No dissipative Lagrangians (or Hamiltonians)
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- เผยแพร่เมื่อ 30 มิ.ย. 2024
- Can Lagrangian (or Hamiltonian) mechanics really describe dissipative forces? What about time varying Lagrangians/Hamiltonians? In this video we show how some Lagrangians/Hamiltonians recover the kinematics of a dissipative system, but not the dynamics.
#classicalmechanics #lagrangian #hamiltonian #newtonianmechanics #determinism #dissipativeforces
Thanks to Phillip Kaufman for help with the slides! These are results from our research project Assumptions of Physics. More details on this topic in our open access book: assumptionsofphysics.org/book/ . If you are interested in our active research, see our other channel www.youtube.com/@AssumptionsofPhysicsResearch .
How about electrical circuits with resistive elements?
"You can have the same equations but the system is different. Equations by themselves are not enough to tell you what system you have".
Thank you for this, I've been thinking along these lines for many years and it is the first time I have encountered someone to speak it. The instance of such a mismatch between systems with the same equation that bothered me was a derivation of the Hamilton-Jacobi equation from the Schrodinger equation. Indeed we can derive it mathematically, but the original Hamilton-Jacobi equation is written for the particle that moves through the trajectory described by the equation, while the derived equation describes the argument of the wave function and this wave function prescribes to the particle to be in every point of the trajectory simultaneously.
Thank you for your efforts. Had always had the feeling that students of sciences were not precise in their description of similar and equivalent systems.
Great video, keep up the good work. Dark background is A+ makes it easier to watch and LEARN! 😊
This has always puzzled me, thank you for the clear discussion of the difference between the kinematics and dynamics of these systems!
Wonderful Video. I have known about a lot of these ideas separately and individually when trying to work through them in the course of my Ph.D. but I haven't found a good way to think about, understand, or explain what is going on. I've had to deal with non-conservative and constraining forces within Hamiltonians where these forces are characterized with some kind of potential. Specifically this type of analysis is necessary when dealing with rigid bond constraints rather than realistic bond constrains. Rigid bond constraints are a common constraint in molecular dynamics. These expressions are necessary to when using the molecular dynamics simulations of materials to extract the elements of the elasticity tensor. I needed a Hamiltonian framework because the Hamiltonian framework is what is used to characterize the canonical thermodynamic free energy, so everything must be in the context of the Hamiltonian.
The problems you point out here are exactly the problems I was running into but couldn't adequately describe. I was getting all these time dependencies in my equations, which I found strange, and I couldn't find an adequate canonical transformation that dealt with the time dependencies while also preserving the continuum mechanics symmetries we rely on. Ultimately that part of the project was shelved to focus on pairwise conservative forces (real frame independent forces) and making sure all of that works before going back to look at these constraining and non-conservative forces, but we do intend to go back and revisit this eventually and having the knowledge that the strange time dependencies in my equations are artifacts of the fact that I'm in some kind of non-inertial frame (and its very clear now that using rigid bonds puts you in a frame like that!) helps us communicate the mathematical technique and how to interpret the equations and results.
Thank you so much for clarifying exactly what these time dependencies in Hamiltonians actually mean because that was one of our greatest hurdles to understanding what was happening when we put these constraints into our simulations. We would get accurate results and measurements reproducible in both the lab and from other different methods (like Monte Carlo), but we couldn't explain the added complexity nor did we want to wander down the rabbit hole trying to deal with all the issues that we knew would arise by taking this path. This really helped make me more confident that our mathematical results were, in fact, correct, but we were not interpreting our results correctly from the physics perspective.
Is there a place where I can get more formal writing about this? I'd like to cite this if we ever publish our results for the constraining and non-conservative "potentials"
Actually, check out the paper "Relativity and Particle Mechanics" by Gabriele Carcassi and Christine A. Aidala, from April 23rd, 2023. I believe this is the citation you seek.
There is also a new video about this on the TH-cam channel Assumptions of Physics which is Carcassi's second channel and I'd also highly recommend that one
Thank you so much for sharing this! You see, one of the problem that I have doing this type of research is that people want "new effects," so papers that go and try to figure out these details precisely get little to no interest. So, having actual examples of people that are helped by the better and more careful understanding is proof of the value. As if in science a search for a better understanding should need justification... 🤣
Anyway, everything in this video is in our open access book Assumptions of Physics ( doi.org/10.3998/mpub.12204707 ). If you use bibtex, the reference is in ( assumptionsofphysics.org/assumptionsofphysics.bib ). Citing the book is essentially like citing the project. We currently do not have a publication with this specific point (I wouldn't even know where to publish it!).
