New video: Variational Calculus and Lagrange Equations

[cooperrc]

Variational Calculus and Lagrange Equations

note: AI helped summarize the video transcript, feel free to submit PRs to fix typos/fixes

Variational calculus has solved a number of problems over time. One of the first questions was how to find functions to minimize the brachistrochrone problem.

Have you heard this before? Have you seen brachistochrone solutions? What Differential equation defines the problem?

[ZakMatt0802]

I have heard of minimizing pathways with things like control systems and through the use of state space models although I have never personally heard of the brachistochrone solution specifically. I have seen problems like brachistochrone solutions specifically in trajectory calculations where the intent was to find the fastest pathway to reach a certain location, with the primary differential equation being that of trajectory of projectile. Would something like sliding mode control, or and LQR be effective for brachistochrone determination as well in terms of control or would it be redundant? I see certain scholarly articles that seem to touch on it but not a broader discussion.

[cooperrc]

those control+optimization algorithms are another great technique to minimize functions, but they usually require some type of assumption about functions/data,

• sliding mode is not continuous
• LQR can be continuous/discrete, but its meant to optimize weights of control parameters. this would be like the first case of brachistrochrone optimization where I assume the path is a parabola and optimize the midpoint.

Variational Calculus assumes the functions are continuous and differentiable, but they can be any function. The finite(discrete) version is the weak form where the integrals are evaluated with shape functions.

[danny-kruzick]

I have heard of this exact principle / problem / question a few times. One time would have been during my undergraduate. I can’t remember exactly what class but probably dynamics or differential equations.

In this class we looked at the same derivation we did here but that was a few years back now. It is sort of a breath of fresh air reviewing diffy Q problems solved with integration by parts during this lecture (ultra violet voodoo is such a great mnemonic) and it has been a while since I have applied these.

The differential equations that defines the problem is a second order lagrangian differential equation where inputs of kinetic and potential energy are used to optimize.

The second place where I have seen this problem is on a video on YouTube by vsauce. This guy has awesome physics and mathematical based videos and he breaks down the brachistochrone problem we are faced with here. Definitely worth a watch. https://youtu.be/skvnj67YGmw?si=QjFBZrQV1OSKeDWQ

This videos breaks down snells law which is an interesting application to speed, angle and optimization and how that was used to derive the cycloid or the answer to, what is the fastest time from point a to b for the problem at hand.

From there, vsauce and Adam from mythbusters create a physical experiment to test the theory. They create a few different paths with a rolling ball to test in real life the brachistochrone principle. The results show exactly what they aimed to do, that the shortest path is not always the fastest!

[mohammadmundiwala]

love vsauce and that video . also when i first heard about the brachistrochrone problem

[LaythDuval]

Another great video going over the brachistochrone solution step by step and finding the differential equation that defines the problem:
https://www.youtube.com/watch?v=ovhyyjeAv7U

[shainaerrico]

Thanks for the video! I had not heard of the brachistochrone problem before and this was a great crash course. I found the example of running through mud vs. pavement to be especially helpful. It really illustrated how the principle applies to situations beyond just an object falling under gravity.

[alw14027]

Thank you all for the videos. Some interesting properties I picked up from the video is no matter where on the curve items are released they all end in the same spot and the brachistrochrone curve is always the fastest. Another interesting point is this is a cycloid problem. Reading up into some pros and cons of this approach is that assumption of the following:
Continuous curves
Motion under gravity
No constraints on direction
No stops, turns, traffic, or discrete choices

With this being said it is harder to apply this to something like a delivery schedule or optimization of straight line paths (ex. driving). Another interesting thought of usage was when it comes to re-entry of a rocket ship or aircraft that has no other forces besides gravity acting on this. Rescue teams can often find where the aircraft will land and pick it up as long as no other forces act on it. This really is interesting concept.

[RMoczek13]

I remember watching that vsuace video back in the day! thanks for sharing it. It does such a good job of explaining and showing the problem in real life. I also really enjoyed the video Vsauce linked by 3blue1brown.

[shainaerrico]

Until now, I had not heard of the brachistochrone problem. Although after looking into it (and thanks to Danny's Youtube video shared earlier!), my understanding is that it asks what path between two points results in the shortest travel time. I also realized that an important distinction is that the shortest path is not always the fastest. My initial thought went to a straight line as the solution, which appears to be a common intuition, but turns out the actual solution is a cycloid. The solution initially caught me off guard, but it made more sense after seeing how a steeper drop would allow the object in motion to gain more speed. Despite the longer path distance. Overall, it was very interesting example of variational calculus can be used to optimize time.

