Least action and numerical integration

[cooperrc]

Numerical integration and principle of least action

I ended up recording x2 sessions for this topic because I ran into numerical instabilities for the brachistochrone differential equations. Sometimes its a nice reminder that no matter how long you've been using these tools, you can hit an error that you didn't expect.

Try your own functions for the harmonic oscillator solutions to see what changes in total action are here,
06_least-action.jl notebook
https://github.com/cooperrc/dynamics/blob/main/tutorials/06_least-action.jl
paste the link into your Pluto notebook and paste the describe the result here.

[ZakMatt0802]

I tried a number of different functions with the same initial conditions that used the same IC, that being several different harmonic expressions as well as an exponential function just to see how that would impact the calculation for the integral of the action. Here are the basics of what I had found:

• The exponential function experienced the largest area for the action integral which is unsurprising but I ran it just to test for fun.
• When introducing more non harmonic elements to the system this generates much more area (again makes sense with the general motion we are observing)
• While other functions can fit to this desired initial conditions and can have a harmonic motion but the complexity of that motion can greatly increase area for the L curve. Simple harmonic motion assists in keeping this lower but I am sure there are probably exceptions.
• The smallest area I obtained with functions that were not just another combination of pure cos() and sin() was A*cos^2(wt), which had a value of ~31 for its area and I can only assume that is because the general motion is not inherently complex for x of v compared to the other functions I had looked at.
  Regardless it was a fun exercise and it was fun to play around with the function to see the different responses. I imagine there are probably other functions that you could get much closer and I am interested to see what others come up with.

[LaythDuval]

I additionally used A*cos^2(wt) to gave the smallest action (about 31), likely because its position and velocity remain relatively smooth compared to the other trial paths.

[LaythDuval]

I tested several different trial functions that all used the same initial conditions as the harmonic oscillator and compared their total action by integrating the Lagrangian. As expected, the exponential function produced the largest action, which makes sense since it deviates strongly from the system’s natural motion. Adding non-harmonic features to otherwise oscillatory functions also increased the action noticeably, showing how sensitive the action is to added complexity in the motion. it was interesting to experiment with how different choices affect the result.

[Afazzino2026]

I keep getting an unavailable server issue, and cant open the notebook. Has anyone else gotten this or have ideas how to fix it?

[cooperrc]

started a new thread, #6

[cjdipietrantonio]

Instead of trying a handful of functions to test the principle of least action, I decided to automate the approach to this problem using Julia. I started with the analytical solutions in the lecture:

$$ x_{an} = A\cos{\omega t} + B\sin(\omega t) $$

$$ v_{an} = \omega \left(-A\sin(\omega t) + B\cos(\omega t)\right) $$

I then created a general form for a trial path that adds a scaled perturbation onto the analytical solution while still matching its boundary conditions:

$$ x_{trial} = x_{an} + a\sin\left(\frac{\pi t}{T_{total}}\right) $$

This works because the sine term is zero at the initial time, $t = 0$, and the final time, $t = T_{total}$, so every trial path will equal the analytical solution at those endpoints. Next, I created 101 equally spaced values of $a$ between -1 and 1, which produces a family of different trial paths. Graphically, this looks like:

For each trial path, I then calculated the velocity, which allowed me to find the lagrangian. Finally, I numerically integrated the lagrangian, as shown in the lecture video, to find the action for each trial path. Plotting the action for each trial path versus its respective scalar, $a$, gave the following plot:

According to the principle of least action, we would expect that the action is minimized for the analytical solution $\left(a=0\right)$. However, my plot shows that although the action of the analytical solution is closest to zero, it seems to be a local maximum. I originally thought I had made a mistake, but upon further research, I learned that the particular solution/physical path occurs where the action is stationary (i.e., stops changing to the first order when you perturb the inputs), which could be a local minimum, maximum, or inflection point. This is also consistent with the negative action we were seeing in the lecture video. I haven't yet had a chance to investigate what affects the second order behavior of the action, so if anyone has any insights on this, let me know!

[alw14027]

Based on my research I also agree that the principle of least action does not necessarily mean "minimize" but rather stabilized or stationary. In this case whether a local min or local max point these are locations in which the overall function is stationary. Looking further into the secondary order of this same scenario I was able to find that determining local max/min/inflection point is dependent upon potential and kinetic energy. More kinetic energy would lead to more of a maximum, more potential would lead to more of a minimum and inflection point I would assume is where these balance out. However, this does not stop there we also need to understand the time interval. In a shorter time there might be a minimum but grow to maximum. Very interesting concept you brought up. If anyone else has insight please let me know.