@@gcarcassi You certainly said it! The work I'm doing has required me to dig up papers from the 50s and dead citations from the old soviet union (I did end up finding them but I found them in places outside of official university or library channels, nobody official had the papers, but they were in lots of citations!). Its been an absolute nightmare finding anything precisely because the work I'm doing isn't really "New" in the sense that any of the physics is new, but rather now that computation is good enough to actually start using some of these techniques it is possible to start implementing some of these old techniques to actually model real materials and fully characterize them. However, there are a lot of fiddly details that need to be sorted out because it wasn't deeply explored. The lack of these details has really inhibited my understanding and our progress because our research group is being very careful because even though we aren't doing anything radical, we are proposing that the methods we are using are in fact, general, and in fact have broad application, while trying to renew interest in the methods, but that requires pushing back on naysayers and really digging into details about pairwise force decompositions, path independent work functions, the nature of equilibrium states, the properties of equilibrium ensembles and how they apply to the Helmholtz free energy vs a volume constrained Gibbs free energy vs a Full Gibbs free energy. These details are critically important when studying any matter that doesn't have a well defined crystal structure, or has gel-like behavior.
Glad I could provide a concrete example of how this is useful!
Reading this interaction made my day; thank you for partially restoring my faith in the ability of humans to communicate true and useful knowledge to one another! (Said faith has taken a beating lately lol)
Also I worked on some dissipative molecular systems a lifetime ago and had more than one argument with colleagues and professors about the Hamiltonians of those systems, if I had been able to articulate these issues half as clearly as this at that time I would have avoided a lot of unproductive miscommunication.
@@josephbrisendine2422
>(Said faith has taken a beating lately lol)
Mine as well! 🤣🤣🤣And thanks for sharing your experience!
Once again you are making me astonished by explaining in such a simple way!
Great video! Very enlightening!
This video is so interesting!!
Great explanation. Even in the side comments you are making while discussing the main point (e.g. about the difference between conjugate momentum and kinetic momentum: very appropriate note).
Good one.
Thanks.
I believe when people say that the Hamiltonian is not always the energy, they (we) mean that it also encapsulates the interactions between systems. In the case of a particle undergoing drag it clearly interacts with another system (e.g., water or air), and a full description must take this into account. I have not done the calculation for the classical particle, but introducing an auxiliary field that can absorb energy via the interaction with the particle should be able to reproduce both the kinematics and the dynamics. We do this in the description of dissipative quantum systems all the time.
Dissipative quantum systems are different case (at least, what I have seen: Lindblad master equation, ...): it would be like doing statistical mechanics in classical phase space and then tracing away the bath. This is not the same as writing those particular Lagrangians/Hamiltonians.
Concerning classical mechanics, during my second year of university, I thought that Newtonian formulation is superior to Lagrangian since the latter can not handle friction. But later I heard that there is such a thing as a Rayleigh dissipation function that can put friction into Lagrangian mechanics. But I've never really dug deep into it. Does Rayleigh's dissipation function have its weaknesses?
With the "non inertial" frame will a Galilean transformation still work to change the reference frame?
Hamiltonians and Lagrangians are built on closed systems i.e. where time partial derivatives of the energy holding position and momentum constant are zero. Dissipative forces take energy away so this is not the case. The mass change math like the time change math is forcing a round peg into a square hole.
I never tried to understand dissipation as a change in frame of reference, but I think this is indeed fascinating.
Maybe the problems of reference frames mattering were staring at us the whole time and the lack of physical precision was the problem.
It's really interesting that you could find a recipe to link t to t' as two different reference frames, but one of them simply was noninertial. I think perhaps this was deeper than I actually realized and I constantly found the equivalence of Hamiltonians and Lagrangians with Newtonian mechanis somewhat forced. I never really quite got what was my holdup, but... This might be some of the reasons: they actually aren't in general for reasons you gave in another video.
I always understood Hamiltonians as not energy in general, and I solved a few problems in which they weren't in the past. Probably, a change in reference frame could be enough to make that happen, and, if true, that means, PERHAPS, you can ALWAYS find this transformation and maybe in general it's all you need.
Obviously though I'll have to ponder on your other video again, because I was thinking about those things lately just for leisure and I really hot bothered by not really finding the crux of the problems I was detecting.