[juliamachaj]

I haven't heard of the brachistochrone problem until now either - aside from maybe seeing the headline for the VSauce video, but I don't believe I watched it. Similarly to what Shaina was saying, as I watched this lecture video I had the sense that I have seen similar problems searching for the shortest path, but not necessarily the path that will take the least amount of time. It is interesting to take this into the time domain, and the calculus that is involved in finding the solution. I wonder - what alternative methods exist for solving the problem, and is this simply the most effective? I'm also stuck wondering what other applications might benefit from this kind of an approach and struggling to come up with anything based on my life and industry (aerospace) experience. Open to hearing if anyone else has any thoughts!

[mohammadmundiwala]

this problem specifically may not directly translate to aerospace (althought i bet it does somewhere), but I think the method of solving it using the principle of least action is game changing. An another great youtube video
(https://youtu.be/Q10_srZ-pbs?si=eZVAssM1vTyScUJX) that focuses more on the history and significance of this principle. As this video and its sequel get into, action might be even more fundamental than forces and classical mechanics stuff that we engineers are used to. We take conservation of energy and momentum as obvious truths but its not obvious why they are true

[alw14027]

You mentioned the principle of "least action" and for someone who was not familiar with it I agree and this is specifically driven from the Lagrangian equation that is constraining the output of this differential equation.

[PeterLucido23]

I have not really heard about the Brachistochrone problem prior to this discussion and topic. My general assumption was that the straight line would be the fastest path. It is pretty cool to see how a steeper initial drop that gains kinetic energy outweighs the longer distance of the cycloid. Since I have done some ship control work on submarines with Electric Boat, I am curious if these variational calculus principles are used with the optimization of diving or ascents for submarines to reach specific depths in the shortest times possible.

[jadongs]

To be honest, I had not heard of the brachistochrone problem either. If it ever came up in school, it must have been one of those concepts that flew right past me or was mentioned in passing during my undergrad years. After digging into it a bit more and connecting it with the principle of least action from the video, I can see how this problem could be a foundational example of variational calculus in action.
I definitely felt some déjà vu with the idea of finding the shortest path between two points or optimizing energy in mechanical systems. But the twist here is that the brachistochrone isn’t about minimizing distance at all; it’s about minimizing time, which is a much more interesting way to think about motion. Like others mentioned, my instinct was also that a straight line would be the fastest path. Learning that the actual solution is a cycloid felt almost counterintuitive at first, but it makes sense once you see how the steeper initial drop gives the bead more speed early on.
What really helped tie everything together for me was the Lagrangian perspective from the video. The idea that nature “chooses” the path that minimizes the action made the brachistochrone feel like a natural example of that principle. It also helped me see how the same mathematical framework shows up in pendulum motion, harmonic oscillators, and even some of the optimization problems that appear in systems engineering.
Like others have said, I’m curious about alternative ways to solve the brachistochrone problem. The calculus of variations seems like the most general method, but looking at the bigger picture, it’s interesting to think about how this approach might show up in robotics path planning or aerospace trajectory optimization. I’m not entirely sure how to implement something like this in practice, but it definitely seems relevant.
Overall, it’s been interesting to see how this classical problem connects directly to the broader idea of least action. It’s one of those concepts I somehow missed before, but now that I’ve seen it, I’m noticing echoes of it in a lot of places I didn’t expect.

[alw14027]

Prior to this class I have not heard of brachistrochrone problems but I have heard of similar problems where you are determining the shortest path from point A to point B. Because I have not worked with this problem before I am not familiar with its solution but I have done LaGrangian equations before in my math courses in undergraduate where we learned the following type problems: LaGrange Problem.

The first-order differential equation defines the brachistochrone curve and it is derived from a Lagrangian equation (which is typically second order). This is concept is important as Lagrangian although not a motion problem plays a key role in problems like the design of a rollercoaster. By applying this amusement parks can reduce the time it takes for each ride, increase the number of rides and hopefully profitability of their built. Also want to add since many have asked about aerospace. Here is a link of one example and explanation.

Some math behind this concept.

[jhngms]

Like some folks have already said, my only experience with the brachistochrone problem is through a YouTube video by Vsauce and Adam Savage. Albeit this was very long ago so I didn’t remember much..