Here are a couple of videos to explain this point and further shed light:
https://www.youtube.com/watch?v=fi5uQdoE49c
https://youtu.be/M05ixbSOY80

[Coen-uc]

I have not put as much effort into this as either of you has, but as you both said, you are seeking stabilization rather than a minimum. In the plot of the Action for trial paths, I see energy entering or exiting a system, not a minimum/maximum. I hope that I make sense, and I am nervous that if I try to explain my thinking further it will reveal how little I know.

[jordanbollacke]

I tested out 2 different trial functions that involved slightly tweaking the original functions while continuing to satisfy the initial conditions. I found that the closest solution I got to the original was an action value of -.544, while the original value we found was -0.588. I did this by using the following functions for x and v in two different trials:


^Where the "Analytical Solution" is from the original trial

I plan on testing more variations to see if I can get closer to the original action, although I know that it will not be the exact same. I'm interested to see what other functions people have tested, and let me know if you get similar results!

[snowman383]

I started testing various mathematical functions, all from the same initial conditions, to see how they affected the calculated value of the action integral.

The key results were:

The exponential function produced the largest action integral area. This was an expected outcome and was primarily tested out of curiosity.

In general, introducing more non-harmonic elements into a system's motion tends to significantly increase the area under the action curve.

While many functions can be made to fit the required starting conditions, those describing simpler, more harmonic motion like basic combinations of sin and cos usually result in a lower action integral. Increased complexity in the motion usually leads to a larger area.

Among the non-pure harmonic functions tested, A*cos²(ωt) yielded the smallest action integral area which I got around 30 for. This is likely because its resulting motion for position and velocity remains relatively simple compared to the other trial functions.

Overall, the exercise demonstrated how the choice of function influences the action and highlighted that simpler harmonic motions tend to minimize it. There is interest in exploring other functions that could potentially achieve an even lower value.

[PeterLucido23]

I experimented with several different functions that fit the initial conditions, including an exponential function and a few non-harmonic expressions, to see how they would impact the total action. While intuition suggests that simpler motions stay closer to the minimum, I was surprised to find that A*cos^2(wt) yielded one of the lowest areas, likely because its velocity and position curves remain relatively 'smooth' compared to the others. It’s been cool exercise in seeing how adding even small complexities to the motion starts to increase that action integral, confirming that the physical system really does prefer the most efficient path.

[JulianPojano]

I experimented with adding a high-frequency wiggle to the harmonic oscillation. I found that increasing either the frequency or the amplitude of the wiggle caused the total action to increase. This makes sense physically, because the additional fluctuations push the system away from the ideal path that satisfies the equations of motion. The faster or larger oscillations increase the velocity, which raises the kinetic energy. This larger kinetic energy disrupts the balance between kinetic and potential energy, resulting in a higher overall action. Its interesting to see that the ideal motion of an object could essentially be verified numerically.

[danny-kruzick]

It’s interesting that even though we approached it slightly differently (you explored frequency and amplitude, while I  varied amplitude), we both saw that any deviation from the smooth harmonic solution increased overall action. When you think about it that does make sense. Kinetic energy depends on velocity squared, even small added oscillations raise the kinetic energy significantly, which then increases the overall action. Cool stuff!

[matlabkid]

I experimented with different functions without changing the initial conditions, and found that the total action always go back a forth if iterate from a range of values. I did not choose to include any exponentials and found that the action always got higher from where ever I added more terms and values. I find it interesting that less complex is more efficient. Also, I found it interesting for using Julia for the first time since I usually use Matlab, love how linear algebra is very efficient like Fortran.

[med17018]

We know that the least action curve comes from solving the Euler-Lagrange differential equations for the action integral of kinetic minus potential energy. The differential equations simplify to trig functions representing oscillating curves. Knowing this, we can predict how the action integral will change in correspondence with manipulating the oscillation functions.
For example, with the help of @jordanbollacke’s post, I was able to figure out how to print multiple integration action values at once to more easily view various studies. I experimented with many harmonic inputs. In the first study I increased the value of the second term, but as you can see, it had no effect on the integral area. It most likely just shifted the integral but still maintained the value of the area. In my second study, I tested incorporation of an exponent. I found that if I changed both position and velocity equations, then there wasn’t much effect of the integral area. However, if I only changed one, there was significant impact. Finally, in my third study, I increased the value of the first term. This would have made the amplitude of the integral larger, therefore increasing the area under the integral.

