The cool part about Lagrange's formulation of classical mechanics is that it is actually quite extensible to other places outside of physics. For example, the main premise is about minimizing the action, which essentially creates an inherent optimization problem. This means that the lagrangian can be extended to other domains where some appropriate definition of "action" needs to be minimized. The Euler-Lagrange equations actually show up a bit in economics and control theory as a result.
I wonder if it's in game theory too, perhaps related to the MTQ(material, time, quality) framework Maybe it's also applicable to the "pick 2: good, cheap, fast" conundrum?
@@RyanApplegatePhD its more that the euler lagrange equation gives a solution to a very general optimazation problem and often times you need to solve an equation that looks identical to the Action in form and you can then just use the euler lagrange equation. For example in the brachistochrone problem you can define the time to fall as an action and then solve the problem with the el equation.
As a high school student I entered a physics competition way above my level and saw many examples of langrangian used to solve complicated problems. I researched into it afterwards, but your video really explains things so well!! Thanks :)
Glad to have given you the intuition! No longer does LM feel like a magic box anymore, eh? Of course, the real power of LM is the stupidly huge range of problems it simplifies, so go and test it out yourself!
The Euler-Lagrange equations can also be used to solve other cool problems like finding minimal surfaces! (e.g. the shape of a soap bubble given some boundary conditions)
I am a physics graduate and even though I did my Master's degree in astrophysics, I had a short delve into theoretical physics because it really interested me and I wasn't sure at the time which direction to take. So I learned all that stuff in university, and found it really fascinating and beautiful, but I have to say that I've never seen an explanation of this as tangible and simple as yours. Also, the genius connection to Noether's Theorem and its motivation by symmetry was genuinely eye-opening. Also, Noether's theorem's beauty really shines when you know its mathematical side and it's connection to symmetry without all the necessary but complicated stuff around it. Also I totally agree with you about teaching or showing stuff like this much more and also much earlier in education, it would do wonders to theoretical physics by attracting a lot of talented people (and makin them better at their work) and by making it to accessible to much more people, from physicists in other fields that might never have gotten much more than a short introduction into Lagrangian (like me if I hadn't gone out of my way to visit non-astrophysics lectures) to common people that would be able to understand things like quantum field theory at least on a surface level, instead of getting it only presented on a "this is very hard and you won't understand a thing of the math behind it so I'm not gonna bother to show you at all" level. This was a really nice video and of the best some2 videos I've seen until now, bravo. Also, I'm also really thankful for 3b1b to do this thing once again, as it kind of brought me back into math youtube. Thank you, mindmaster!
It warms my heart reading all that :3c I don’t plan to stop making videos, so keep an eye out and more should be on the way! I have too many of these explanations to keep to myself.
I always see Largangian mechanics and Newtonian mechanics as an example of the proper use and statement of Occam's razor in science. The proper statement is that if two theories generate identical results they are both valid. The proper use is that given two theories that give the same result you should use the one that is simplest (or easiest) for the given problem.
One thing to mention is that computationally forces are easier to work with because they are vector quantities. You can very easily write code to represent the change of an object in time using forces, this is because forces behave linearly so we can take advantage of numerically efficient linear algebra algorithms. Solving complicated physical systems computationally using the lagrangian is quite hard if you haven't already solved all the equations yourself iirc. Moreover you'd need some way to translate the result into a position vector, and it would be a lot slower than arriving to the same position iteratively. You are better off just using Newton.
@@mindmaster107 it is overall a much stronger theory because it allows for more interesting extensions and formulations. Particularly Hamiltonian mechanics lends itself in this way into the world of modern physics and Quantum Mechanics. You should make a video on that as well!
Forces are just a lazy students way of coming to quick and approximate solutions for a majority of problems. Making for lazy research and no real progress in any field.
@@Number6_ I'm not sure what you're talking about, forces are rather useful and Lagrangian and Hamiltonian mechanics still respect this. Newton says: F = ma Lagrange says: ma = dp/dt and Hamilton says: dp/dt = -dU/dx These are all equivalent formulations of forces, the main difference is that Newtononian mechanics treats forces as the center focus and explicitly constructs them first, while Lagrangian and Hamiltonian mechanics deal with them implicitly as tangent bundles over manifolds.
not gonna lie, I was definitely reccomended this video because I love your maps, and I've specifcially gone on the page for mindmaster's hard dif of lagtrain so often that google picked up on it and reccomended this channel, but unironically, this video blew my mind and is genuinely going to help me a bunch in class. You, mindmaster, are a literal deus ex machina, thank you for it.
Lol, osu has taken over. I plan to make more physics videos, so sticking round (and maybe sharing it to people who will enjoy it) will help me wonders! Thank you for the comment!
I would just say, man you explained it extremely extremely well. Keep up the good work. Normally every TH-cam video explaining concepts would leave somethings vague and it's hard to grasp the concept of it.
I specifically was searching for a SoME video on this, only to find this was the second result when I searched #SoME2 ! This is a fantastic explanation
Hey, just found your channel through front page recommendations and lovely video! It's the perfect amount between "details" and "intuition" to not be too messy with the specifics, and also be specific enough to learn those concepts. I'll be subscribing and waiting for the next videos. Keep it up!
God, videos like these make me appreciate math and physics even more than I initially had. Part of me wishes my engineering (biomedical) program would allow me to go more in depth with some of these topics. At the very least, there are tons of great resources online to passively get familiar with some of the topics. Great video by the way! The way you explain topics such as Lagrangians or tensors (throwback to last years 3B1B video) are very accessible :)
While a physics degree is epic, it is also 3 years XP Thats what my channel is aiming to do. Make university physics accessible to anyone with secondary school maths. Stick around for more :D
Very insightful approach to the Lagrangian, least action and Noether’s. Good job! More videos please. I liked the way you mixed insight and math. I liked your graphics and your voice/presentation and I’m sure you’ll step it up as your channel grows. I would like to see more juggling of equations (I think a lot of viewers do), so please don’t spare the math. Can you do an intuitive and mathy explanation of the standard model Lagrangian? Subscribed!
I might make a video on the Standard Model Lagrangian one day, but I would like to go through the easy ideas first :P Certainly something for the end of the road
Just found a gem. Pretty sure this channel is gonna have thousand of subscribers in no time. Love the animation. Increase the upload frequency a bit mate 📈.
Quaternion and octonians allow for vector multiplication and vector division. Good luck to you. I was never taught Lagrangian Mechanics but I understand the concept of Least Action. That tends to be nature's way always.
Quarternions and octionions are subalgebras of Geometric Algebra. If you ever wonder how tjey work, Geometric Algebra will make it simple and obvious to you.