Great Video! There's a typo in the video description mentioning dynamics twice (I assume it should say kinematics instead of dynamics)
Thanks so much for letting me know!
I had no idea Pupo/Enzo Ghinazzi knew so much about this subject!
This is super interesting!
And I think starting to highlight this might be pedagogically a good idea too: Even though this isn't directly special, let alone general relativity, it motivates relativity from a different angle.
Certainly makes a lot of sense that a particle under drag is equivalent to one that isn't in an inertial frame. It's a system where everything (unseen, just assumed to be there) around the particle *is* considered at rest and uniformly grabs onto it to incrementally bring it to the same velocity, right?
If you had spatial transformation, yes, you can accelerate toward and then it looks like it stops. However, if that is the transformation, then you are also accelerating away from other particles that had different initial conditions. Here, in the ma = -bv case, ALL particles must stop, no matter the initial conditions. So, in a way, the transformation makes "time stop", so everything moves infinitely slowly.
What about non-Hermitian Hamiltonians in quantum mechanics? It is my current understanding that they appear whenever we encounter a gain-loss system, since their energy spectrum is in complex domain and the imaginary parts of eigenvalues are the losses/gains. Do you consider this also to be a sort of "equivalence mechanics"?
Non-Hermintian Hamiltonians are a bit different. Those do not generate a unitary operator, therefore they do not correspond to deterministic and reversible evolution. I haven't done a full investigation on those... What I'd like to understand, for example, is whether they can be seen as a "symplification" of the Lindblad master equation (which, to my understanding, is the "proper" way to handle dissipation. But I am working on other things at this moment... Too many things! 😄
This is a very interesting example! I come from the QFT side, and there, dissipation always comes from a bath. I am questioning what energy means more and more. From Schwinger-Keldysh formalism, one can find that the conservation of energy as the average of the Hamiltonian is more related to a specific factorisation of time translation symmetries. They will always guarantee probability conservation and sometimes imply conservation of the average of the Hamiltonian. But back to the classical setting, you managed to construct a Hamiltonian, which generates the correct equation of motion. Is that generally possible? And can it tell us something about the bath, where the dissipation comes from?
>I am questioning what energy means more and more.
I have been doing that for a decade now... 😁 In classical (particle) mechanics, the Hamiltonian is the time component of the vector potential of the flow of states. Next video I will show how it is a potential, and when I'll deal with Lagrangians I'll show how/why is the time component of a vector potential. I think this generalizes to field theories and QM (though I haven't done the work yet). What I do not have is the link between Hamiltonian and energy in thermodynamics. I have some ideas which I think may work, but I can't honestly say that I have it yet.
>you managed to construct a Hamiltonian, which generates
>the correct equation of motion. Is that generally possible?
I do not know the answer to that. The issue is whether there is enough freedom in the coordinate transformations to get to all possible kinematics. I really do not know.
>And can it tell us something about the bath, where the dissipation comes from?
As you saw in the video, the same equation can be used to describe different settings. You can imagine the system losing energy because of air friction, EM forces, rolling friction, etc... As long as it can be modeled in the same way, the equations holds.
If you are asking, more in general, how do we get entropy production... well, that's an open question. I have seen different proposals in our of equilibrium thermodynamics, and some even in the realm of quantum gravity... It could also be that different mechanism apply in different circumstances. Personally, I am not convinced that you can simply put a Hamiltonian on system+environment and expect everything to work. But in the end, I do not know. So I try to be sure that everything else that I say is independent from the thing I do not know. 😁 Like the founders of thermodynamics did! 😁
I haven't completely watched the video yet, but dissipation and hamiltonian together reminded me of Port Hamiltonian Systems.
How do you differentiate the forces that do not change entropy from the ones that do? 24:43
In the next couple of videos I'll show how Hamiltonian mechanics can be derived from entropy conservation. That is, Hamiltonian mechanics is exactly the subset of classical mechanics that conserves entropy (and independence of DOF). It will take some time to get all the diagrams ready 🙄 But all the info is already in a couple of our papers and in our open access book.
How does the system with uniformly accrediting mass relate to a system where you count the momentum transferred to the medium? Like, I can imagine a cone of air that is being accelerated by the particle as it passes through the medium, and if you squint that kinda feels like adding mass to the particle, as long as you count anything moving as part of the particle. It's been a while since I've done any Lagrangian mechanics so I'm probably missing something subtle.