[danny-kruzick]

To investigate how the total action changes here, I worked with a third-harmonic dynamic change to the analytical harmonic oscillator:
xtrial​(t)=xtrue​(t)+a(cos(3ωt)−1).

This formula keeps the initial position and velocity while introducing a higher-frequency oscillation (a “wiggle”) on top of the true solution. I then looked at the total action for several values of the amplitude parameter a.
The true solution produced an action of approximately
Strue​≈6.66×10−16,

which is effectively zero. This confirms that the analytical solution is correctly implemented. Woo hoo!
As the magnitude of a increased, the change in action ΔS increased smoothly and significantly:
a=0.05→ΔS≈0.047
a=0.1→ΔS≈0.188
a=0.2→ΔS≈0.754
a=0.4→ΔS≈3.016
a=0.6→ΔS≈6.786
The second plot (ΔS vs a) shows a clear nonlinear, perhaps quadratic growth in action as the changed amplitude increases. This shows that even small deviations from the true trajectory increase the total action, and larger deviations increase rapidly.
The oscillator trajectory plot helps visualize what is physically happening. As the value of my input a increases, the motion develops sharper dips and additional curves compared to the smooth sinusoidal motion of the true solution. 
The results show that the original smooth harmonic motion has the smallest action compared to the modified motions I tested. When I add an extra oscillation, the system moves more sharply and with higher velocity, which increases the kinetic energy and therefore increases the total action. Both the numerical results and the plots show that the true motion is the most “efficient” path within this group of trial functions.

[bp-thibodeau]

As I simple change, I removed $\omega$ from the $v_{an}$ equation. The KE, V, and action all change as shown. The integrated action takes on a large negative value. I played around with other functions and also saw the magnitude of the integrated action grow.

[joshuajimenez1]

When I introduced a slight perturbation to the true harmonic oscillator solution by adding a small extra term with something like x(t)=2cos(wt)+0.2sin^2(wt), it the motion still satisfied the same initial conditions x(0)=2 and v(0)=0, but its overall behavior changed noticeably. One trial function started at 2 with a smaller effective amplitude than the original cosine solution, while another variation began at 0 and resembled a reflected sine curve with much larger peaks reaching approximately + or - 20. Although these perturbed functions can look dramatically different, especially in amplitude, the key result is seen in the total action value which gave an output action of about −0.34 differing from the original solution’s value of -0.58. This demonstrates the principle of least action where even small deviations from the true cosine solution change the balance between kinetic and potential energy over time and change the integrated Lagrangian. The original harmonic oscillator solution is special because it minimizes or more precisely stabilizes the action and any small perturbation, even if it satisfies the same boundary conditions, results in a different total action, confirming that the original solution is the physically correct and natural most efficient path for the system.
http://localhost:1234/edit?id=369d51ce-087d-11f1-9f62-73bc111bc320#

[cooperrc]

great summary here, one note: the localhost is for your computer. We can't see the notebook. You can put it in a gist if you want,

[LRand7]

I tried a number of different trial functions that satisfied the same initial conditions, including several harmonic expressions as well as an exponential function just to see how it would affect the value of the action integral. I then compared the total action by integrating the Lagrangian over the same time interval.

As expected, the exponential function produced the largest value for the action. That wasn’t surprising, since its behavior deviates strongly from the natural oscillatory motion of the system, so the kinetic and potential energy terms don’t balance in the way they do for true harmonic motion.

[whteCobra]

I experimented with several trial functions using the same initial conditions from the harmonic oscillator to see how the total action changed when integrating the Lagrangian. Like others have observed, the functions that deviated more from harmonic motion tended to produce much larger action values.

For example, I tested a few polynomial paths, and all of them gave noticeably higher action values compared to simple sinusoidal functions. The exponential version consistently produced one of the highest action integrals in my tests as well.

Similar to what others found, the lowest action I obtained outside of sin/cos combinations came from a smooth, low‑complexity function—A·cos²(ωt). Its relatively gentle variation in position and velocity seems to help maintain a smaller integral overall. It was interesting to see how even small changes to the assumed trial path can shift the action quite a bit.