@Garry Simpson I guess everything from David Hestenes et al. That you can get hold of is good start, I also like the approaches of Joan Lasenby and Anthony Lasenby and their groups. There are different applications now with quite some introductory textbooks, depends if you are interested in general physics, space-time physics and quantum mechanics, electrodynamics, mechanics, and for computer graphics there is a separate world with tons of books, papers, videos and software libraries.
Man, I really love your videos. You have a great style and I also enjoy the topics you are talking about. One can really notice that you are extremly passionate about these topics yourself which makes your videos so incredible and fun to watch. I also love your avatar :D
This is such a wonderful way to introduce Lagrangian Mechanics! Is there any connection between the Laplacian and the Lagrangian? P.S. I subscribed! Make more cool math and physics videos!
Unfortunately no, despite the similar name. The Laplacian is like a type of equation (or operator) in physics, while the Lagrangian is a energy of a system.
@@mindmaster107 I understand, however, that the harmonic function which solves the Laplacian minimizes the energy given some boundary conditions (Dirichlet's Principle). That's why I was wondering whether there's a relationship, since the action seems to minimize the lagrangian (integral of T-U).
The relationship might just be they are both optimisation problems. Physics can be seen as one big optimisation problem, whether optimising for least action, fastest compute time, or most funding.
There are people who touched the topic of lagrangian that went (oh dang, so the concept of it was *that* easy?) Then there is me, never actually know what Lagrangian is, who doubt "in practice its not gonna be that easy innit?" just because how easy the video is to be understood. Well done!
Dude, very well explained. I have been watching PBS Space Time for a while now and in some occasions Lagrangian mechanics and Noether's theorem are mentioned, but too briefly for a `higher understanding` behind the real deal. This video contextualized me about a lot of things were still confusing, a wonderful video. Definitely worth of a sub.
I was about to comment _"quietly hides my vector product."_ Nice to see someone else beat me to the punch! Now I'm honestly curious if someone has tried to do formulate something normally done only with Lagrangian Mechanics in Geometric Algebra using Newtonian Forces.
Great vid! Btw, it's not that vectors are *inherently* annoying, but that the cross-product (which to start is non-associative 🤮) is pretty awkward and makes vector-multiplication awkward. Geometric/Clifford algebras introduce a really nice way to multiply vectors making them great again!
This was a really insightful video! Thank you! Also, if it isn't too personal, where is your accent from ? It sound's American but with a British 'A'? It sounds really cool.
I grew up in Asia interestingly enough, and am now in the UK. My accent is very American simply because I learnt English predominantly through the internet.
A bit late to the party but loved the video! I personally wish you had spent a bit more time acknowledging how unintuitive and "hacky" working in Lagrangian Mechanics can feel. When you're used to Newtonian Mechanics (and/or grew up in a western-influenced society - can't speak personally about people in or from other societies), you are reasoning from a cause-and-effect worldview most of the time: an object possesses a mass and acceleration which are acted on by forces, all of which causes it to move/behave a certain way. This lines up nicely with our human experience of the world. Lagrange formulating via equations "it doesn't matter exactly what causes what to move as long as all the constraints [on the Lagrangian] are satisfied and the action is minimized" feels like cheating, like you're ignoring how reality "actually works". What I really love about Lagrangian Mechanics and want to share are the following thoughts: - wave function collapse, and more generally the fundamental probabilistic nature of measurement/observation in Quantum Mechanics seems to follow a similar behavior of constraint satisfaction through (statistically) minimizing the action involved. It seems like Newtonian Mechanics are the "real" hack, and feel closer to reality to us solely because, at a human scale, most of the possible "least action solutions" are physically indistinguishable. Just think about how many atom-widths you'd have to displace an object for your eyes to be able to pick up the difference! - some physics simulations solve collision detection by basically using a Lagrangian approach - using a constraint solver/satisfier to find a valid post-time-step state that effectively minimizes the action of each particle/object involved without needing to understand the concrete evolution of the system within the time step itself, *and then* deriving the new velocity & acceleration from the difference in position with the pre-time-step state (for those interested, look up "Verlet integration" and more generally "symplectic integration"). It's funny to me that what feels like a hack or cheat to avoid figuring out what "actually" happens is potentially just as rigorous if not more so regarding how things actually "work in reality". If I had to attempt to express the core utility of Lagrangian mechanics in my own words, it would be something like this: You can neatly do away with having to exhaustively explore an unknown solution space by trying different solutions until you are reasonably certain you are "good enough". Instead, you can prune the possible solution space until you are reasonably certain a random choice from within will be "good enough". Note that pruning the possible solution space is also routinely done in Newtonian Mechanics. That is effectively what we do every time we use an inherent symmetry of the system we're considering to simplify certain equations by "realizing" terms cancel out or are constant. Similarly for when we chose a particular boundary for the system we are considering because it allows us to invoke certain invariants (like total electromagnetic flux through a sphere always being zero). We just end up reverting to "cause-and-effect forces" thinking once we've simplified the problem enough to be mathematically manageable!
A bit more context for the second of the two thoughts: As you cited in the video, the more classical approach to deriving the physical state of a system in the next time step is by blindly following what the derivatives predict for an entire time step's worth of movement, in a straightforward "x(t + 1) = x(t) + dx/dt" manner. We then need to check for collisions by making sure nothing has ended up overlapping or passing through anything else. This is not trivial and can get very computationally expensive. It also does not necessarily tell you where or when the collision happened. If we realize at least one collision has happened, we need to adjust/recalculate until things don't overlap anymore. If we are lucky there is a simple mathematical derivation we can do to find the correct "implementation" of the collision. For example, if a rigid ball on a linear trajectory collides with a wall, we can just reflect the trajectory vector around the collision point. In a general case, we have no better choice than to chop up this time step into smaller sub-steps and progress through them until we find a more precise estimate for when and where the collision happened. Effectively, this parallels the Newtonian perspective of needing to understand the precise chain of events (or cause and effect) that happen inside the time step to be able to know how things "end up" at the end of the time step. This has several unfortunate consequences as both relative speeds between entities get larger (i.e. some things get really fast compared to others) and/or the distances between them get smaller. One such consequence is that the amount of attempts (and thus calculations) required to find the most precise "correct" next state is difficult to accurately predict and can vary wildly from time step to time step as the objects in the system speed up or slow down relative to each other. If the simulation needs to run at a constant rate - which is the case for video games and more broadly most interactive physics visualizations - this is undesirable. Another consequence is that, by varying the length of the time step with which you are integrating your physical quantities, you can much more easily break conservation of energy or momentum without intending to in the actual calculations you do. Furthermore, when you only chop up certain time steps it becomes even harder to correct for these two problems. And yet the whole point of only doing so for *some* time steps is that you cannot afford to simply reduce the size of *all* time steps! Constraint satisfaction will generally provide a more direct mathematical method to finding the state of the next time step, and the right choice of constraints will outright prevent numerical inaccuracies from accumulating from time step to time step. The only potential danger is that, because the "trajectory" of objects during the time step is of no concern, you can technically write a simulator using this approach that allows objects to phase through each other as soon as they're traveling fast enough to enter and exit each other within the same time step! This can usually be solved by taking a small enough time step that nothing ever moves fast enough to cleanly "exit" anything else in a single time step. Constraint solvers being very fast and efficient in general, means you can **in general** shrink your time step further than with an iterative sub-step approach without exceeding your computational budget for the combined steps.