You are going to have a force acting on the system in the inertial system, which is given by the change in momentum. That force is, obviously I hope, not conservative, so you are going to bump into similar problems.
Is there an implication for quantum Hamiltonian? In particular for cases where dissipation is added, e.g. as a non hermitian component.
See other reply...
Very enlightening. Could the variable mass description be connected to the need for energy dependent mass in renormalization in QFT?
No idea! 😁 I don't think I'll fully understand QFT until I have complete understanding of classical field theories and quantum mechanics... And Lord knows how long is that going to take! 😁
@@gcarcassi speaking of quantum mechanics, you pointed out some important differences between Newtonian mechanics (or "Equations of Motion") approaches and Lagrangian/Hamiltonian mechanics. Now, for QM, we've historically quantized either the Hamiltonian or Lagrangian (path integral), but what about quantizing the equations of motion? If what you point out is true, this would give a meaningfully different quantum theory. I did a literature search yesterday on this topic and it appears there's a few works pursuing this.
@@1Adamrpg Other people have asked similar things. I do not think it works. In my view, what you are really quantizing in quantum mechanics is the entropy: you are putting a lower bound at zero. Therefore evolution in terms of pure states always conserves entropy and independence between DOF (i.e. equal partition of entropy). Hamiltonian mechanics is exactly that evolution in both classical and quantum mechanics. If you take a classical damped Harmonic oscillator, for example, ensembles will shrink more and more. This is exactly what quantization doesn't allow you to do. You get to a minimum and you stop (e.g. the electron does not fall on the nucleus). That said, one may still get something going in some special cases (I wouldn't know).
Another way to actually preserve the Lagragian would be to use generalized forces.
Hi Gabriele, I watched your recent appearance in the demystifying science podcast (great stuff) in which you said : “In classical mechanics we have Position and momentum variables and in thermodynamics we have Pressure and volume variables “
If we are to continue this logic what do we have in quantum mechanics; energy and frequency variables?
There is a relationship between energy/time, but it is a bit different. Position/momentum, pressure/volume, spin/angles are pairs that identify an independent degree of freedom (in classical mechanics, thermodynamics and quantum mechanics). Energy/time define the "temporal" degree of freedom, but it is not an *independent* degree of freedom. In fact, time is not even a state variable, like position, volume, angle... Apart from that gigantic "detail"... you do have the pairing energy/time.
@@gcarcassi thank you
Is it possible to buy your book as well?
You mean as a hard copy instead of a pdf? Not at this point. Ideally, we'd be writing something more meant as supplementary book for a classical mechanics course... Ideally I'd also have a machine to duplicate myself and do all the things I'd ideally should do🙄
At 13:27 you introduce the force on a variable mass system. The first equation depends on velocity. The velocity will vary depending on reference frame so this seems to suggest different observers will measure a different force. Is this your intention?
Yes, different observers will see a different force. This is exactly what happens in the third part, when we have a non-inertial observer. For the variable mass system, there is only one observer, the inertial observer
@@gcarcassi You may not have explicitly examined them, but in principle there could be any number of *inertial* observers all observing a different force. This to me seems implausible.
@@iridium1118 I am not sure what you mean. In the variable mass case, the system is not closed... there is a force acting on the system that depends on velocity. Even for a system with linear drag, the expression of the force changes between inertial observers. In fact, there is a special inertial observer, the one comoving with the source of drag.
If you can transform to non-inertial time and map the system to a free particle, does this mean that the energy is conserved in the non-inertial time?
We need to be careful about what we mean by "conserved" and by energy... The Hamiltonian changes value in time, so in that sense no, it is not conserved. But I don't think that's the right way to think it. I'd say, in the inertial frame you have a balance of forces and therefore a balance of energy. When you go to the new frame, the expressions of the forces and the energy changes, so the balance is expressed in a different way. Some of the apparent forces will not be conservative, therefore the energy will appear to no be conserved... but by the exact amount required by the non-conservative forces. Essentially: in both cases the energy lost/gained by the system is exactly the one given by the apparent forces. In the rest frame, no conservative forces, no energy is lost/gained. In that sense, energy is "conserved".
