Hamiltonian mechanics in 12 equivalent characterizations

[@gcarcassi]

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 .

[@EnterTamed]

Thank you so much❤ Intuitively many of these things always seemed connected, but I have never seen anyone explain it in such a beautiful way, until today...

[@areyoushitting5]

Amazing ！Thank you very much for your effort on these analyses 👍👍👍

[@scottychen2397]

It’s philosophically important to be speaking about states : In the sense of feynman ( a robustly Logical appreciation of just 2 orthonormal system components) , this is enough to be speaking of the ammonia maser , and there is no immediate association between that and the distinct philosophical interpretation of reinterpreting a point as a wave complex . Indeed the wave equation as a classical event , is an approximation … Why would one choose to place ones intuition on this exact differential operator … My own opinion is that the Logical analysis of ‘ states’ Should emerge from all diversity of comprehension of the Newtonian Data complex : ( force , acceleration , mass ) Or ( Momentum , Force ) This dichotomy - indeed one may accuse things here of being ‘meaningless’ - of Law may indeed yield a diversity of meanings to the ‘force ‘ concept: A diversity of meanings to the pressure concept . Cauchy’s stress theorem is a pure mathematical thing . By this I mean to say that the algebra / analysis of states is not a priori the same study as the Schrodinger equation.

[@GabrielNunes-mg7jt]

plz dont stop make videos, this type of content isnt in the textbook so directly.

[@gcarcassi]

As long as people keep leaving comments that ask to make more videos, I'll keep making more videos. ;-)

[@RogueRunna]

@@gcarcassiYes!

[@miguelaphan58]

..a Master class, what marvelous discovery !!!

[@106ohm]

These videos are likely the best content on the subject here on KZbin. Thank you so much for sharing them

[@rajinfootonchuriquen]

I never thought of poisson bracket as a determinant. Now it seems so obvious.

[@gcarcassi]

You can imagine my surprise when I noticed! 😂

[@Henry77680]

This is a super juice of knowledge that one can watch again for years!! Thanks

[@gcarcassi]

LOL 😂

[@justalittlestretch9404]

This was so wonderful. Such a comprehensive tour of important ideas in a short time and well explained.

[@Kralasaurusx]

You make this seem so simple and easy - and that's a testament to your way of teaching! I had several "Ohhhh interesting" moments while watching this. I've encountered most of these ideas before, but never in such a coherently interconnected and elegant manner.

[@gcarcassi]

>I had several "Ohhhh interesting" moments while watching this. That's what I am going for! 😁

[@mbabb]

A treasure trove of information. Thank you so much!

[@SemiStableM]

Fantastic Video!

[@brankoras5947]

beautiful geometric representations

[@joeybasile545]

Great, Gabe.

[@scottychen2397]

This is excellent .

[@DarkTouch]

this pisses me off. I am a retired physicist. a geophysicist. and when i was in school, waaaay back in the jurassic, I only learned classical newtonian mechanics. I never even heard of hamiltonian or lagrangian mechanics at university. Now in my advanced age, I am struggling to wrap my head around this. so... back to school. this and other videos will be an invaluable to resource for me. I am a born nerd, and I love revisiting old math and physics topics.

[@gcarcassi]

Even today, a lot of physics curricula just go through Newtonian mechanics and then directly to quantum. But quantum mechanics is in the Hamiltonian formulation, so you are essentially trying to wrap your heap among multiple things at once. Which does not help. 😐

[@tinkeringtim7999]

However Jurrasic you are, I promise Hamiltonians were around and mature long before you were born. School and university syllabuses generally absolutely suck at everything science, but especially maths and physics. Especially maths; its not hard if you spit out the formalist pill.