Oooo, a juicy response :D For your first point of Lagrangian mechanics (specifically the optimisation step) feeling like a hack, is actually physically founded. This video (th-cam.com/video/WFbgfknnWqA/w-d-xo.html) goes over how Quantum Mechanics “considers” all paths at once, and off-optimal paths simply destructively interfere. I have a video planned going deeper into this, but while it was originally a beautiful hack, Lagrangian mechanics is a precipitation of modern physics. I don’t have too many points to say on computation itself, since I specialise in particle physics, but it’s interesting to know about the approximate models of motion used by different game engines to fit their simulation needs. I’ll leave that or you to make a video on ;)
This was a very nice explaination, I am going to share this with my math club! Thank you, I feel like I actually learned something. Criminally underrated. Also, finally I can write a physics simulation without a DES
Bruh. This is so cool. Also, this is the first video on lagrangians that I can understand. Too bad I'll never be taught it and will have to try teach myself
I genuinely believe that with some practice questions (which you can steal from Lagrangian Mech textbooks), you can fully understand it yourself. It’s only one topic. You can do this.
Add dissipation of any complexity and crush the Lagrangian/Hamiltonian dreams. Tho it is still useful for physics in general. You just don't want Hamiltonians that keep coupling systems with each other.
Alas, Friction is what makes all this maths grind to a halt. Aren’t coupled systems where LM/HM most useful? I guess clever PDE skills would get around those.
@@mindmaster107 I'm thinking of many particles. The collective states get a lot more challenging the more collective they are, because finding a basis gets a lot more challenging. In principle, dissipation is just the result of time dependent coupling of Hamiltonians, so it's not an easy fix for most cases with anything like those terms, taking time into account is fundamental in those cases (there's time ordering explicitly in the time evolution of those systems). Tho, very simple cases even of those are something within reach for a few particles (personally never done anything with more than 100 particles myself, and with very low dissipation at that).
This was a brilliant video - super engaging! As an educational video creator myself, I understand how much effort must have been put into this. Liked and subscribed, always enjoy supporting fellow small creators :)
I’m going to make another video hopefully in Christmas, so you can look forwards to that then! It is going to be on Quantum Field Theory, and how the maths is surprisingly accessible.
Very nice video, but I think it would have been worth emphasizing that you're free to choose the scheme (Newtonian or Lagrangian) which is most convenient for the problem at hand. A lot of simulations handle Newtonian PDEs easily enough but would become totally intractable with the Lagrangian.
I think after I make my conceptual videos, I will make videos addressing how to use them effectively. Right now though, I would rather share the tips I’ve collected over university.
Geometric Algebra offers a third way of Mechanics with a whole new point of view. There is plenty of work covering that, for example by David Hestenes, from the 1960s on.
8:17 iirc, the time-component of the momentum 4-vector in relativity is proportional to energy, which implies the two symmetries (t- and x- independence) that relate to two conservation laws (cons. of energy and cons. of momentum) are simply the time- and space-components of a single *spacetime translational symmetry* that conserves 4-momentum!
Something that keeps me awake at night is: Why L = KE - PE though? How to derive that? Compared to the Hamiltonian, the Lagrangian gives you more information about the path some particle is going to take. How to explain all that?
The Lagrangian being KE - PE… is because it matches observation lol. Trust me, I tried digging deeper as to why, but aside from the intuitive step in the video, Lagrangian mechanics works because it works. I showed in the video how it reduces to F=ma, so an equivalent question is where does F=ma come from, which is equally as founded. The Hamiltonian is actually equal in power to the Lagrangian, but easier to calculate for non-relativistic (Special relativity) situations. They both give a global view of mechanics, so I didn’t include it in the video to avoid overloading people.
Geometric algebra gives the best answer I know to how to multiply vectors in way that isn't totally annoying. It's a pity it isn't well known. Also, I've always been more at home with Hamiltonian mechanics than Lagrangian. Probably because I'm not really comfortable with the calculus of variations. However neither of them can elegantly handle inelastic processes. I'd love to see if there's a similarly general, abstract model for inelastic mechanics that we could draw insight from (and maybe make better physics simulators from too). Thermodynamics is always mucking up our nice theoretical models, so maybe we should invite it to be part of the picture?
Inelastic statistical physics would be one of the most powerful revolutions in physics, you are absolutely right. As a physicist myself, I pray that a mathematician does the maths for me. I'm not smart enough for that lol
@@mujtabaalam5907 I haven't watched all of it yet, but th-cam.com/video/60z_hpEAtD8/w-d-xo.html seems like a pretty good intro. There are also quite a few resources linked from the wikipedia page for geometric algebra.
With the falling ball example, aren't we just doing (total energy before) = (total energy after)? If there's some initial speed, you have more energy budget, so the final speed is going to be faster than the final speed of the first case. If the end is higher than 0 height, that remaining potential energy is using up some of the energy budget, so the final speed is slower. Where does this weird difference come into it? It's just going to be equal to some number at the start, and some greater number at the end?
The Lagrangian is a mathematical “energy”, which all physical interactions try to minimise. Imagine a ball is in a gravitational potential. The potential wants to give the ball kinetic energy and pull it down right? But, the ball can only get this kinetic energy, by actually moving down the potential. There is a balance between the amount of potential energy that can be given at any point in time, due to the minimum kinetic energy required to move to the position of lower potential energy. This can be mathematically written as minimising the Lagrangian at every point in space, or minimising the Action across an entire path.
@@mindmaster107, but the example energy method that you said is secretly lagrangian doesn't have us minimising something. just accounting with equalities. Also, if the kinetic energy increases and the potential energy decreases, wouldn't the lagrangian increase rather than be minimised?
