Double pendulum: equations of motion for small oscillations

Using Lagrangian mechanics to derive the equations of motion for a double pendulum undergoing small oscillations. We’ll go on to solve them and find the normal modes next time.
About me: I studied Physics at the University of Cambridge, then stayed on to get a PhD in Astronomy. During my PhD, I also spent four years teaching Physics undergraduates at the university. Now, I'm working as a private tutor, teaching Physics & Maths up to A Level standard.
My website: benyelverton.com/
#physics #mathematics #doublependulum #pendulum #lagrangian #lagrangianmechanics #eulerlagrange #kineticenergy #potentialenergy #dynamics #mechanics #angles #smallangleapproximation #calculus #differentiation #matrices #vectors #oscillations #smalloscillations #maths #math #science #education

[@paveltsi1220]

Hello, Sir! I have not seen you for ages! As I can see you have been doing only the right things, thank you very much!!! Glad to watch what you do here!

[@DrBenYelverton]

Nice to hear from you, thanks as always for watching!

[@dbf72829]

Great video for the core concepts thanks as always

[@mehmetali4626]

Very helpful Thank you.

[@DrBenYelverton]

Glad to hear that, thanks!

[@omarkhalil4676]

Thank you
very helpful

[@DrBenYelverton]

Thanks for watching, glad it was helpful!

[@REMOMosesTV]

Thanks you're a super teacher

[@DrBenYelverton]

Thanks for saying so!

[@mkh6338]

Hello, thank you for the video. Have you possibly made any videos about small scale oscillations?

[@DrBenYelverton]

Thanks for watching. I haven't done any videos on general methods for small oscillations, so this video (and the follow-up video, ruclips.net/video/x8pGtspZrk0/видео.html) might be the closest to what you're looking for. It might also be worth watching my video on oscillations of a diatomic molecule, which considers another special case: ruclips.net/video/NG6nV5u2oBg/видео.html

[@mkh6338]

@@DrBenYelverton ok I'll watch. Thanks indeed.

[@r2k314]

Can you do more complex problems with translations, multiple pendulums with masses attached to springs on them, etc.? And external forcing. Thank you.

[@DrBenYelverton]

Good idea, I will put this on my to-do list and see what I can come up with.

[@r2k314]

@@DrBenYelverton Thanks!

[@srujanm8109]

Could you please explain how did you get to the expression of y2 = -L(1-Theta^2/2)?

[@DrBenYelverton]

Just used the small-angle approximation for cosθ. See for example en.wikipedia.org/wiki/Small-angle_approximation

[@sakia5680]

Thank you for the video! Would we be able to use this to use this to figure out the angle of theta 1 at which the velocity of the second mass is the greatest?

[@DrBenYelverton]

To do that, I think you'd first need to solve the equations of motion to get the angles as a function of time (as in ruclips.net/video/x8pGtspZrk0/видео.html). The answer to your question will depend on the initial conditions and is probably not straightforward to work out in the general case! If the double pendulum happens to be oscillating in one of the normal modes though, the maximum velocity of both masses would occur when the angles are both zero, since they're then both performing SHM either in phase or in antiphase with each other.

[@sakia5680]

@@DrBenYelverton I see. I was wondering if I could use this mechanics to calculate the theoretical angle at which a shoe should be thrown off someone's foot for it to fly the furthest, by simplifying the model into a double pendulum where the first is the swing seat and the second is the person's foot (assuming no extra force is added when the shoe is thrown off). Do you think Lagrangian mechanics could be used to do this? Thanks for your help!

[@karlasharairodriguezramire7259]

Hello, thank you for your video, it is very easy to follow c: I just have a couple of questions, I hope you can help me. In the video, when doing the small angle approximation you said that we were going to work up to second order to "end up with equations of motion that are linear", how can I connect those two ideas? I would really like to understand the importance to do this essential step. Thank you for your work c: have a good day

[@DrBenYelverton]

Thanks for watching! The equations of motion come from differentiating the Lagrangian L with respect to the angles and their time derivatives (i.e. from the Euler-Lagrange equations). So, if we want to get linear equations of motion, we need to find L to second order because differentiating a squared term gives a linear term. If we'd found the Lagrangian e.g. to third order, we would have ended up with quadratic equations of motion that would have been more accurate but much more difficult to solve.

[@karlasharairodriguezramire7259]

@@DrBenYelverton Thank you so much!

[@garykirkham1389]

At the step where you came up with the equation for y2, why isn't cos(theta) approximately 1? That would make y2 approximately -2l, which makes physical sense when the oscillations are small.

[@DrBenYelverton]

Approximating the y coordinates as constant doesn't give enough accuracy to get a useful equation of motion, because you then get a constant potential energy V, which is equivalent to just ignoring gravity. If you try working through the rest of the problem with V = const, I think you'll come to the conclusion that θ₁ and θ₂ just increase linearly with time, consistent with the idea of having no external force.

[@garykirkham1389]

@@DrBenYelverton Thank you. That makes sense.