[@joshwalker7460]

this is my jam

[@mpcformation9646]

Excellent and delightful. You’re a gem representative of the excellence of the Italian (I guess from your discrete accent) school which is particularly expert in differential geometry. But I’m surprised that you didn’t use any animation which are particularly suitable in 2D and which can be spectacular, thus motivating for beginners. And particularly if you’re indeed Italian, you may be pleased to discover that contrary to what is mainstreamly thought, so called « Poisson brackets » have in fact been undoubtedly brilliantly discovered by Louis Lagrange (French academist of Italian origine, after he replaced Euler in Berlin), very soon after he discovered the so called « Lagrange brackets ». Which makes him the full discoverer of Symplectic geometry. The two types of crucial brackets being « only » the covariant and contravariant form of such symplectic form. All that has been proven at length by the great French mathematician Jean Marie Souriau, despite it is still unwell known. And talking about that, I’m surprised also that you didn’t even mentioned the background Geometric structure that are concerned here, i.e. cotangent bundle (for Hamiltonian mechanics), whereas Lagrangian ones with Lagrange bracket lives in the tangent bundle. And furthermore I wonder if you are aware of Clifford-Hestenes Geometric Algebra, where all what you are laying the emphasis on, namely that it is more surface and volume dynamics than points one, and which is indeed a crucial remark, are fully treated in a fully unified dimension free way with an invertible product, from which « curl » (in fact wedge) and scalar ones are only incomplete part of it. The suprême advantage of Geometric Algebra representation, instead of matrix and vector field ones, is that it is fully intrinsic, coordinates free. And indeed all what you are here playing with, essentially oriented area, are actually Bivectors (which are also pseudo scalar in 2D). Your 90° rotation from gradient to flow, being also a crucial « passage by DUALITY », by simply multiplying by the unit bivector which squares to minus one. More generally shrinking and rotations being performed algebraically by Motors which are exponential maps of such Bivectors. Furthermore, upgrading to PROJECTIVE Geometric Algebra, allows to unify translations with rotations, by adding a projective base vector which squares to zero and which allows to se translations as rotations with projective center « at infinity ». Which makes you work, in Affine 2D (1 degree of freedom), in the « 3D » graded Geometric Algebra, which has in fact of dimension 8 : 1 scalar, 3 vectors (two spatial squaring to 1 and one projective squaring to zero), 3 Bivectors, and one pseudo scalar trivector. Thus in such intrinsic setting, the Physic will change in higher dimension since the type of graded Geometric Algebra changes, bringing new stuffs, but surprisingly the formula don’t. They are dimension free. Truly intrinsic, truly geometric. Which is a breathtaking advantage. Making the coding for calculations and animation hell easier. Check this out. Finally it would be very interesting to see you developing an entire video to the limits, lacks and weaknesses of Lagrangian and Hamiltonian approach. What it canot do, exactly. Or what can be modified for them to model non conservating systems, etc. In particular the first quantification, which relies (to) heavily, and in quite folkloric and « magic » way, on Hamiltonian mechanics.

[@gcarcassi]

Thanks for the detailed comment! As I say often, I get more useful/in depth comments, such as yours, on KZbin than through peer review. 😁 I'd love to have animations, but I do not have time, both to acquire the skills and then to plan them. They take time. And most of my time is spent on my research. I am aware of Souriau: I tried to read an old book of his, but the notation was too different and it took too much time to "translate." I am also aware of Clifford-Hestenes Geometric Algebra, though the scalar product is not always defined in physical spaces. I'll probably make a video on the subject at some point since many people point out GA. Was not aware of Motors or projective GA (I'll have to look those up). As for the limits of Lagrangian/Hamiltonian, we have a few results in those areas, and (time permitting) I'll make videos about those! Thanks again!

[@mpcformation9646]