The minimisation for the example question is cleverly avoided, by focusing on only the start and end point. If you want the path between the two, you would need the full lagrangian equation, or to use newton’s equations but thats the cop out. Also, you might be interpreting the Lagrangian incorrectly. (My bad, i’ll do better next time) L = KE - PE For all common situations (excluding wormholes and quantum mechanics), the Action (lagrangian summed over time) stays the same. (Minimised is a can of worms, and I should have made it clear in the video) L increases as PE converts into KE. Due to energy conservation, the Lagrangian is basically double the KE if the particle came from infinity (PE=0). Hence, yes, you are correct the Lagrangian increases, but it is the Action that is minimised. We want the path which has the least total L. We want the path which has the least total Kinetic energy over the path. In other words, we want the shortest path. This is similar to Snell’s law, and should be thought of as such. The minimised action is the shortest (energetically speaking) path from the start to end state.
You would use an E-L equation on the lagrangian of the system. I went over that explicitly in the E-L equation section in the video, though solving the 2nd order differential equation requires semi-university maths. ( th-cam.com/video/tFQggKCdt_c/w-d-xo.html is a nice introduction example, though you will need to know calculus)
I understand that this a simplified version to entertain more than anything but the part on the Standard Model is plainly wrong. While you could get away with (with some stomach churning) saying the Dirac kinetic term is akin to the kinetic energy term, you'd also have to interpret at least part of the Yang-Mills piece F_{\mu u} F^{\mu u} as the "kinetic piece of the gauge bosons" just like the kinetic piece for the Higgs. Also to say "the rest is just potential" is pretty bad - maybe with some squinting you could away with it but the analogy really falls flat since the classical interpretation of "potential" doesn't really have a quantum mechanical equivalent, especially for the Yukawa terms.
I plan in the deep future to do a full review of the SM Lagrangian, having worked through it for university. I think the Higgs mechanism (and now scattering amplitudes are calculated in general) is so fun to learn. For this video on LM, three years before anyone would even touch gauge theory, I decided to flubb the details for the sake of a narrative. It was one of the many things I had to do to cut it under 10 minutes. It’s still lovely to know there is an appetite for postgrad topics though. Thanks for the comment!
great video! tiny recommendation, not sure if it was only for me, but the BGM was kinda annoying and doesn't really fit the theme. sorry for the complaining!
@@marcrindermann9482 In a system where KE = PE of course the diference is 0 (conservarion of energy) more generally you could have KE-PE=H (for heat perhaps).
It should say, given a particle’s start and end point, we can plot out the path the particle HAD to take to get there. Also, the euler lagrange would have zero solutions if the particle couldn’t make it at all.
![ค้างคืนโรงเก็บกระสวยอวกาศร้างกลางทะเลทราย [Around The World Ep. 3]](http://i.ytimg.com/vi/E5bWZQhABJE/mqdefault.jpg)
The cool part about Lagrange's formulation of classical mechanics is that it is actually quite extensible to other places outside of physics.
For example, the main premise is about minimizing the action, which essentially creates an inherent optimization problem. This means that the lagrangian can be extended to other domains where some appropriate definition of "action" needs to be minimized. The Euler-Lagrange equations actually show up a bit in economics and control theory as a result.
Oooooo nice ty
I wonder if it's in game theory too, perhaps related to the MTQ(material, time, quality) framework
Maybe it's also applicable to the "pick 2: good, cheap, fast" conundrum?
Yo hi wasn't expecting to see you here
Oh Lagrangians are a live savior in control theory. Newton simply does not work without sin and cos which are none linear.
@@RyanApplegatePhD its more that the euler lagrange equation gives a solution to a very general optimazation problem and often times you need to solve an equation that looks identical to the Action in form and you can then just use the euler lagrange equation.
For example in the brachistochrone problem you can define the time to fall as an action and then solve the problem with the el equation.
As a high school student I entered a physics competition way above my level and saw many examples of langrangian used to solve complicated problems. I researched into it afterwards, but your video really explains things so well!! Thanks :)
Glad to have given you the intuition! No longer does LM feel like a magic box anymore, eh?
Of course, the real power of LM is the stupidly huge range of problems it simplifies, so go and test it out yourself!
The Euler-Lagrange equations can also be used to solve other cool problems like finding minimal surfaces! (e.g. the shape of a soap bubble given some boundary conditions)
The EL equation can generally minimize any integrand.
Crazy that random youtubers are now able to make TH-cam videos that are hard to distinguish from 100k+ subscriber TH-cam videos.
Omg thank you so much!
Exactly 💯, now you can't put the criteria of no. Of views and likes to get the best video on a topic ...
Exactly
That's an expectation from the platform. You can't just do a 2012 call of duty video anymore.
@@mindmaster107 Wait what I really thought it was a 100k+ subscriber channel before I read this comment wth haha
I am a physics graduate and even though I did my Master's degree in astrophysics, I had a short delve into theoretical physics because it really interested me and I wasn't sure at the time which direction to take.
So I learned all that stuff in university, and found it really fascinating and beautiful, but I have to say that I've never seen an explanation of this as tangible and simple as yours. Also, the genius connection to Noether's Theorem and its motivation by symmetry was genuinely eye-opening. Also, Noether's theorem's beauty really shines when you know its mathematical side and it's connection to symmetry without all the necessary but complicated stuff around it.
Also I totally agree with you about teaching or showing stuff like this much more and also much earlier in education, it would do wonders to theoretical physics by attracting a lot of talented people (and makin them better at their work) and by making it to accessible to much more people, from physicists in other fields that might never have gotten much more than a short introduction into Lagrangian (like me if I hadn't gone out of my way to visit non-astrophysics lectures) to common people that would be able to understand things like quantum field theory at least on a surface level, instead of getting it only presented on a "this is very hard and you won't understand a thing of the math behind it so I'm not gonna bother to show you at all" level.
This was a really nice video and of the best some2 videos I've seen until now, bravo. Also, I'm also really thankful for 3b1b to do this thing once again, as it kind of brought me back into math youtube. Thank you, mindmaster!
It warms my heart reading all that :3c
I don’t plan to stop making videos, so keep an eye out and more should be on the way! I have too many of these explanations to keep to myself.
The one thing I remember from mechanics,
"instantaneous rest" = Energy equations
"equilibrium" = Forces
I always see Largangian mechanics and Newtonian mechanics as an example of the proper use and statement of Occam's razor in science. The proper statement is that if two theories generate identical results they are both valid. The proper use is that given two theories that give the same result you should use the one that is simplest (or easiest) for the given problem.
This is the most cleanest explanation of what Lagrange mechanics is.
