I’ve had a lot of success with Taildrive (still alpha) to share directories between devices in sync. e.g my keepassxc database lives on my homeserver, all devices access it through taildrive, this is impossible with iCloud drive because there is no linux integration.
I just put dotfiles directly where they are and majr a .git directory in my $HOME. .gitignore everything and git add -f files when I need to. no symlinking or anything.
I’m currently in using sway on fedora, I’m considering moving to macOS because of tighter integration of icloud (everything is on my iphone) and generally better polish of the OS. This is another reason to switch
The Hamiltonian formulation of classical mechanics is such a beautiful way of describing classical motion compared to the Newtonian formulation. See Laundau & Lifschitz book 1! All the Hamilton-Jacobi equations are derived from observing symmetries in space time, even Newton’s 3 principles are derived (F=ma an the rest). All of this has the added benefit of transposing well into quantum mechanics, where forces are anyway replaced with hamiltonians.
For fluid mechanics I don’t know if Hamiltonians are the right formulation.
Is it not the case that F=ma and other Newtonian laws are encoded in the Lagrangian, and therefore not derived from symmetries? After all nature and her symmetries alone do not tell us how her physics works; there has to be some rule for time-evolution as well, and that's what's encoded in L (L's form is essentially a list of pairings of variables and their costs of evolution, which for classical mechanics is L = T - V = ∫ p·dv + ∫ F·dx, which says "the cost of changing v is p and the cost of changing x is F").
(QFT sort-of has an explanation for time evolution in terms of symmetries alone, but it requires a lot more machinery. But afaik classical mechanics does not.)
> the cost of changing v is p and the cost of changing x is F
I’m sure the equation is right and all but this seems sideways in terms of an intuitive explanation - velocity changes position, and force changes momentum. Force doesn’t directly change position (only indirectly via changing momentum) and momentum doesn’t change velocity, having momentum is consistent with a constant velocity. It doesn’t even make much sense to me to think of integrals as being about “costs of changing”, would that not be a derivative?
Although not the same, I remember the first time I encountered the description of dynamic systems using Lagrangian mechanics and it was beautiful. It finally made sense.
Instead of doing what seemed like some mumbo-jumbo sarting from a static system, that, eventually and through convoluted ways, lead to the equations we were kind of looking for, here was a clean and logical way to look at what was relevant to the dynamics of the system, with an infaillible, straightforward, mathematical way of getting the equations of motion.
I'm not that familiar with Hamiltonian formulations, but its conservation properties could bring some important improvements to the current way the Navier-Stokes equations are treated. Conservation of linear and angular momentum, for a start, could be nice..
Now, let's see if I understand anything from this paper..
I have tried to teach myself both Hamiltonian and Lagrangian classical mechanics and there’s one mental hurdle I have not been able to get over. The problems are generally set up with starting position and momentum known, and ending position and momentum known, and then the math tells you the path taken along the way. But what if the ending position and momentum is unknown? How does one use these formulations of mechanics to predict the future and not just postdict the past? Is this just how beginner problems tend to be set up?
Hm, well typically in Lagrangian formulations, you're right that you define a functional called an "action", and this involves some integral from an initial to a final position in configuration space. Then the idea is that the system will evolve in such a way as to minimize this action. A stationary point (like a maximum or minimum) has gradient 0. So a condition for this "principle of least action" to be realized is that the "gradient" (called the variation here) of the action is 0 for the real path. That is, if you take a path the system carves out through the configuration space and perturb that path, and you compute the total effect on the action from that small perturbation, you find it should be 0.
You do this by actually taking this derivative and you find that you can guarantee that the differential of the action is 0 if the system takes a path which is the solution to a set of differential equations, and you can generally find the solution to those differential equations only with information about the origin, ie you don't need both the start and end conditions to find a unique solution.
So you're right, it's a bit weird conceptually. You sort of start saying "the system obeys a path that minimizes the action between it's initial and final positions" and then find that this produces a set of conditions which form a system of diff eqs that you can find general solutions for and select out a unique solution just with the initial conditions, no need for the final condition.
The answer is, unfortunately, boring. You solve equations to find the unknowns. If there are too many unknowns there are too many solutions and the whole thing is useless.
Please read "Story of your life" by Ted Chiang. A beautiful story that involves a discussion of Newtonian and Hamiltonian formulations.
One of the best stories I have read.
Not the person you were responding to, but: the Lagrangian formulation describes physics in terms of Least Action (so minimizing (or maximizing) S = ∫ L dt) on a manifold in (x,v) coordinates, and its equations of motion are like L_x - d_t L_v = 0. The second term tends to be second-order in time.