@@gcarcassi Yes I very well understand. That’s the problem with representation of théories, everyone has its own and need to be learnt and it’s exponentially time consuming and confusing. Too much translation to be made all the time. And dimension dépendance makes it worse. This is precisely why GA solves all that once for all! All Geometry and Mechanics, therefore all Physics, has been built on a wrong starting point assumption : supposing that the geometric atom was « the point ». Unfortunately even Euclide failed to give any clear definition of such délicat concept. Nevertheless the Greek kept perfect account of Pythagoras disciple discovery, that sqrt(2) was incommensurable, therefore not a number for them. And on that deep vision they based all their geometry, not on « length » but on AREA. And THIS is much much deeper than we thought for 2000 years. This is why GA is so powerful, because by letting go this wrong base built on « points », it built everything on REFLEXION, which allows to build all geometric objects, with lines, surfaces, etc, and intersection or reflexion of those. And points are then included, but in a intrinsic way, and dimension free. Thus everything is based on a universal solid ground well defined and optimal. Geometric objects and transformations are not two separate world interacting, but geometric objects of the same space. Dirac operators become simple VECTORS. And no need for representations anymore, unless for final computations of needed. Everything is intrinsic. Coordinate free. Dimension free! It’s the greatest revolution since Copernicus. All the power comes from a tremendous unseen before unification of three types of vectors : those squaring to +1, those squaring to zero and those squaring to -1. Which was for millennia written but unseen in the famous equation : x^2+1=0, which in fact has to be seen as : x^2+1^2=0^2 Thus relativistic hyperbolic spaces are unified with euclidien ones and the bridge between rotations and translations, is made by projective objects driven by the null vectors. Those allow in supplement to make EXACT differential calculus, because Taylor series end exactly at the linear term! That gives the solution to the old nebulous « infinitesimal concept », without falling in the dead end of Schwartz distributions, who don’t really work. It’s thus a total revolution. And every manifold can be projected into a geometric algebra vector space. So there is no limitation to work in GA. Lasenby, Doran from Trinity achieved building a Gauge Theory of Gravity in flat space. Usual curvature is being described by a Gauge. And all physical theories fit in this GA framework, even weak and strong « forces », chromodynamics, etc. Souriau is difficult to read, since he hated Bourbaki and created almost his own langage the « Sourien »! One of his most remarkable idea for me is to have decided to forget about configuration and phase spaces, because they are not intrinsically relativistic. And create a new view point based on the MOUVEMENT SPACE, where there is no evolution! The entire evolution and dynamic of the system is represented by a single point of such fiber bundle. This is a master piece, a conceptual tour de force. It seems crazy. It’s utterly brilliant. But as Poincaré, he missed something crucial. Which is to forget about the « point » as a starting point, and start instead with more universal areas, from which lines and point can be universally and rigourously defined. Forget about the tricky « length » and start with areas, I.e. determinants or wedge product. So where do you see a problem with cross product? There isn’t since it is defined as the symmetric part of the geometric product. It’s always defined because the geometric one is always defined as a fundamental axiom of GA. And so is the wedge product which is the antisymmetric part of the geometric product. This last one being inversible. Which allows true unified algebraic calculus. There is an ALGEBRA. And this is even better than a field, because it’s not as restrictive, but richer enough to have it all. This liberty allows to find not only linear algebra and quaternions as special sub algebra, but also « twin quaternions », which are more useful than quaternions. In particular they have a 2 by 2 real matrix representation of dimensions 4, whereas quaternions has only a 2 by 2 complex matrix representation, or 4 by 4 real one, thus more complex. I truely think you would enter a intellectual revolution by driving deep in GA. Mind blowing. So much to be done to rewrite all 500 centuries of Maths and Physics in GA! Check the excellent videos of Steven and Leo on Projective Linear Algebra, about reflexions and mirrors. They are funny but brilliant. then the one of Lasenby on application of GA to physics. In particular Gauge Theory of Gravity in flat space! Finally on the video of steve, which is a programmer making GA animations, you will find link to his coffee shop where there is hundreds of of animations with code. Just paste it and try it, include it in your video, making some needed adjustments. Or simply ask chat GPT to write the code for your animation. He’s getting stronger and stronger in Maths since Wolfram joined, and even better in programming. It’s time to work together in a new era, before it’s to late… 🙏

[@eduardbuletsa9485]

Support your channel 👍

[@RogerPowell]

Many thanks for making this so clear and emphatic. I love your videos! Just three observations: The first is that using differential forms and geometric algebra, your 8 points could be made with even more elegance (the wedge product automatically gives you symplectic skew-symmetry and your areas become bi-vectors; the second is essentially that the second law of thermodynamics tells us real processes are irreversible and entropy increases whenever work is done. I guess you are saying that any attempt to use a Hamiltonian representation of a dissipative system is plain wrong, because it breaks all your 12 rules. Then we should find some other name for that representation, and kiss goodbye to some elegance. (I am an engineer not a physicist.) Finally, as soon as you get rid of 0-D "points" and admit other dimensions, the equations must include orientations and rates of change thereof.