I really like that there are youtubers that are interested in non-trivial physics topics and are able to explain it well! pls make more
i adore how youtube is showing videos from smaller math content creators
Glad you decided to watch it too :D
Hope you enjoyed!
@@mindmaster107 it was great!
-Virality? (Maybe I did not notice in 2021.)-
Either way, agreed!
that's probably due to youtube noticing you like the #some2 hashtag.
at least that's what I think is happening to my recommendations and I love it
One thing to mention is that computationally forces are easier to work with because they are vector quantities. You can very easily write code to represent the change of an object in time using forces, this is because forces behave linearly so we can take advantage of numerically efficient linear algebra algorithms.
Solving complicated physical systems computationally using the lagrangian is quite hard if you haven't already solved all the equations yourself iirc. Moreover you'd need some way to translate the result into a position vector, and it would be a lot slower than arriving to the same position iteratively. You are better off just using Newton.
Thats a pretty good write-up of the weaknesses of Lagrangian mech, nice.
It relies on energy conservation, so with friction, it fails spectacularly.
@@mindmaster107 it is overall a much stronger theory because it allows for more interesting extensions and formulations. Particularly Hamiltonian mechanics lends itself in this way into the world of modern physics and Quantum Mechanics. You should make a video on that as well!
@@mindmaster107 Yes indeed EL dealing with heat would be quite something.
Forces are just a lazy students way of coming to quick and approximate solutions for a majority of problems. Making for lazy research and no real progress in any field.
@@Number6_ I'm not sure what you're talking about, forces are rather useful and Lagrangian and Hamiltonian mechanics still respect this.
Newton says:
F = ma
Lagrange says:
ma = dp/dt
and Hamilton says:
dp/dt = -dU/dx
These are all equivalent formulations of forces, the main difference is that Newtononian mechanics treats forces as the center focus and explicitly constructs them first, while Lagrangian and Hamiltonian mechanics deal with them implicitly as tangent bundles over manifolds.
a nice intuitive explanation, I love the context it gives for Noether's theorem. Emmy N is my hero.
"vectors are cringe"
~ joseph louis lagrange, 1788
Amen
not gonna lie, I was definitely reccomended this video because I love your maps, and I've specifcially gone on the page for mindmaster's hard dif of lagtrain so often that google picked up on it and reccomended this channel, but unironically, this video blew my mind and is genuinely going to help me a bunch in class. You, mindmaster, are a literal deus ex machina, thank you for it.
Lol, osu has taken over.
I plan to make more physics videos, so sticking round (and maybe sharing it to people who will enjoy it) will help me wonders! Thank you for the comment!
I would just say, man you explained it extremely extremely well. Keep up the good work.
Normally every TH-cam video explaining concepts would leave somethings vague and it's hard to grasp the concept of it.
I specifically was searching for a SoME video on this, only to find this was the second result when I searched #SoME2 ! This is a fantastic explanation
one of the videos of all times
this was awesome, fist time understood the meaning of lagrangian clearly enough to speak about it somewhere, awesome work brother
Just blew my mind.
this was the best explanation so far. why I know ? because finally I understood ❤
Glad to know I was the straw that broke the camel’s misunderstanding :D
One of the best video I've ever seen on youtube, the animation is so good too!! Good Luck for your future.
Hey, just found your channel through front page recommendations and lovely video!
It's the perfect amount between "details" and "intuition" to not be too messy with the specifics, and also be specific enough to learn those concepts.
I'll be subscribing and waiting for the next videos. Keep it up!
Never understood a bit Langrangian mechanics. Now I got a sense what it is.
God, videos like these make me appreciate math and physics even more than I initially had. Part of me wishes my engineering (biomedical) program would allow me to go more in depth with some of these topics. At the very least, there are tons of great resources online to passively get familiar with some of the topics.
Great video by the way! The way you explain topics such as Lagrangians or tensors (throwback to last years 3B1B video) are very accessible :)
While a physics degree is epic, it is also 3 years XP
Thats what my channel is aiming to do. Make university physics accessible to anyone with secondary school maths. Stick around for more :D
@@mindmaster107 Id say you’re achieving that goal extremely well! Excited for more :)
This channel is AMAZING
we need more of these
This is the way we need to attract young people to physics
one of the best videos on the lagragian out there!
I just love your videos, man, thank you so much!
Very insightful approach to the Lagrangian, least action and Noether’s. Good job! More videos please. I liked the way you mixed insight and math. I liked your graphics and your voice/presentation and I’m sure you’ll step it up as your channel grows. I would like to see more juggling of equations (I think a lot of viewers do), so please don’t spare the math. Can you do an intuitive and mathy explanation of the standard model Lagrangian? Subscribed!
I might make a video on the Standard Model Lagrangian one day, but I would like to go through the easy ideas first :P
Certainly something for the end of the road
Just found a gem. Pretty sure this channel is gonna have thousand of subscribers in no time. Love the animation. Increase the upload frequency a bit mate 📈.
Great Video! Really enjoyed the motivation. One of my favorites from the SoME2 peer review I have seen so far.
Quaternion and octonians allow for vector multiplication and vector division. Good luck to you. I was never taught Lagrangian Mechanics but I understand the concept of Least Action. That tends to be nature's way always.
Quarternions and octionions are subalgebras of Geometric Algebra. If you ever wonder how tjey work, Geometric Algebra will make it simple and obvious to you.
@@patrickstrasser-mikhail6873 Can you recommend a textbook for self-study of Geometric Algebra? I was educated as an engineer.
@Garry Simpson I guess everything from David Hestenes et al. That you can get hold of is good start, I also like the approaches of Joan Lasenby and Anthony Lasenby and their groups.
There are different applications now with quite some introductory textbooks, depends if you are interested in general physics, space-time physics and quantum mechanics, electrodynamics, mechanics, and for computer graphics there is a separate world with tons of books, papers, videos and software libraries.
@@patrickstrasser-mikhail6873 Thanks
Man, I really love your videos.
You have a great style and I also enjoy the topics you are talking about. One can really notice that you are extremly passionate about these topics yourself which makes your videos so incredible and fun to watch.
I also love your avatar :D
My god it's like a lightbulb just went on in my head. Can't wait to watch more of your videos.
More will be made :D
TH-cam recommended me this video and I had the feeling it was going to be a good one.
I wasn't wrong
Great vid. Thanks for uploading!
This is such a wonderful way to introduce Lagrangian Mechanics! Is there any connection between the Laplacian and the Lagrangian? P.S. I subscribed! Make more cool math and physics videos!
Unfortunately no, despite the similar name.