This problem is a great and “simple” way to think about optimizing the travel time between two points using variational calculus. While I don’t think the brachistochrone problem pertains to my line of work directly, concepts of using variational calculus for optimization definitely do. Many military jet engines use variable vanes. These are designed to open/close depending on flight condition in order to boost engine efficiency. I feel an approach of time optimization can/is used to design the kinematic systems of the variable vanes, allowing for the quickest transition between transient engine conditions.

Thanks,
John

[Afazzino2026]

I have not come across brachistochrone solutions prior in depth, but have heard of them. Personally I found it a bit tricky to follow through the derivation and would love to see visual representation side by side with the derivation with things color coded to what they all are for clarity. These solutions have many applications I am considering, such as when we would see this play out naturally. I was thinking about what situations would cause the fastest point of contact to happen before the mid-point and how that gets shaped or if it is even really possible in nature without human interference.

[joshuajimenez1]

I haven't heard of the name before but I have seen the visual representation of a solution of a ball going down a straight slope versus a curved line and the curved one reaches the end faster. This proves the shortest path is not necessarily the fastest path. The problem is made by minimizing a time functional and leads to Lagrangean differential equations. It starts minimizing time using difference between kinetic and potential energy and involving arc length. The resulting solution comes from deriving second order differential equations and it is not an intuitive result and is important in the development of variational methods.

[ericbrown1118]

I have heard of LaGrangeian before but I haven't encountered them until now academically. Similar with the brachistochrone curve, I have heard about it before and I knew that it was the path of least time but up until now I had no idea how it was discovered. This is incredible useful as now we can build systems that can be 100% optimized on paper at least, though i do wonder if LaGrangeian mechanics can be applied to physics outside of kinematics?

[matlabkid]

I found the lecture quite interesting for variational calculus. There should be many applications for this in industry such as projectiles with guidance, and aerospace trajectories.

How to find functions to minimize the brachistrochone problem would be to assume multiple paths or variations.

I have heard of this before on social media through various STEM accounts, and went over this in undergraduate during a matlab course, but didn't use the Euler-Lagrange differential equation more till grad school.

[LRand7]

No I haven't heard of the brachistochrone problem, the lecture was interesting to me since I do enjoy calculus and seeing different ways of it applied is interesting. I looked it up and saw some interesting videos about it that I might watch to learn more about it.

[whteCobra]

I have heard about this problem in a couple of places, maybe not by name exactly but the theory of the fastest decent between two points. It has come up in physics and my undergraduate dynamics course but only in a quicker overview.

In that case I probably have seen the solution and it looks familiar but might need to jog my memory a bit to really bring it back.

I have seen this problem before from YouTube videos and clips of stem related videos. It is a interesting topic because it challenges you to think critically.

[cjdipietrantonio]

Like many others in this feed, I have heard of the brachistochrone problem from the Vsauce video. But, it had been a while since I had watched it, and revisiting the video after the lecture definitely gave me a greater appreciation for the problem than the first time I watched. The brachistochrone problem starts with the time functional; applying the Euler-Lagrange equation to this functional produces a second-order non-linear ordinary differential equation, the general solution to which is known to be a family of cycloids. The particular solution (depending on your boundary conditions/endpoints) is the brachistochrone curve we're all now familiar with. If we continued the example we started in lecture, we would find the following defining second-order non-linear ordinary differential equation:

$$ f'' = \frac{1 + (f')^2}{2(h-f)} $$

Again, this ODE is known to have a general solution of a family of cycloids that can be written parametrically as:

$$ x(t) = r(t - \sin(t)) + C_x $$

$$ y(t) = r(1 - \cos(t)) + C_y $$

However, when we substitute our boundary conditions, we're left with a transcendental equation. Nevertheless, we can write a quick script in Julia to numerically solve for the solution to our particular brachistochrone problem (h=1,L=1). After doing so, we can create the following plot of the brachistochrone curve for the example we started in class!

Finally, although I had not heard of the Calculus of Variations prior to this course, I can see how it may be applicable in other cases, such as control theory, to optimize a cost function!

[HerminioV]

I had heard of the Brachistochrone problem before, but I never really understood how it could actually be approached mathematically until learning about the Calculus of Variations and the Euler Lagrange equation. Before this, I probably would have guessed that the fastest path between two points would just be a straight line because that seems like the most direct route. What makes the problem so interesting to me is that the actual solution is not the shortest distance, but the path that minimizes the travel time by allowing the object to speed up more quickly at the beginning.