[@gcarcassi]

@RogerPowell Spot on! I AM trying to understand how to use bivectors as a more primitive object and see the simplectic form as simply a linear function of a bivector. When I try to do that, however, I get a counting issue, some factorials start appearing and it becomes less nice. I haven't been able to get around that (suggestions welcome!). As for the second, yes: I do believe that Hamiltonian mechanics is the wrong tool for describing anything that truly increases entropy. You can move entropy across DOFs, and you can create correlations which increase the entropy of the marginals, but you are essentially kicking the can. As for the third, we do have results on "directional degrees of freedom". Each direction becomes a point on a sphere. But the 2-dimensional sphere (i.e. sphere in three dimension) is the only symplectic manifold. Therefore 3D is the only dimensionality that allows an independent directional DOF. At some point I'll do a video on that as well.. 😁

[@RogerPowell]

@@gcarcassi I'm reading your book (AssumptionsOfPhysicsV2.0) as well. I will make an attempt to rewrite the equations in your slides using differential forms and GA in order that we can have a more meaningful conversation. I don't know if I can help. Please don't hold your breath! I may be retired but I seem to be very busy all the time. 😅

[@gcarcassi]

@@RogerPowell LOL! 🤣 Differential forms I am fine with. That works well, though I use a different notation (that I think makes more sense for physics). It is the multivector that I have issues with... If/when you are ready for a conversation, do contact me. I am not hard to find... 🤣

[@sensorer]

With how much you focus on geometric understanding in your videos, I wonder if you've read Needham's VCA and VDGF. If you haven't, I think you'd love those. The last two chapter on Polya vector fields contain ideas which are similar to things you touch on here

[@gcarcassi]

Thanks for pointing those books out! No, I wasn't aware of them. I may finally have a book to suggest when people ask me for one on differential topology/geometry!

[@sensorer]

​@@gcarcassi it's worth it going through those books even with a thorough understanding of complex analysis/differential geometry. It's a delightful read, Needham makes you imagine and picture stuff every step along the way, it is way more enlightening than just confirming something is true with a calculation. For VCA, a nice companion is Visual Complex Functions by Wegert since it covers phase portraits, a way of visualizing complex functions that is missing from VCA(it is missing because this approach actually grew out of a review of VCA's first edition!). By the way, there's a quote by Michael Atiyah in the preface to VGDF that I think you'll love :)

[@jaydenwilson9522]

"As long as people keep leaving comments that ask to make more videos, I'll keep making more videos. ;-)" -gcarcassi You got it Profes-SIR!😇 Can I ask some potentially silly questions!? Can we create an Octree Matrix which SPROUTS and grows!? Those 3D matrices which are overlayed on top of each other to make 9-12D matrices seem crazy to a mathephobe like me! Oh and what do you think of DB Larsons Natural Units of Measure!? Concept - Conventional = Natural distance, length - meters = S duration, time - seconds = t velocity - mph, kph, fps = s/t energy - Joules = t/s acceleration - m/s2 = s/t^2=s/t/t force - Newtons = t/s^2=t/s/s momentum - N.s = t^2/s^2 mass - grams = t^3/s^3 pressure - Pascal = t/s^4 = t/s/s^3 And.... Conventional - Concept = Natural power - watts =1/s dipole moment -seconds =t current -ampere =s/t charge - Joules = t/s resistance - Ohms = t^2/s^3=t^3/s^3/t voltage - Volts = t/s^2 (= force) pole strength - Weber = t^2/s^2 (= momentum) inductance - Henrys= t^3/s^2 (= mass) capacitance - Farad = s^3/t He was a mathephobe and coped a lot of heat for it! But his ontological view seems mostly sound to me! Oh and what do you think of this regarding the Planck Length!? ℓPπ = 5.083 x 10^-35 m But, φπ = 5.083 And, (φπ)/10^35 = 5.083 x 10^-35 m Why does his length multiplied by pi nearly equal phi multiplied by pi divided by ten to the power of thirty five!? (Who is writing this script!?)