The Laplacian is like a type of equation (or operator) in physics, while the Lagrangian is a energy of a system.
@@mindmaster107 I understand, however, that the harmonic function which solves the Laplacian minimizes the energy given some boundary conditions (Dirichlet's Principle). That's why I was wondering whether there's a relationship, since the action seems to minimize the lagrangian (integral of T-U).
The relationship might just be they are both optimisation problems. Physics can be seen as one big optimisation problem, whether optimising for least action, fastest compute time, or most funding.
Thank you for this kind of video
There are people who touched the topic of lagrangian that went (oh dang, so the concept of it was *that* easy?)
Then there is me, never actually know what Lagrangian is, who doubt "in practice its not gonna be that easy innit?" just because how easy the video is to be understood. Well done!
That was my intention, and glad you got something out of it!
Wow, I love the SoME2 videos and I didn't realize that you made one! awesome
Hey Hobbes! Thanks for the kind works!
Very, very well presented. A word of thanks...
Then a word of gratitude is mine to return
Excellent presentation.
In addition to appreciate the clarity of the explanations, I love the cute bear…chipmunk…squirrel…err, whatever…
it was good of you to mention Noether up front, but the video should also have come with an Euler warning.
I quit Physics at uni before getting to this. Nice to see what I missed.
I'm taking ap physics this year and I'm so excited
Don’t be afraid of sharing what you don’t know, and you should to amazingly
Masterful video!
Dude, very well explained. I have been watching PBS Space Time for a while now and in some occasions Lagrangian mechanics and Noether's theorem are mentioned, but too briefly for a `higher understanding` behind the real deal. This video contextualized me about a lot of things were still confusing, a wonderful video. Definitely worth of a sub.
Vectors can multiply. You just need a geometric algebra, and it gets almost as easy as complex numbers. Lagrangians are still cool though
I was about to comment _"quietly hides my vector product."_ Nice to see someone else beat me to the punch! Now I'm honestly curious if someone has tried to do formulate something normally done only with Lagrangian Mechanics in Geometric Algebra using Newtonian Forces.
"The action is called 's' for the sole purpose of confusing people " ~All of physics in a nutshell
Just coming to drop some love in here, this is an awesome SoME2 entry :3
Great vid! Btw, it's not that vectors are *inherently* annoying, but that the cross-product (which to start is non-associative 🤮) is pretty awkward and makes vector-multiplication awkward. Geometric/Clifford algebras introduce a really nice way to multiply vectors making them great again!
Dude why are gems like you still hidden,at least the algorithm revealed you to me today.
just a math major but this vid is amazing
Well, I was going to sleep, but now I'm fully up and studying physics, thanks.
This was a really insightful video! Thank you! Also, if it isn't too personal, where is your accent from ? It sound's American but with a British 'A'? It sounds really cool.
I grew up in Asia interestingly enough, and am now in the UK.
My accent is very American simply because I learnt English predominantly through the internet.
Yayy, physics video!!
A bit late to the party but loved the video!
I personally wish you had spent a bit more time acknowledging how unintuitive and "hacky" working in Lagrangian Mechanics can feel.
When you're used to Newtonian Mechanics (and/or grew up in a western-influenced society - can't speak personally about people in or from other societies), you are reasoning from a cause-and-effect worldview most of the time: an object possesses a mass and acceleration which are acted on by forces, all of which causes it to move/behave a certain way. This lines up nicely with our human experience of the world. Lagrange formulating via equations "it doesn't matter exactly what causes what to move as long as all the constraints [on the Lagrangian] are satisfied and the action is minimized" feels like cheating, like you're ignoring how reality "actually works".
What I really love about Lagrangian Mechanics and want to share are the following thoughts:
- wave function collapse, and more generally the fundamental probabilistic nature of measurement/observation in Quantum Mechanics seems to follow a similar behavior of constraint satisfaction through (statistically) minimizing the action involved. It seems like Newtonian Mechanics are the "real" hack, and feel closer to reality to us solely because, at a human scale, most of the possible "least action solutions" are physically indistinguishable. Just think about how many atom-widths you'd have to displace an object for your eyes to be able to pick up the difference!
- some physics simulations solve collision detection by basically using a Lagrangian approach - using a constraint solver/satisfier to find a valid post-time-step state that effectively minimizes the action of each particle/object involved without needing to understand the concrete evolution of the system within the time step itself, *and then* deriving the new velocity & acceleration from the difference in position with the pre-time-step state (for those interested, look up "Verlet integration" and more generally "symplectic integration"). It's funny to me that what feels like a hack or cheat to avoid figuring out what "actually" happens is potentially just as rigorous if not more so regarding how things actually "work in reality".
If I had to attempt to express the core utility of Lagrangian mechanics in my own words, it would be something like this:
You can neatly do away with having to exhaustively explore an unknown solution space by trying different solutions until you are reasonably certain you are "good enough". Instead, you can prune the possible solution space until you are reasonably certain a random choice from within will be "good enough".
Note that pruning the possible solution space is also routinely done in Newtonian Mechanics. That is effectively what we do every time we use an inherent symmetry of the system we're considering to simplify certain equations by "realizing" terms cancel out or are constant. Similarly for when we chose a particular boundary for the system we are considering because it allows us to invoke certain invariants (like total electromagnetic flux through a sphere always being zero). We just end up reverting to "cause-and-effect forces" thinking once we've simplified the problem enough to be mathematically manageable!
A bit more context for the second of the two thoughts:
As you cited in the video, the more classical approach to deriving the physical state of a system in the next time step is by blindly following what the derivatives predict for an entire time step's worth of movement, in a straightforward "x(t + 1) = x(t) + dx/dt" manner.
We then need to check for collisions by making sure nothing has ended up overlapping or passing through anything else. This is not trivial and can get very computationally expensive. It also does not necessarily tell you where or when the collision happened.
If we realize at least one collision has happened, we need to adjust/recalculate until things don't overlap anymore.
If we are lucky there is a simple mathematical derivation we can do to find the correct "implementation" of the collision. For example, if a rigid ball on a linear trajectory collides with a wall, we can just reflect the trajectory vector around the collision point.
In a general case, we have no better choice than to chop up this time step into smaller sub-steps and progress through them until we find a more precise estimate for when and where the collision happened.
Effectively, this parallels the Newtonian perspective of needing to understand the precise chain of events (or cause and effect) that happen inside the time step to be able to know how things "end up" at the end of the time step.
This has several unfortunate consequences as both relative speeds between entities get larger (i.e. some things get really fast compared to others) and/or the distances between them get smaller.