I was also really intrigued by how the problem can be reformulated so that we are not just comparing one or two paths, but thinking about all possible paths at once. That idea was surprising to me because it shows how powerful the Euler-Lagrange equation really is. Instead of guessing and checking, it gives a systematic mathematical way to determine the optimal curve. Overall, I think the Brachistochrone problem is such a cool example of how math and physics work together, and it definitely helped me better understand the difference between minimizing distance and minimizing time.

[erick220]

I’ve heard of the brachistochrone problem before. It’s about finding the path that lets an object move from one point to another in the shortest time when it’s only affected by gravity. At first, it seems like the straight line would be the fastest since it’s the shortest distance, but that’s not actually true. The actual answer is by using a cycloid. It works better because the object drops more steeply at the beginning, so it speeds up faster and keeps that higher speed for most of the path. From what I understand, you solve this using variational calculus by trying to minimize the total time it takes to travel the path. When you apply the Euler–Lagrange equation, you end up with a differential equation. I haven’t gone through every step of the derivation recently, but I get the general idea of how it works. I think this problem is a good example of how physics and math can go against your first guess. It shows that the shortest path isn’t always the fastest, and it also helped me understand that variational calculus is more about finding the best overall function instead of just solving for one number.

[ktburnash]

Before watching this video, I had never heard of the Brachistochrone problem. It seems logical to assume the shortest distance would also take the least time, but this problem shows that isn’t always true. A steeper initial drop allows the object to gain speed more quickly, which ultimately reduces the total travel time. The video also introduced how variational calculus is used to minimize time by finding an optimal function. Using the Euler–Lagrange equation helps determine the differential equation of the path that results in the shortest travel time.

[Michaelben2004]

I have heard of the brachistochrone problem before. It’s about finding the curve that lets an object move between two points in the least time under gravity, not the shortest distance. The solution is a cycloid, which is a curved path, not a straight line. This is what makes the problem interesting because the fastest path actually dips down first and then comes back up. The differential equation comes from using variational calculus to minimize the time it takes to travel along the curve. So instead of solving a regular equation, you’re finding the shape of the function that makes the time as small as possible, which leads to that cycloid result.

[AidanJak]

I have not heard of the brachistochrone problem before this lesson. At first, like most other people have said, I would have assumed that the fastest path between two points would be a straight line since that is the shortest distance. However, after learning about the topic, it was interesting to learn about how, when gravity is introduced, that including a steep drop in the beginning would allow the object to speed up faster and reduce the travel time.

This problem is solved through variational calculus to find the function that gives the shortest time. Then, the Euler-Lagrange equation is applied to find the differential equation which defines the curve. It was interesting to find out that the curve is a cycloid.

[AuZwick]

I’ve heard of the brachistochrone problem before—it’s the problem of finding the curve that lets a particle travel between two points in the shortest time under gravity. The solution is a cycloid, which is interesting because it’s not a straight line.

The differential equation comes from applying the Euler-Lagrange equation to the time functional. In general form, it is:

d/dx (∂f/∂y') - ∂f/∂y = 0

For the brachistochrone specifically, this leads to a second-order differential equation like:

2y y'' + (y')^2 + 1 = 0

This equation defines the shape of the curve, which ends up being a cycloid when solved.

[gwgezas123]

I knew about the brachistochrone problem not by name but when I was a kid we built it in my Sunday math school. I knew how it worked but not the math behind it. The idea is that you minimize the time functional T = integral(ds/v) and then do an energy conservation v =sqrt(2gy) and then solve the integral. It is kinda cool when your intuition isn’t right and the math supports it!

[bliew52]

I didn't recognize what the brachistochrone problem was initially. Researching it now, I remember my high school physics teacher showing us a demo of something similar. Intuitively, I believed that the line with the shortest distance would travel the fastest, but my thought was quickly disproven once the demo started. I don't recall my physics teacher explaining how to design a curve with the fastest time, but it's really interesting to see the many solutions to this problem.

