Physics Students Need to Know These 5 Methods for Differential Equations
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- Опубликовано: 26 июн 2024
- Differential equations are hard! But these 5 methods will enable you to solve all kinds of equations that you'll encounter throughout your physics studies. Get the notes for free here: courses.physicswithelliot.com...
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Almost every physics problem eventually comes down to solving a differential equation. But differential equations are really hard! Fortunately, there are powerful tools for tackling them, and in this video I'll introduce you to five of them: substituting an ansatz, using energy conservation, making a series expansion, using the Laplace transform, and finally using Hamilton's equations, which give a new way to visualize the solution as what's called a flow on phase space, as well as a way to solve an equation with a matrix exponential.
We'll see how they all work using one of the most important differential equations in physics: the F=ma equation for a simple harmonic oscillator, or in other words a block attached to a spring. You certainly don't need crazy powerful tools to solve such a simple equation, but seeing how they work in a simple problem will help prepare you for the harder problems you'll inevitably meet later on in physics!
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0:00 Introduction
2:20 The equation
4:01 1: Ansatz
9:10 2: Energy conservation
14:17 3: Series expansion
18:23 4: Laplace transform
22:41 5: Hamiltonian Flow
26:48 Matrix Exponential
29:31 Wrap Up
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About me:
I’m Dr. Elliot Schneider. I love physics, and I want to help others learn (and learn to love) physics, too. Whether you’re a beginner just starting out with your physics studies, a more advanced student, or a lifelong learner, I hope you’ll find resources here that enable you to deepen your understanding of the laws of nature. For more cool physics stuff, visit me at www.physicswithelliot.com. 
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I am extremely impressed with the high quality of your talks. It is apparent that you put much thought, and much work, into the script, the examples, the animations, and the presentations. Also, your voice is perfect for narrating videos like this -- expressive, clear, and pleasant to listen to. With this video on differential equations, you have packed a whole semester's worth of learning into a half hour. Your notes are equal to any physics book I've seen, and I appreciate that you provide them for free. I am going to increase my Patreon donation to your channel. Thank you, and best wishes. I'm so grateful for your work.
Thank you so much Michael!
Totally agree
@@PhysicswithElliot you are the Morgan Freeman for Physics!
Yes that is the sad part " Your notes are equal to any physics book I've see" Its al dark and ambiguous as any physic would approach
@@PhysicswithElliot This is excellent even though the pace of explanation is very tough to follow. I got lost after 12 minutes of the video even though I used to be famiIiar with the contents of the video once. I am not a mathematician in any sense. But I studied physics and took calculus a long time ago. I am still studying physics on my own at my own inspiration and times when it overwhelms me. But may I say that even in school one variable always gave me trouble to understand. And it was and still is time. Denoting time (t) we use it in many equations and mathematical formulas. But after years and years of pondering over ''time'' I cannot undestand how ''time'' is being used in mathematics without a definition of time. We know what distance or space are and we can define them in a scalar manner and use vectors or whatever else. But - excuse my coy knowledge (I've forgotten so much that I need to reread a lot of math) of math - I think ''time'' cannot be associated with clocks at all. When I see a clock or even read about atomic clocks I do not apprehend ''time'' in them. They do not show me ''time''. The idea of time flowing in some direction is an erroneous way to approach this elusive entity. Time does not flow niether has a direction. If time flowed (as you hear all over) it would have to be moving. In my opinion ''time'' is some kind of force. After all it forces us to get up in the morning to do things and live. But in the deeper sense if I one says that an hour has passed I cannot grasp that hour and adhere it to some point of reference. In your video of the example of the block oscillating you have to define the initial condition in order to perform differentiation. But I envison that with ''time'' one cannot do that. Might as well start using words like ''I did it then'' and ''I do it now''. But one cannot use these words in mathematics even if you give them symbols. Definition of ''time'' would be so much helpful in seeing the whole picture.
6 - Sturm-Liouville 7 - Green's function 8 - Hypergeometric Functions 9 - Lie symmetry method and similarity invariant 10 - Advanced Perturbation Methods
Hope you like the animations in this one! It's the first video I've made using "manim," the programming library for math animations created by @3blue1brown for making his incredible videos, and further developed by the community of developers who work on the open source project. A huge thank you to them for their hard work!