One such consequence is that the amount of attempts (and thus calculations) required to find the most precise "correct" next state is difficult to accurately predict and can vary wildly from time step to time step as the objects in the system speed up or slow down relative to each other. If the simulation needs to run at a constant rate - which is the case for video games and more broadly most interactive physics visualizations - this is undesirable.
Another consequence is that, by varying the length of the time step with which you are integrating your physical quantities, you can much more easily break conservation of energy or momentum without intending to in the actual calculations you do.
Furthermore, when you only chop up certain time steps it becomes even harder to correct for these two problems. And yet the whole point of only doing so for *some* time steps is that you cannot afford to simply reduce the size of *all* time steps!
Constraint satisfaction will generally provide a more direct mathematical method to finding the state of the next time step, and the right choice of constraints will outright prevent numerical inaccuracies from accumulating from time step to time step. The only potential danger is that, because the "trajectory" of objects during the time step is of no concern, you can technically write a simulator using this approach that allows objects to phase through each other as soon as they're traveling fast enough to enter and exit each other within the same time step! This can usually be solved by taking a small enough time step that nothing ever moves fast enough to cleanly "exit" anything else in a single time step. Constraint solvers being very fast and efficient in general, means you can **in general** shrink your time step further than with an iterative sub-step approach without exceeding your computational budget for the combined steps.
Oooo, a juicy response :D
For your first point of Lagrangian mechanics (specifically the optimisation step) feeling like a hack, is actually physically founded. This video (th-cam.com/video/WFbgfknnWqA/w-d-xo.html) goes over how Quantum Mechanics “considers” all paths at once, and off-optimal paths simply destructively interfere. I have a video planned going deeper into this, but while it was originally a beautiful hack, Lagrangian mechanics is a precipitation of modern physics.
I don’t have too many points to say on computation itself, since I specialise in particle physics, but it’s interesting to know about the approximate models of motion used by different game engines to fit their simulation needs. I’ll leave that or you to make a video on ;)
This was a very nice explaination, I am going to share this with my math club! Thank you, I feel like I actually learned something. Criminally underrated. Also, finally I can write a physics simulation without a DES
This is amazing
Bruh. This is so cool. Also, this is the first video on lagrangians that I can understand. Too bad I'll never be taught it and will have to try teach myself
I genuinely believe that with some practice questions (which you can steal from Lagrangian Mech textbooks), you can fully understand it yourself.
It’s only one topic. You can do this.
@@mindmaster107 🥲thank you
Really interesting - thank you!
Nice video and channel🤩🤩
*Subscribed*
Fantastic explanation.
Read this as 'Lagtrain Mechanics'. Turns out I was wrong (pity; I like 'Lagtrain'), but I'm still interested in seeing how this relates to astonomy.
Waoo... That's great... 👍👍.. kindly make video on Hamiltonian mechanics and rigid body dynamics as well.... Thank you...
NICE ONE MINDMASTEER
Add dissipation of any complexity and crush the Lagrangian/Hamiltonian dreams.
Tho it is still useful for physics in general. You just don't want Hamiltonians that keep coupling systems with each other.
Alas, Friction is what makes all this maths grind to a halt.
Aren’t coupled systems where LM/HM most useful? I guess clever PDE skills would get around those.
@@mindmaster107 I'm thinking of many particles. The collective states get a lot more challenging the more collective they are, because finding a basis gets a lot more challenging. In principle, dissipation is just the result of time dependent coupling of Hamiltonians, so it's not an easy fix for most cases with anything like those terms, taking time into account is fundamental in those cases (there's time ordering explicitly in the time evolution of those systems). Tho, very simple cases even of those are something within reach for a few particles (personally never done anything with more than 100 particles myself, and with very low dissipation at that).
Ooooooooooo ty for reminding me of stat mech
I really wanna make a video on stat mech now
@@mindmaster107 bear in mind that this is more of a simulation centric approach instead of analytical, for the most part.
This was a brilliant video - super engaging! As an educational video creator myself, I understand how much effort must have been put into this. Liked and subscribed, always enjoy supporting fellow small creators :)
I’m going to make another video hopefully in Christmas, so you can look forwards to that then!
It is going to be on Quantum Field Theory, and how the maths is surprisingly accessible.
superb video. got me hooked after only few seconds. subs and like
Very nice video, but I think it would have been worth emphasizing that you're free to choose the scheme (Newtonian or Lagrangian) which is most convenient for the problem at hand. A lot of simulations handle Newtonian PDEs easily enough but would become totally intractable with the Lagrangian.
I think after I make my conceptual videos, I will make videos addressing how to use them effectively.
Right now though, I would rather share the tips I’ve collected over university.
Gonna follow you, good stuff
Well said man!!
Hey guys, did you know that in terms of male human and female Pokémon breeding, Vaporeo-- *head gets chopped off*
(Awesome vid btw!
okey, im falling in luv
you have great potential
Very nice work my dude
I like the video, it gives nice insights on why people thought about lagrangian and why it was defined in this way, but I think the music is too much
Might need to lower the volume whoops
i love this video! Wondering what do you use to animate all this?
Clip studio paint, and Davinci resolve!
@@mindmaster107ah brilliant! Thanks!
How did you get from a=-g to v=sqrt(2gh) at 6:30? Great video!
th-cam.com/video/kwGukY_2qWQ/w-d-xo.html seems to work
Basically using integration to find the motion.
A quick comment before finishing the video, at 3:09. Look at geometric algebra, and the method of actually multiplying vectors. A side note only.
Geometric Algebra offers a third way of Mechanics with a whole new point of view. There is plenty of work covering that, for example by David Hestenes, from the 1960s on.
I plan to make a video on Clifford algebra in physics one day, though I have many other ideas to go through first.
pleasee pleasee make moree videoss!! ❤
Thank you so much!
8:17 iirc, the time-component of the momentum 4-vector in relativity is proportional to energy, which implies the two symmetries (t- and x- independence) that relate to two conservation laws (cons. of energy and cons. of momentum) are simply the time- and space-components of a single *spacetime translational symmetry* that conserves 4-momentum!
I recommend my video on Special Relativity if you want a review ;)
Great video for such a small channel :) Keep it up.
Ty!
Something that keeps me awake at night is: Why L = KE - PE though? How to derive that?
Compared to the Hamiltonian, the Lagrangian gives you more information about the path some particle is going to take. How to explain all that?
The Lagrangian being KE - PE… is because it matches observation lol. Trust me, I tried digging deeper as to why, but aside from the intuitive step in the video, Lagrangian mechanics works because it works. I showed in the video how it reduces to F=ma, so an equivalent question is where does F=ma come from, which is equally as founded.