@@gcarcassi thanks for your reply. I was curious for a few reasons. I believe there is a field of study called Hamiltonian invariants originating with Dirac, in which one can use an extended description of phase space (including time) and define an analogous ‘conserved’ Hamiltonian. I am curious about its implications for defining an equilibrium in such an extended space and whether it is possible to formulate time-dependent Hamiltonian mechanics using purely the extended phase space (i.e. the extended phase space just evolves according to a generalised EoM and has a conserved Hamiltonian invariant). I’m not sure how useful these concepts are though, since there is some non-uniqueness about Hamiltonian invariants. On the off chance you have encountered this obscure thing, please let me know if I’ve got the right idea.
@@solaireofastora4091 I make extended use of extended phase space. Sorry for the pun. 🤣 That is indeed the "correct" space to study time-varying Hamiltonians. At some point, I'll make videos about it. The "Hamiltonian constraint" is what typically the invariant is called. It generates and affine parameter, instead of time, and if the Hamiltonian constraint is 1/2 mc^2, and the affine parameter is proper time.
@@gcarcassi thanks for your reply. One last thing: do you know if there is an established theory of classical statistical mechanics in these contexts? Specifically I was wondering whether for a (non-relativistic) Hamiltonian of the form p^2/2m+V(x,t), where x, p may be N dimensional vectors, there exist transformations to non-inertial time such that t->tau(t) and d/dtau(H)=0, and a probability density in the canonical ensemble of the form e^(-beta*H)/Z. It seems this probability density ought to also inherit tau as part of the extended phase space. I know that there are ways of formally writing down partition functions for time dependent Hamiltonians via Wick rotation t->-ihbar*Beta in the path integral formulation of QM (then a classical limit) but I don’t know what the corresponding expressions mean since -beta*H is usually replaced with integrals of the form \int_0^\Beta d\Beta’ H(x,p,\Beta’).
@@solaireofastora4091 No, I do not know where such a thing exists. But I think I see where you are going with it. Interesting thought.
you mean when you slow something down in time there’s a correspondent apparent force?
Correct. You can see it as an apparent force or as apparent change in mass.
@@gcarcassi Okay so silly question but could all forces be explained in some way like this? Great video by the way! I love your work
@@lawrence-1 Ah! Great question! The short answer is no. The long answer is far more interesting. There is a theorem in Hamiltonian mechanics that states that, locally, all Hamiltonians are equivalent to a free particle. That is, in a neighbourhood, you can make a canonical transformation from any Hamiltonian to the one of a free particle. Now, the catch is that canonical transformation is not just a coordinate transformation. That is, momentum does not change like a (co)-vector like it has to when you make a coordinate transformation. I do not know whether this result has any other physical implications, but I find it very interesting. 😁
Hmmm... and why Lagrangian & Hamiltonian mechanics can only treat conservative systems can be understood from your previous work on the geometry of the Hamiltonian/action principle?
Right. I am making newer videos that go into a bit more detail for that.
Is this the origin of the concept of "effective mass"?
Not for all cases. In solid state physics and particle physics there are other things called effective mass and they are due to other effect. In general, "effective X" in physics is whenever the original X needs to be modified by some sort of effects. Effective charge, effective theory, ...
Who are you, so wise in the ways of science?
LOL! 😁
Well, he's not all covered in s--t, so draw your own conclusions.
Interesting!
If you added another part of the system which would be meant to take on the momentum that the main part loses to drag,
ah, I suppose it still wouldn’t really be enough to model the dissipation correctly…
or… would it?
if you have a force between the main particle being considered and another pretend particle (or, an aggregate of all the particles providing the drag)
so as to produce the correct drag on the main particle,
uh… I am not well enough versed in Lagrangian mechanics to ask my question properly without getting out some scratch paper..
There other results that lead to the same point. For example, I can derive Hamiltonian mechanics just from requiring that entropy is conserved. Those are points that are, to me, more conclusive. I'll talk about them in the next few video, where we completely reverse engineer Hamiltonian mechanics (and then Lagrangian mechanics as well).
I think the end remark is a bit of a mischaracterization of mathematical precision. After all, if something can be made "physically precise", it means it can be made mathematically precise with respect to a more physically meaningful mathematical structure.
At the end of the day, even in physics, if something can be made precise, then it can be expressed rigorously by the appropriate piece of mathematics (I'm sure even concepts like "having the same dynamics" etc).
[This is not to detract from the video, which is really great!]
Now you are confusing me... as a comment to another video you said 'a physical theory is not something that can be fully formalized or fully "made rigorous" in the sense of mathematics'. 😁 So there you seemed to agree! Things like meaning, connection to experiment, assumptions, etc.. can't be made "precise" in a formal way, but can be made "precise" in the sense that practitioners of the craft will understand each other.