Thank you dear Dr Schneider 🙏💚
Animations look amazing! Very smooth, love it
Very nice. Thank you! 👍
Just one thing. The animation at ~24:45. The red ball is swimming against the flow. I’m told that phenomenon occurs only in Australian toilets. 😁
Great video, thanks. 3B1B is excellent!
@@orsoncart802 I see the flow going the right way, I’m pretty sure just depends which way u look at it
This channel is going to blow up in the future.
Thanks Bruh!
I cannot express how grateful I am for these videos. Your content has single-handedly changed my outlook towards physics work, and my ability. Your easy to digest videos and worksheets talking about the mathematical rigour of such a broad range of physics is just breath-taking. And it's certainly done a lot for me. Thank you for what you do, Elliot, and I'm excited to see what's in store for the future.
Here before this channel gets millions and millions of subscribers. Keep doing these animations, they are invaluable when you show the concepts. It really helps visualising the physics and the math.
I had a bit of trouble following along at the end of the video, but just because the material was tough for me; the explanation was outstanding. Thank you so much for taking the time and effort to make these really high-quality videos and then sharing them for free!
I'm so grateful for this video. I've been trying to self-study Differential Equations and kept getting stuck early on. This really helped clarify not only what to do to solve Differential Equations but WHY the methods work. Thank you!
Very interesting! It was definitely instructive to see all 5 techniques applied to the same example.
You're my favourite physics tutor! I can't tell you how much it was painful looking for information for months and being unable to find one that make you content. But with your videos you've answered to a lot of my questions so I can't tell you sir how grateful I am. Thank you for your clear explanation and representation, and for feeding my curiosity and growing my knowledge, I owe that to you.
Going over an E&M course, and the boundary conditions cannot be undervalued. Good stuff! Glad to see this content on RUclips!
Maxwell's Equations are the best; but it's all fun 'n' games until boundary conditions are imposed!
After that trial, someone imposes mixed Dirichlet and Neumann boundary conditions.
@@douglasstrother6584 Very true! It's enlightening though when you finally understand the physical implications/meaning of boundary conditions. This of course applies to many fields of study. Acoustics was another fun area to see these applications!
@@curiousaboutscience E&M is my favorite Unified Field Theory; the collaboration between Faraday and Maxwell is sorely underappreciated.
Learning to visualize charge and current distributions and field patterns is invaluable, even with the existence of numerous E&M computation tools. The boundaries are where most of the interesting stuff in happening.
@@douglasstrother6584 There is so much to say about the power and accuracy of this theory.
My first class I didn't appreciate how much was related to the importance of the boundaries.
Elliot, that was a beautiful, clear and concise presentation of these important core concepts. The time, effort and intelligence you put into your videos is very much appreciated; you are a natural born teacher.
I studied physics for many years and I wish I had these videos back in the day. So clear !
Would love to see a similar video on partial differential equations :) Thank you for your content very well explained!
This video is for physics students, but math students or anyone with an eye for math might be interested in some of the technicalities. For the first proof, although it is easy to verify that sine and cosine functions solve the equation, it might not be obvious how we know that a combination of a sine and a cosine with the same phase is guaranteed to give the _general_ solution; that is, it might not be obvious that every solution to the differential equation has that form. But remember that the equation is _linear_ with continuous coefficients, and so the uniqueness theorem for initial value problems for nth order linear ODEs (which seems not to have a name) ensures that the solution is unique. The two coefficients A and B account for the two initial values. We know sinusoids solve the general equation, so a specific solution must be a combination of sinusoids, which just turns out to be another sinusoid. So the general solution is a sinusoid, with an amplitude and phase shift determined by the initial values. You can write this as A cos(ωt+φ) or as A sin(ωt) + B sin(ωt), which you should remember from trig or precalc as an identity of sine and cosine. Here, ω is fixed by the differential equation, but A, B, and φ are pairwise independent and depend on the initial values. Also, you may see this equation applied to pendulums, but keep in mind that this relies on the small-angle approximation sin θ = θ and so is only a good approximation when the pendulum makes a small angle to the normal. As a final note, the nature of sinusoids is such that you will typically only see solutions like this for second-order ODEs, because these functions have a period with respect to differentiation of 2 up to a constant and correspondingly have just two degrees of freedom (like an exponential, of which they are special cases).