[NiFoley]

I tried a few different functions and also found that a combination of harmonic functions sine and cosine were the closest to least action which should be expected give the absence of damping from the equation of motion the general solution to this is X(t)=acos(wt) + bsin(wt) however the principle of least action gives a new prospective on why specifically there is only one solution when seemingly infinite solutions fit the initial conditions.

[juliannnatran]

I tested different trial functions for the harmonic oscillator to see how they affect the total action. The principle of least action says the true motion makes the action stationary, so I compared different paths to the sinusoidal solution. When I used functions close to the true sine-wave motion, the action stayed relatively low. As I changed the function shape by adding distortions and changing the curvature, the total action increased. It shows that small deviations from the correct trajectory lead to a higher action value.

[willhennessey428]

Using the Pluto.jl notebook I found that no matter what function was input, with the same initial conditions, the original function for v_an and x_an provided the smallest total action. Both when adding/removing different terms from the function the total action, while keeping the same initial conditions, the total action always increased. Plugging in various equations that others provided above also proved the same. Small deviations from the original equation will cause a larger action value. In dynamic systems we can infer that an equation with the smallest value of total action will be the most accurate representation of the equations of motion.

[AuZwick]

I tried changing the harmonic oscillator function by adjusting the amplitude and frequency. When the trial function was close to the expected sinusoidal motion, the total action stayed lower. When I used a function that was less physically realistic, the total action increased.

This helped show the main idea of the principle of least action. The actual path taken by the system is the one that makes the action stationary compared to nearby possible paths. I also noticed that small changes in the function did not always cause huge changes, but larger changes away from the natural motion made the action noticeably different.

This graph shows how the total action changes as the amplitude of the trial function varies. The minimum occurs near the physically realistic motion, while values farther from that increase the total action.

[AidanJak]

I tested a few different trial functions for the harmonic oscillator while keeping the same initial conditions and then compared their total action. I noticed that the analytical sinusoidal solution produced the smallest action and when I modified the waveform or added extra sine or cosine terms, the total action increased. My observations were similar to some of the other responses, especially how the functions deviating from the simple harmonic motion led to significantly larger action values. I also noticed how the closer the function was to sinusoidal form, then the smaller the total action would be.

[agambhir-uconn]

To begin investigating this problem, I decided to focus my efforts on one particular function--the cosine function. In particular, I investigated how changing the frequency of the function effected the action. I began my testing a number of discerete frequencies. After a while, I decided to utilize the copilot tool to help parametrize the investigation. The following plot visualizes the results of testing many cosine functions that each had a frequency that was a scaled version of the original frequency. The best action was achieved with the original frequency. scaling the frequency up tended to increase the action while scaling it down lowered the action.

[jaydawg-young]

I tried working through the notebook and playing around with the harmonic oscillator part, and I ran into something pretty similar. Even for a system that is “simple” like a harmonic oscillator, once you start modifying the trial functions, the behavior of the total action can change a lot more than you expect. When you plug the notebook from Pluto.jl using the file from 06_least-action.jl, the baseline solution that satisfies the boundary conditions gives the minimum action, which is exactly what you would expect from the principle of least action. But when you start trying your own functions, like adding small perturbations or using slightly different shapes between the endpoints, the total action always increases compared to that optimal path. What stood out to me is how sensitive the action is to those changes. Even small deviations from the true solution can noticeably increase the action, which really reinforces the idea that the physical trajectory is not just any solution, it is the one that minimizes the functional. It also makes it clear why numerical issues can show up. If your trial function or discretization is not well behaved, the computed action can jump around or give unstable results.

[cbull0]

I used the Pluto notebook to test a few different functions for the harmonic oscillator while keeping the same initial conditions. No matter what function I tried, the original harmonic oscillator solution for x​ and v always gave the smallest total action. When I added extra terms or changed the shape of the function, the action value increased, even when the changes were small. Some functions stayed closer to the normal sin/cos motion and had lower action values, while functions that varied more from harmonic motion had much larger values.

[tjs21014]

For this exercise, I used the provided notebook to test several trial functions that satisfied the same initial conditions as the harmonic oscillator and compared their total action by numerically integrating the Lagrangian. Like many others found, the analytical harmonic oscillator solution consistently produced the smallest action. When I tested functions that deviated from simple harmonic motion such as exponentials, polynomials, or added high‑frequency “wiggles” the total action increased noticeably. This makes sense physically since added complexity increases velocity variations and therefore raises the kinetic energy term which accumulates in the action integral. Even small perturbations that visually look similar to the true solution still resulted in larger action values. One interesting result also seen by others was that smooth and low‑complexity alternatives like Acos⁡2(ωt)A\cos^2(\omega t)Acos2(ωt) produced relatively small action compared to most non‑harmonic functions likely because the motion and velocity remain fairly smooth. Overall, this exercise made the principle of least action much more concrete by showing numerically that the physical trajectory is the one that the action is stable around while any deviation leads to a higher action.