@@gcarcassi: I guess when one says precise/formal/rigorous in math, there are two senses of the term:
A) to construct a well-defined mathematical object.
B) to write down a bunch of axioms in a language of first-order logic.
They're not the same (even when A might happen on the background of an axiomatic theory that is precise in the sense of B).
Also, one may or may not use the term referring to the whole package of a physical theory as a philosopher of science would intend it (that is, including the practical "pointing to the world" and putting it in relation with the math part of the theory).
In the case of this video (which I watched and commented before the other one), I was merely referring to sense A and _not_ including the whole package.
What I wanted to convey was: if a Lagrangian is missing something in the description of a system, maybe one could "decorate" the Lagrangian with further pieces of structure, like, I don't know, an energy functional that you can call "the dynamics" or something like that. [ I realize this specific suggestion is probably nonsense or wrong, I'm just trying to convey the type of thing I meant]. That is, tinkering with the math (in sense A) to make it more descriptive of physics, one might formalize (sense A) things that were previously only part of the "surrounding discourse" but not the math.
Yes, it's not a very subtle (nor detailed) point I was making here.
---------------------------
Returning to the other video, I totally agree with you that a whole physical theory ( _including_ its explicit connection with the actual physical world) can't be fully mathematically formalized, in either sense A or B, cause you have to "point at things" in the world and the "pointing" doesn't happen within pure mathematics.
Cheers :)
@@rv706 the "pointing" doesn't happen within pure mathematics.
LOL! 🤣 We are in vehement agreement.
So, in general yes: if you figure our the "physics requirement" better, you can make a better encoding in the math. That is exactly what we are trying to do in our research. For example, units and physical dimensions are currently not really encoded in the math (e.g. different physical quantities must form separate vector spaces because you cannot sum them). I am playing around with a partial order to capture those relationships. But, in this particular case, the math is fine, it is just the "pointing" that is not done well. As I say in the video, you can't just say "the Hamiltonian is not always energy" without saying what it actually is... I mean, you can, evidently, but it's not a "precise" answer in the physical sense. 😉
@@gcarcassi: I used to think "physical dimensions" had something to do with an action of a real algebraic torus* (or other Lie group) on some algebra or other structure. This would encode not a single unit, but the way expressions change when you change units, which is the intrinsic piece of data.
You may also want to have a look (if you haven't already) at the concept of a torsor**. Have a look at the page "torsors made easy" by John Baez.
* ("torus" in the sense of algebraic geometry: see the wikipedia page. In this case, it's just a fancy way to call a power of the multiplicative group of the real numbers)
** (torsors are essentially principal bundles. But when they're over a point they're essentially the same thing as a principal homogeneous space)
@@rv706 I looked at the definitions, and there are already 3 or 4 terms I do not know and have no physical intuition for. 🤣 It's too abstract. The structure I am working on is much simpler (i.e. a partial order on the Borel sets) and the physical motivation is straight forward (i.e. does one set contains fewer cases than the other?). The order is partial because not all sets are comparable (i.e. 1 meter of cases in position is not comparable to 1 Kg m/s cases in momentum). The problems that I am having is finding necessary/sufficient conditions to construct measures that are monotone with respect to that partial order.
The subtitles superimpose over your equations. You should not use the bottom part of the screen for equations.
25:52 No mathamatical formal system can proove that sugar is sweet unless you take it as an axiom, so the source of this axiom is human tounge which is physics or imperical truth.
Well, you would have to define what is meant by “sweet” first?
(Perhaps in terms of the human tongue?)
@@drdca8263
The experience the word only try to describe it.
Right: the axioms for physics are, in the end, grounded in experimental verification, which is outside of the formal system. I am a preparing a series of video on these issues (i.e. what can and cannot be part of the formal system in physics).
Nice. Great content. Please invest in a decent mic and spend a day setting it up correctly.
You are assuming I haven't already tried and failed...
I used to really question your approach, but the more I dug into these systems I found your questions to be valid.
Do you have a lecture on decoupling time dependence? I'd be really interested in a formalize that didn't privilege 1 dimension.
I do not. Also, note that you can have actual time dependent forces in an inertial frame, so I am not sure you can always take that time dependence away with a simple coordinate transformation.
A great alternative interpretation ！It inspired me a lot！