For the second example, this is a purely mathematical consequence of Newton's laws, as the video says, but I don't have time to explain it. Technically, it is a consequence of the work-energy theorem. One way you might get insight is from the kinematic equations (which themselves are purely mathematical), one of which is (v²-u²)/2=aΔx. Multiplying by mass and defining F = ma and T = ½mv², we get ΔT := T₁ - Tₒ = FΔx := W, which in this loosey-goosey world means that work equals the change in kinetic energy. From this, we define potential energy U for conservative forces such that the difference in U between two positions equals the work done by going from one to the other. Then it is simply necessary, by definition, that energy be conserved. It's slightly more complicated for nonconservative forces, but in the end, it is always possible to define potential energy in this way. That's what teachers mean when they say potential energy is the "ability to do work": it is literally defined as the work done to go from one state to another. For some people, this might demystify potential energy a little; it's not some ethereal, nondescript substance, just a property of a state defined by what happens when you change it, much like temperature or stiffness.
For the third example, you may know that not every function equals its Taylor series at every point. First, the function must have derivatives of all orders at that point for the Taylor series to even be defined. Second, a Taylor series will typically only converge on some neighborhood of the point, so you have to pick a close enough reference point. Third, in pathological cases, the Taylor series will be converge at a point but not equal the value of the function there. And fourth, even when the Taylor Series does equal the value of the function at a single point, it might fail to equal it on any neighborhood of the point (i.e. given any open set containing the point, the function's Taylor series will either fail to be defined, will diverge, or will converge to a different value than the function at at least one point in that open set). In these cases, the function is said not to be "analytic" at that point, and this method will not apply. Elementary functions (functions created by composing complex numbers, +, -, ×, ÷, exp, and log) are all analytic on their respective domains. But other functions don't necessarily have these properties, so you cannot assume this approach will work for every function you come across. In physics, however, functions are almost always at least piecewise analytic, so this is rarely an obstacle.
For the fourth example, the condition is far milder. A Laplace transform will always exist when the function in question is locally integrable, i.e., whenever its absolute value is Lebesgue integrable over any compact set. Essentially, if you force the function to be always positive, but the integral around any point is still finite, then the function is locally integrable. This is a weaker condition than L₁, which requires that the integral of the entire function be finite; some functions have finite integrals over finite parts but the integral over the whole function is still infinite (e.g. f(x) = x). But also, even if the integral converges only conditionally (i.e. the Lebesgue integrals over the positive and negative parts both diverge, but the appropriate conditionally convergent integral has a finite value), the equation still holds as expected. The inverse still exists and the formula remains correct. This is the most general method of them all. (Of course, the inverse Laplace transform won't always be elementary, so you might not be able to simplify at the end, and even if you can, the simplification might be far from obvious.)
The fifth and final example is the narrowest and the most physically-inclined. This method only applies to systems satisfying Hamilton's equations, which were specially designed for Newtonian mechanics (but which are also applicable in an extended form to quantum mechanics). The method will work precisely when these equations hold, which is to say, precisely when they describe a general sort of dynamical system. It is not a general fact of mathematics that this is the case, but it is simply the case for physical systems. There are various "deeper" reasons one can provide for this relating to symmetry and Noether's theorem.
I finished my degree about 4 years ago, and this reminded me of so much. What a great presentation! Such a clear delivery with great perspective to relatable concepts
I'm glad to find a high quality content explanations about basic physics, it's harder to solve cubersome problems skipping the bacics, thank you from Brazil 🇧🇷
Method 0: use Mathematica
Method 0: go to mit open courseware
Ask wolfram alpha
@@StuffinroundWolfram Alpha is weak compared to Mathematica (and this is also logically comprehensible)
Hi from Argentina, I am preparing for a very hard physical chemistry final exam in March, and I found this tutorial very valuable. I know a 30 minute video won't replace hours and hours of differential equation solving, but I got to say the laplace transform and hamilton parts are brilliant, because your approach has an integral view, it is perfectly edited and explained, and it shows the beauty and simplicity underlying these concepts. Too often as students we lose track of this global view because we are alienated with calculations and exercises. I found your explanation beautiful. Beauty serves as a path to a deep understanding of anything, that's my opinion. I am subscribing right now!