[jaydawg-young]

Yeah, this is one of those classic problems that basically kicked off the calculus of variations. The brachistochrone problem was originally posed by Johann Bernoulli, and it asks a pretty simple question: what path gets you from one point to another in the shortest time if you are just sliding under gravity? At first, you would probably think the straight line is fastest, but that is actually wrong. The real solution is a curve called a cycloid, which drops down steeply at the beginning so the object speeds up quickly, then gradually flattens out. It sounds a little unintuitive, but it works because you gain speed early and carry it through the rest of the motion. What makes this problem important is how it is solved. Instead of solving for a single value, you are solving for an entire function, meaning the shape of the path itself. That leads to a differential equation that describes the optimal curve. That idea of optimizing over functions instead of just variables is basically the foundation of variational calculus. So it is not just a neat physics problem, it is one of the first examples of a much bigger idea that shows up all over math and engineering.

[agambhir-uconn]

It is a strange thought that curiosity over a problem as seemingly benign as the brachisterone lead to the development of a whole subfield of mathematics. But the solution to this kind of question is practical for so many more applications related to optimization. I appreciate that we also started learning about the math involved with this type of investigation in our work in the course, bringing it full circle. I had not known about the brachisterone problem but quickly went to watch the YouTube video on the subject that others have mentioned. As I mentioned in another comment, I think it would be neat to see solutions to this problem animated in Julia in future iterations of the course.

[HCastillo03]

I knew what the concept of the brachistochrone was before but never knew the actual name for it. I had seen examples where a curved path gets an object to the bottom faster than straight paths even though there was more distance involved. Now I know the Euler-Lagrange equation defines this problem, and solved by minimizing time using variational calculus. The equation solution when solved is a cycloid, which I find really interesting that less distance does not necessarily mean a faster time in this scenario.

[bgoncal22]

Variational calculus is something I have heard of before, but I had not really looked closely at the brachistochrone problem until this topic. The brachistochrone problem asks what curve allows an object to slide from one point to another in the shortest amount of time under gravity, assuming there is no friction. At first, I would have expected the answer to be a straight line because that is the shortest distance, but the fastest path is actually a cycloid. This makes sense because the curve drops steeply at first, allowing the object to gain speed quickly, and then uses that speed to travel across the rest of the path. The differential equation that defines the problem comes from minimizing the total travel time using the Euler-Lagrange equation. In words, it relates the shape of the curve to the slope of the curve so that the travel time is minimized. Solving that relationship leads to the cycloid path. This shows how variational calculus is useful because it does not just find a shortest distance, but instead finds the function or path that best satisfies a physical condition, such as minimum time.

[cbull0]

I haven't heard of the brachistochrone problem before this class, but I did explore a similar topic in a high school physics class lab. I would have assumed the fastest path between two points would be a straight line, but the solution is actually the cycloid which is a curved path. The problem is solved using variational calculus and the Euler–Lagrange differential equation to minimize travel time under gravity. I think it is interesting because it shows that the shortest distance is not always the fastest path.

[tjs21014]

Before this lesson, I had seen the brachistochrone problem mentioned in videos and examples but never worked through the actual theory behind it. Like many others, I initially assumed the fastest path between two points would be a straight line. However, learning that the optimal path is a cycloid helped clarify the difference between minimizing distance and minimizing time. Using variational calculus to define and minimize a time functional made it clear why the solution is not intuitive. A steeper initial drop allows the object to gain speed quickly and maintain a higher velocity, which leads to a shorter overall travel time despite the longer path. Applying the Euler–Lagrange equation to this time functional results in a nonlinear differential equation whose solution is the cycloid. Overall, this was a great example of how variational calculus and Lagrangian methods are used to solve real physical optimization problems, and it helped connect energy-based modeling to broader ideas like least action and optimal paths.

[Coen-uc]

While I have never seen it called the"brachistochrone," I have seen the clip from VSauce where he experiments on the fastest path of a ball down a slope with Adam Savage many times on Instagram. This path is called the cycloid, and it can be defined using the differential equation below:

$$y(1 + y'^2) = C$$

[JulianPojano]

The brachistochrone problem asks what curve allows a particle to slide between two points under gravity in the least possible time. While a straight line seems intuitive, the true solution is a cycloid, the curve traced by a point on the rim of a rolling circle. The result comes from balancing steep initial descent, which builds speed quickly, against excessive path length.
I had a little familiarity with this before, but the Vsauce video other students were discussing helped make the geometric intuition much clearer.
The problem is solved using variational calculus by minimizing the travel time functional. Applying the Euler-Lagrange equation from the lecture video to this functional yields the governing differential equation:

$y(1 + y'^2) = \text{constant}$

This emerges from taking the first variation, integrating by parts, and requiring the bulk terms to vanish. The brachistochrone remains one of the earliest problems that motivated variational calculus as a formal discipline.