The Hamiltonian is actually equal in power to the Lagrangian, but easier to calculate for non-relativistic (Special relativity) situations. They both give a global view of mechanics, so I didn’t include it in the video to avoid overloading people.
Good stuff. BTW, “VECTORS ARE CRINGE” should absolutely be your thumbnail :-)
Does this work for somthing with axles gears and motors?
Yes, if you use circular coordinates
It's good!
What happens when space change? With the Lagrangian I say
Geometric algebra gives the best answer I know to how to multiply vectors in way that isn't totally annoying. It's a pity it isn't well known.
Also, I've always been more at home with Hamiltonian mechanics than Lagrangian. Probably because I'm not really comfortable with the calculus of variations. However neither of them can elegantly handle inelastic processes. I'd love to see if there's a similarly general, abstract model for inelastic mechanics that we could draw insight from (and maybe make better physics simulators from too). Thermodynamics is always mucking up our nice theoretical models, so maybe we should invite it to be part of the picture?
Inelastic statistical physics would be one of the most powerful revolutions in physics, you are absolutely right.
As a physicist myself, I pray that a mathematician does the maths for me. I'm not smart enough for that lol
Where do you learn geometric algebra?
@@mujtabaalam5907 I haven't watched all of it yet, but th-cam.com/video/60z_hpEAtD8/w-d-xo.html seems like a pretty good intro. There are also quite a few resources linked from the wikipedia page for geometric algebra.
With the falling ball example, aren't we just doing (total energy before) = (total energy after)? If there's some initial speed, you have more energy budget, so the final speed is going to be faster than the final speed of the first case. If the end is higher than 0 height, that remaining potential energy is using up some of the energy budget, so the final speed is slower.
Where does this weird difference come into it? It's just going to be equal to some number at the start, and some greater number at the end?
The Lagrangian is a mathematical “energy”, which all physical interactions try to minimise.
Imagine a ball is in a gravitational potential. The potential wants to give the ball kinetic energy and pull it down right?
But, the ball can only get this kinetic energy, by actually moving down the potential.
There is a balance between the amount of potential energy that can be given at any point in time, due to the minimum kinetic energy required to move to the position of lower potential energy.
This can be mathematically written as minimising the Lagrangian at every point in space, or minimising the Action across an entire path.
@@mindmaster107, but the example energy method that you said is secretly lagrangian doesn't have us minimising something. just accounting with equalities.
Also, if the kinetic energy increases and the potential energy decreases, wouldn't the lagrangian increase rather than be minimised?
The minimisation for the example question is cleverly avoided, by focusing on only the start and end point. If you want the path between the two, you would need the full lagrangian equation, or to use newton’s equations but thats the cop out.
Also, you might be interpreting the Lagrangian incorrectly. (My bad, i’ll do better next time)
L = KE - PE
For all common situations (excluding wormholes and quantum mechanics), the Action (lagrangian summed over time) stays the same. (Minimised is a can of worms, and I should have made it clear in the video)
L increases as PE converts into KE. Due to energy conservation, the Lagrangian is basically double the KE if the particle came from infinity (PE=0).
Hence, yes, you are correct the Lagrangian increases, but it is the Action that is minimised.
We want the path which has the least total L.
We want the path which has the least total Kinetic energy over the path.
In other words, we want the shortest path.
This is similar to Snell’s law, and should be thought of as such. The minimised action is the shortest (energetically speaking) path from the start to end state.
@@mindmaster107, For the start and end we have (L.start = 0 - PE.start) and (L.end = KE.end - 0). How do we use that to get KE.end=PE.start?
You would use an E-L equation on the lagrangian of the system. I went over that explicitly in the E-L equation section in the video, though solving the 2nd order differential equation requires semi-university maths. ( th-cam.com/video/tFQggKCdt_c/w-d-xo.html is a nice introduction example, though you will need to know calculus)
I understand that this a simplified version to entertain more than anything but the part on the Standard Model is plainly wrong. While you could get away with (with some stomach churning) saying the Dirac kinetic term is akin to the kinetic energy term, you'd also have to interpret at least part of the Yang-Mills piece F_{\mu
u} F^{\mu
u} as the "kinetic piece of the gauge bosons" just like the kinetic piece for the Higgs. Also to say "the rest is just potential" is pretty bad - maybe with some squinting you could away with it but the analogy really falls flat since the classical interpretation of "potential" doesn't really have a quantum mechanical equivalent, especially for the Yukawa terms.
I plan in the deep future to do a full review of the SM Lagrangian, having worked through it for university. I think the Higgs mechanism (and now scattering amplitudes are calculated in general) is so fun to learn.
For this video on LM, three years before anyone would even touch gauge theory, I decided to flubb the details for the sake of a narrative. It was one of the many things I had to do to cut it under 10 minutes.
It’s still lovely to know there is an appetite for postgrad topics though. Thanks for the comment!
The ball doesn't "think" or "want" anything. The laws of nature determine what it does.
It's a good exercise to do so anyways. It helps to place you in the object's perspective to solve problems.
Wah wah poor baby
great video! tiny recommendation, not sure if it was only for me, but the BGM was kinda annoying and doesn't really fit the theme. sorry for the complaining!
It’s a personal touch of mine ;)
It is a bit too loud in this video, so more recent videos might be better for you
Yo wait a second, you are an osu! mapper? i swear dude, all osu! mappers are just geniuses
Oh god don’t tell anyone else lol
Meanwhile, glad you enjoyed my maps!
When I was in school I used to think that vectors are cringe.
Amen brother
@@mindmaster107 time has passed since then, I don't find it cringe anymore, but an extremely useful thing.
Tiny nitpick but at 4:24 isnt the potential energy subtracted from the kinetic energy? You say the reverse in the video
I’m pretty sure it is kinetic minus potential.
yes, you're right. Even though it's correct on the slide, he says it the other way round.
@@marcrindermann9482 In a system where KE = PE of course the diference is 0 (conservarion of energy) more generally you could have KE-PE=H (for heat perhaps).
I'm gonna use Lagrangian mechanics to calculate my sliderball velocities
4:56 this cracked me up.
5:30 this doesn't make sense. The particle will only ever go to one place, but this lets you make it go anywhere.
It should say, given a particle’s start and end point, we can plot out the path the particle HAD to take to get there.
Also, the euler lagrange would have zero solutions if the particle couldn’t make it at all.
Informative lesson. ❤ And I'm stealing your teddy bear.😆 📲
When is the general relativity video coming?Very nice channel.
Next video? Maybe.
Will it be made? Absolutely.