You could argue the ability to express complex ideas in a simpler manner is what defines a great teacher from a sufficient one. The ability to understand a person's abilities and limitations to such an extent that you can translate the most obscure information that your target audience can easily understand and utilize is the most important factor. It's not what you know but what you can convey to others.
Awesome work, I wish we had this around when I was studying physics and maths. This really accelerates learning and understanding. I’m envious of current students of physics having such great educational tools available!
Man this is high quality, easy some of the best physics educational content on youtube. Do you still plan on uploading any problem sets for this video? Thanks a lot for the notes btw
You are a terrific educator, sir. Thank you. This was superbly constructed.
Appreciate your effort and pedagogical skills
Found this through RUclips recommended, and I have to say this video is a masterpiece. Instantly subscribed and looking forward to more videos from you
I am just starting to learn classical mechanics and this was a great simplified bird’s eye view of all the techniques! Thank you sir 🙏🏼
Elliot, that was excellent and solving same problem different ways important for many different reasons from educational to checking a solution. Thanks. Have been looking at your videos on lagrangian. Again, very enjoyable and very informative. And thanks for access to "notes" .. Your students must really appreciate you.
I have studied economics and maths was part of that. This explanation really brought home some concepts I always grappled with in an easy to understand way. Thank you.
Excellent explanation of these 5 core concepts used to solve differential equations using the Manim animations. I like the whirl pool analogy and animation you used to convey a visual intuition of the Hamiltonian Flow. The matrix exponential construct is interesting. Thanks for sharing your work.
I struggled mightily through this stuff in college. Not only was that before RUclips but it was before electronic calculators. This is so much easier to understand.
4th & 5th methods are mind blowing especially Hamilton's Flow. Thank you for sharing.
lovely intro about not only the physics but also for the math and general engineering. Great video!
Great stuff 🙂I know you already did a video on Hamiltonian mechanics, but a deeper explanation of the Legendre transform involved would be nice.
Thank you so much, especially to see the Laplace transform in use was an eye-opener
So high quality! Thank you!
Brilliant as usual! 👍 One fun thing about the Ansatz: English-speaking world tends to solve, for example, the harmonic oscillator differential equation as A cos(omega t) + B sin(omega t), which is very sensible in from a maths point of view (you find a basis of two independent vectors in 2D vector space of solutions of this linear second order ODE and you express any solution as its decomposition on this basis). French way - for example - would be lean towards a physicist strategy and write A cos(omega t + phi), since in physics, amplitude and phase are much clearer to interpret than A and B from previous sentence. 😊 You arrive on this second writing in a very natural way with the energy reasoning, though, which is very interesting.
Thank you for these wonderful videos ! Are you planning one CFTs?
Bravo! One of the clearest and detailed lesson I have ever seen...
This is absolutely a fantastic explanation of this subject. Many thanks for this
Just came across your video. Holy, the best I have ever seen in explaining and summarizing in such concise and clear terms! Thanks!
I'm so glad that I found your channel I've been looking for such channel that explains physics in english. Tysm for your hard work!
Splendid! Nicely presented and generous in content for introducing the concepts. You have a new subscriber.
Very good video! You've definitely won a subscriber here! I can't wait to see what's coming up next! Thank you!
Brilliant lecture! Thank you!
Amazing video. I saw this topics before but this video really makes me enjoy what I couldnt while taking these classes...
Great video, certainly some of the best math animations and exigesis I have seen.
Beautiful and concise. Thanks Elliot.
Thank you very much! The video is gorgeous and very clear. For the first time i have connected better my knowldege about differential equations in a way i have never thought! Thank you a lot very much!!!
that was so enlightening. thank youu!