The Hamiltonian formulation performs a Legendre transformation[1] on L giving H = v L_v - L, which is essentially a convenient trick: it reparameterizes L in (q,p) coordinates, where p = L_v, and writes S as ∫ (pv - H) dt. This changes the E.o.M. to (qdot, pdot) = (H_p, -H_q), which is (a) first-order in time and therefore easier to deal with and (b) geometrically elegant because it is a rotation in (q,p) space, which is easy to think about.
At least those are the reasons everyone gives why it's important. I think the real reason is that QM is formulated in terms of H so you need to know it, and also that this (q,p) thing makes statistical mechanics easier because it has good geometric properties: it amounts to saying that time evolution conserves area in (q,p) space, which means that you can treat the evolution of many-particle systems as being in a whole block of states at once, treated as a geometric object that flows over time.
I've never been able to understand if there is something "truly fundamental" about H compared to L, or if H is more of a mathematical convenience for making the equations first-order.
In classical mechanics the Lagrangian and Hamiltonian formulations are mostly equivalent (though, confusingly the Lagrangian formulation is due to Hamilton, who has shown that the Lagrange equations can be derived from a variational principle, while the complete Hamiltonian formulation is due to some later phycisists; for what has become the Hamiltonian formulation, Hamilton has shown only how to obtain the system of first order equations from the system of second order equations, but that had already been done before by Cauchy in 1831 in a journal that few were reading, so it was ignored).
Still, in classical mechanics some consider the Hamiltonian formulation to be more fundamental, because the first order equations can be applicable to some problems that have discontinuities incompatible with second-order equations, though such problems are artificial (real systems are continuous enough, strong discontinuities appear only through approximations).
However this changes completely in relativistic mechanics, where the Hamiltonian is not invariant, while the Lagrangian is a relativistic invariant quantity.
This makes the Lagrangian formulation a far better choice in relativistic mechanics and it is a strong argument to consider the Lagrangian formulation as the fundamental one and the Hamiltonian formulation as only an approximation that can be used at small velocities or only as a mathematical trick for numeric solutions.
When the Lagrangian formulation is used, after a coordinate system is chosen, it is always possible to use the Legendre transformation to obtain a Hamiltonian system of first order equations. However, in the relativistic case the system depends on the coordinate system. Therefore, if the coordinate system is changed, the Hamiltonian equations must be derived again from the invariant Lagrangian formulation.
The reason why the Lagrangian is a relativistic invariant is that this scalar value is the projection of the energy-momentum 4-vector on the trajectory curve in space-time. The Hamiltonian is just the temporal component of the 4-vector, which is changed by any coordinate transformation. Therefore L is more fundamental than H, in the same sense that the magnitude of a vector is more fundamental than any of the components that the vector happens to have in some particular coordinate system.
The traditional formulation of the quantum mechanics using H is a serious inconvenience for extending it to the relativistic case. Coherent formulations of the relativistic quantum mechanics must also use L instead of H.
Thank you! The Lagrangian as projection of energy-momentum actually makes sense, unlike the "let's just subtract potential from kinetic. No reason, it just works" story. I'd been idly wondering that for a while (and this is as someone with a physics degree, though in astro, which is a good bit more applied).
Hamilton has introduced what he has called the "Principal Function S", which is used in his variational principle on which the Lagrangian formulation is based.
Nowadays this function is frequently called "Hamilton's action", though this is not a good idea because it causes confusions with what Hamilton, like all his predecessors, called "action", which is the integral of the kinetic energy.
The "Principal Function S", which is a scalar value, i.e. a relativistic invariant quantity, is the line integral of the Lagrangian over the trajectory in space-time, i.e. it is the line integral of the energy-momentum 4-vector over the trajectory in space-time.
Like any line integral of a vector, the line integral of the energy-momentum 4-vector is equal to the line integral over the trajectory of its projection on that trajectory.
This is why the Lagrangian is the projection of the energy-momentum 4-vector. Hamilton has found the correct form of this line integral in relativistic theory, even if that was about 3 quarters of century before the concept of 4-vectors became understood.
The "Principal Function S", i.e. the integral of the energy-momentum, can be considered as a more fundamental quantity than the Lagrangian, which is its derivative (the energy-momentum vector is its gradient). In quantum mechanics the "Principal Function S" is the phase of the wave function, so it is even more obvious that it must be an invariant quantity.
In an earlier answer I gave more information about that resource. To find that earlier answer: go up to the entire thread, and search on the page for my nick: Cleonis
This is the exact point that confused me a lot (and still confuses me) when I tried to read the "The Theoretical Minimum: What You Need to Know to Start Doing Physics" : "Hey, let's just fix/define the lagrangian as T - V and you'll see that after some magical math stuff in the following chapter, we'll find back newtonian equations. Trust me for now".
If anyone has a reference/book/paper that allows you to learn this concept more intuitively, I'd be grateful.
I have created a resource for the purpose of making Hamilton's stationary action transparent.
It is possible to go in all forward steps from F=ma to Hamilton's stationary action; that is what I present.
