2) instead of -m1/m2 A1 ^(2)?
@Rushunnh Fernandes Ahh thanks for pointing this out! Actually, the general expression is correct. I did make an error at 31:40, however, where the second solution should be A2=-m1/m2 A1 and B2=-m1/m2 B1 (I just double checked the math to be sure). Apologies for the mistake!
@@zapphysics hey we all make mistakes.. The video was very useful.. Thanks .. You need to do more of these detailed videos...