What a masterpiece. Please continue with this excellent work
Extremly good video, perfect refresher for some, superb intro to others. Very, very good content. Thank you very much.
Now I can finally say I am enjoying Physics. Hats off to you!!!
First time I understand what a Laplace Transform a Hamiltonian are! Very clear explanation. Thank you.
Bro, u are giving away this high level of knowledge FREE!
Man I'd pay the $$ to attend your courses, the content is simply awesome!!
What a wealth of knowledge!... thanks for sharing this Doc, this was truly helpful.
A very excellent presentation. Thanks a lot Elliot👍
Nice examples! It would be interesting to do the same with a more difficult DE, too.
I enjoyed this much more than i could, thank you a lot for your effort, this was very thoughtful, im an absolute fan
Very clear explanation, bravo!
Thank you so much for this video, now it's really clear in my hand. I have just make tremendous progress with this video! Again thank you !
What a nice simple explanation of Hamiltonian mechanics!
This is an incredibly helpful video
Really helped me review some necessary content
This was brilliant! You've gained a new subscriber!
Hey Elliot, I am so glad I found your channel today (subscribed!) and that you have the time and opportunity to release someone of the finest "math physics" videos on RUclips that are on the same superb level of quality like 3b1b's math videos! Please feel free to dive more into details, but easier said than done I guess as it must take quite a while to create such a high quality video and maybe I am not your main target-audience 🙂
Excellent Work!!!!
Super interesting video, as always! The quality of these videos is really great. I wonder, the Hamilton equations kind of reminds me of a cross product. Is there a relation there, or am I imagining things?
Wow! No distractingly unnecessary music over your excellent narrative skills and important information??? I’m exponentially impressed!!!!👍😃
I love the video. Very useful and easy to grasp
Keep doing this amazing work 👌👌 You are just different and unique👏👏
Absolutely love this.
This is super interesting ! Never had such a bird eye view on the way to resolve such a canonical system whilst having studied the harmonic oscillator for 5 years at uni !
Amazing stunning mesmerising. Being an electrical and electronics Engineer from the most reputed university in my country I have been struggling to fathom the inner meaning of the differential equations and its solutions. Finally I have got to understand it. Thank you awfully
Love the videos! What program do you use to make such videos?
Beautiful way explained
Excellent notes.
Increadible explanation! I would like to recomended this video to my students later on. Thanks :)
Great insight to see everything together... thanks!!!
As engineer I'll keep with Laplace but uncle Hamilton was incredible! Nice...
Excellent video, man, thank you :)
Thanks for the explanation, would love to see the Poisson Equation on gravitational field on next video. It would be great!
Very well done. Thank you.
You've just earned another subscriber. Brilliant and elegant.
Beautifully explained ❤️❤️❤️
Thank you very much. Good content. Greatly appreciated. Keep up the good work🎉
Awesome Video. Thank you very much.
What I like to do in class is connecting the hamiltonian flow with the Eigenvalue Problem and find a solution in terms of Basis functions.
Btw: The oscillating Block is by far my favorite example as well 😊
Incredible... This is "Quality Education". Great.... Thank you 🙏🙏🙏😊
💓 thanks 🍻 especially for you acknowledging others' contributions
Hi Elliot, many thanks for the video. Kudos!
Fantastic. Thank you
Brilliant. Thank you.
Great work! Thanks!
I need to pose and play again a few times... sorry I am a little slow...😁
*a few (thousand) times
I am rooting for you. Slow but constant.
That's totally fine and normal dear
This was such a brilliant video
Thankyou so much for this precious knowledge and explanation 🙏🙏 I don't have words to express my gratitude for such an amazing lesson.
This is a great video. Thanks for your nice effort 🙂
Finally, a channel that I can watch without torturing my eyes! Show me a black text on a light background, and I’m yours! Just subscribed.
Brilliant animations and stunning video
This is honestly fantastic
Thank you very much for this video
You did a great job and I like how Manin library is used.
Great explanation appreciate it
Thank you. Enjoyed the 30 minute wholeheartedly.
That was sick! Gonna try to master these methods now