The path from F=ma to Hamilton's stationary action consists of two stages:
(1) Derivation of the work-energy theorem from F=ma
(2) Demonstration: when the conditions are such that the work-energy theorem holds good then Hamilton's stationary action will hold good also.
These presentations are illustrated with interactive diagrams. Each diagram has one or more sliders for manipulation of the contents of the diagram. That way a single diagram can offer a range of cases/possibilities.
About my approach:
I think of Hamilton's stationary action as an engine with moving parts. To show how an engine works: construct a model out of translucent plastic, so that the student can see all the way inside, and see how all of the moving parts interconnect. My presentation is in that spirit.
Unfortunately there are tons of books about the history of physics that get many details wrong, because apparently the authors have not actually read many of the primary sources, especially when they had not been written in English, but in German, French, Latin or other languages.
Even the original works written in English, like those of Hamilton, pose serious problems when you are not careful, because many words used in physics have changed in meaning during the time, some of them multiple times (e.g. "energy", "action" or "force"). Those who are not aware of this frequently reach wrong conclusions about who has said what.
A few authors have actually read the primary sources, but those typically do not understand physics so well as to be able to distinguish the important concepts from those of little importance, so they are not able to trace the evolution of the important concepts :-(
The only foolproof method to understand the history of physics is to read very carefully the original works (carefully, because the words and the notations used may be very different from those used now). Fortunately, that has become much easier now than before, because there are many online repositories with digitized scientific works from the previous centuries.
For example, "Sir William Rowan Hamilton (1805-1865): Mathematical Papers":
The most important for this thread are (especially the 2nd, which introduces the modern Lagrangian formulation):
1834-04: “On a General Method in Dynamics”, and
1834-10 (but published in 1835): “Second Essay on a General Method in Dynamics”.
The first who has shown how to rewrite the second order system of Lagrange equations into the first order system of equations now called Hamilton's equations (i.e. by using the equivalent of the Legendre transformation) was Poisson in 1809, but the last time when I have searched for that work online I could not find it.
After Poisson, Cauchy has presented in 1831 a method equivalent with that published by Hamilton in 1835. I also could not find online the 1831 work, but an extract of it has been republished in 1837 and this can be found online in many places, for instance at:
Note sur la variation des constantes arbitraires dans les problèmes de Mécanique,
Cauchy, Augustin, pp. 406-412.
You can read some books about the history of physics for a general acquaintance with the authors and the published works from the previous centuries, but you must remain skeptical about any opinions presented there until you read yourself the primary works to verify if they really contain what is claimed about them, or they contain something else.
I have found the reading of many old scientific papers, especially from the 19th century, surprisingly useful for a better understanding of the modern theories that I use now.
It's worth mentioning that Feynman's dissertation was on a Lagrangian formulation of quantum mechanics. However, just because Feynman thought it was interesting doesn't mean it's a good idea for the rest of us (although it's a relatively short 69 pages and an interesting read).
> I've never been able to understand if there is something "truly fundamental" about H compared to L, or if H is more of a mathematical convenience for making the equations first-order.
Actually, Hamiltonian formulation, being equivalent, offers more room for finding solutions. Lagrangian formulation of the Least Action principle allows you to search for a solution employing arbitrary smooth re-parameterizations of the configuration variables `q`. The Hamiltonian formulation, on the other hand, allows you to re-parametrize the entire phase space (q,p) and find solutions that are much harder to get in Lagrangian formulation.
That sounds interesting. Do you know of a particular example where that helps?
I guess maybe it's 'all of them'. The pedagogy on Hamiltonian mechanics had been strangely hard for me to learn beyond the elementary level, like it doesn't make enough sense for my brain to organize it in a memorable way.
The benchmark is specifically done on the encoding/decoding steps on the formats. But Cap'n Proto does not have that, so it is, in effect infinitely faster for this step.
Weird as it sounds, put a mirror in front of another, same size and shape better. The reflection will vound "infinitely" from one mirror to the other creating a "never-ending" effect of mirrors inside other mirrors.
In my university gym changing rooms there were two mirrors in front of each other like that.
First time I saw it it kept me thinking about infinity :D
No need to be snarky, clearly monitoring end user connections is a must. But the general idea of using RSS for monitoring is new to me, thanks for sharing!
Looks great. I remember using bpftrace at work to debug a nasty performance issue, I went down the rabbit hole only to find a certain syscall was being called much too often. We managed to trace it back inside the sourcecode to a sleep(1 second), which was some sort of manual io scheduling commited by the CTO when the startup was early stage.
Removing those few lines and installing kyber fixed the issues!
- usage is uniform on all platforms through the tailscale cli or platform specific integration (iOS taildrive shows up in system files app)
- ACLs makes managing permissions a breeze
- no need to think about ports/vpns/configuration, it just works
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