[GoateT33]

It’s always a bit of a reality check when even the Professor runs into numerical instabilities, but it highlights how sensitive these solvers can be to the physics we’re modeling. For this exercise, I played around with a few different trial functions in the Pluto notebook to see how the total action (the integral of the Lagrangian over time) responded compared to the "true" path of a harmonic oscillator.

I started by testing a simple linear path and a cubic function that satisfied the initial and final boundary conditions. As expected, these paths yielded significantly higher action values than the standard sine and cosine solutions. It really illustrates the Principle of Least Action in a tangible way—nature effectively "chooses" the path that minimizes this integral. When I strayed from the natural frequency of the system by injecting higher-frequency harmonic terms (like adding a sin(5wt) component), the kinetic energy terms spiked, causing the total action to climb rapidly. It’s a great visual reminder that any "unnecessary" motion or extra curvature in the path adds a penalty to the action.

The most interesting part was seeing how small variations in the trial function could lead to vastly different results in the action calculation. I tried a damped harmonic expression just to see the effect, and while it stayed closer to the true path than the exponential function, the "cost" in terms of action was still notably higher than the undamped, natural oscillation. It’s fascinating to see that while many paths can technically connect point A to point B, the physics of the harmonic oscillator is perfectly tuned to find that one specific "path of least resistance" where the difference between kinetic and potential energy is optimized over time.

[Coen-uc]

I tested doubling the frequency of the analytical solution. The true solution had a total action very close to 0, which is anticipated. When I doubled the frequency, the mass moved faster, so the total action went up to 379 because there is more kinetic energy.

[dugganjake]

I tried messing around with some different functions but didn't change the start and end points. I mainly noticed the action seemed to be smallest when the function closely resembled the normal harmonic oscillator motion.

With functions that are more curved (exponential looking) the action increased. I think this is because those types of paths don't jive with the path the system wants to take. I also saw that small changes around the expected solution didn't change the action as much as I thought it would. This sort of made the least action idea click, since it's more about the real path being the closest to the most natural path compared to the other nearby ones.

[HCastillo03]

I used the notebook to test damped trial functions and compare it to the actual oscillator solution. I expected to see that changing the motion would increase the action compared to the true solution. After running the calculations, the result showed the paths having a larger action than the oscillator. The added damping changed the shape of the motion over time and reduced the range of the oscillations. The motion remained smooth, but no longer had the same behavior as oscillator. The trials show how changing the motion affects the position and energy balance compared to the actual solution.

[HerminioV]

I tried testing a different trial function while keeping the same starting conditions, then compared the action value to the original harmonic oscillator solution. The numerical solution lined up really well with the analytical solution, which helped confirm that the ODE solver was giving the expected response for the equation

x=−k/mx

The position plot showed the normal sinusoidal motion, and the velocity followed the same harmonic pattern but shifted in phase. Looking at the energy plot also helped because the kinetic and potential energy kept exchanging over time. When displacement was largest, the potential energy was highest, and when the mass passed through equilibrium, the kinetic energy became highest. For the trial function part, I noticed that changing the function changed the action value, even when the initial conditions were still satisfied. The action was calculated from the Lagrangian,

L=T−V

so any change in the shape of x(t) and v(t) affected the total integral of the action. The value I got for the alternate function was close to the original, but not exactly the same. That made the least action concept make more sense to me because the real oscillator path is not just any path that works mathematically. It is the path that makes the action stationary compared to nearby possible paths. Overall, seeing the numerical solution, energy exchange, and action calculation together made the concept a lot easier to understand.

[franbelle]

Similar to my peers, after testing the harmonic oscillator with the same initial conditions, the original harmonic oscillator solution for x and v produced the smallest total action. After adding extra terms or modifying the function shape, the total action increased. I observed that the functions that deviated from the simple harmonic motion led to larger action values. The total action would be smaller depending on how close the function was to the sinusoidal form. Changing the terms of the function shows us the relationship between change in motion and the position and energy balance in relation to the actual solution.