[@mathiaschaves7604]

No Idea why KZbin recommended your channel... I am a biologist and never heard of Hamiltonian anything hahaha. Well, It seams like a cool thing to learn even though I didn't grasp much of it.

[@gcarcassi]

The algorithm is imperscrutable... 😁

[@josequispe4513]

Great job! Its posible a hamiltonian mechanics characterization using calculus algebra / geometric algebra? I would like to see, thanks

[@gcarcassi]

To my understanding, no. The geometry is really given by the symplectic form, which is neither an inner product nor an outer product. Mathematically, you can probably re-encode the information in a different way, but you are going to lose the geometrical picture.

[@monkerud2108]

Yeah dissipative forces don't conserve energy, but you can add in some energy sink and it is well behaved again. Point being, the properties you reference can be reconciled with a lot of theories that are dissipative, by well making them non dissipative by some extensions. But you can also assume some dissipative dynamics that from the inside, looks lile it conserves energy in a different way, namely if for example all velocities got halfed right now, and the physics changed with it, we would not be able to observe that change from looking at the velocities, so we would assume there is no friction at all, and could assume a Hamiltonian just neglecting the friction because we are using changing coordinates without being aware of it, and from the perspective of the dissipative dynamics, you could just assume some mechanism that makes the friction dump its energy into heat or whatever else, and you could assume a different Hamiltonian, so now we have two different Hamiltonians for the same system, one considering more details and has its own notion of conserved energy, the other doesn't account for those details and has a notion of energy conservation that differs from ours in that their unit shrinks with respect to ours over time if we use the dissipative coordinates and formulation of the physics.

[@ChaineYTXF]

Very interesting, indeed

[@hinova77]

Wow, best utube, Thank you.

[@Arthur-so2cd]

god i wish i was a physics grad student

[@naturalphilosiphers9848]

Would you consider creating problems for us to try?

[@gcarcassi]

Interesting idea! I'll think about it.

[@manfredbogner9799]

Sehr gut

[@gcarcassi]

Danke

[@tanchienhao]

awesome video! shouldn't 6:43 be S^q \omega_{qq} + S^p \omega_{pq} ?

[@gcarcassi]

Thanks! I don't see that equation at 6:43... but \omega_{qq} is zero

[@tanchienhao]

@@gcarcassi ah that makes a lot of sense, because it's a symplectic form. thanks

[@nas8318]

I'm a condensed matter physicist who never learned Hamiltonian or even Lagragian mechanics and I feel like something is missing in my education. I cannot find any good lectures or videos, or even books that bridge the gap between regular mechanics and Hamiltonian mechanics. I know many physicists like me. I try to watch your videos but they are too advanced. If you feel like it, please consider making a video for people who are comfortable with newtonian mechanics who want to learn Hamiltonian mechanics and undersrand how it ended up in Schrodinger's equation!

[@gcarcassi]

Interesting. The basic idea is not that complicated: Schrodinger's equation studies a closed system (i.e. deterministic and reversible). If you are a condensed matter physicist, you may know that you use the Lindblad equation for open systems in QM. So what you ask should be doable... Can you tell me in what way this video is too advanced? I mean, I know I go through a LOT of things quickly (and it was still 40 minutes!). Apart from that, it is basically vector calculus in a slightly different notation... If we can figure out what the gap is, I could make a video with that in mind.

[@SystemsMedicine]

OutFuckingStanding!!!

[@andreasxfjd4141]

and only 16 GB Ram

[@scottychen2397]

Of course electromagnetic interference is the most precious empirical event in Quantum Mechanics : I’m not saying this algebra [ Poisson brackets ] implies anything a priori . Nor does probability theory imply unknowability a priori … Of course , one doesn’t appreciate unknowability whatsoever: so a principle is not a big problem . @17:50 this is your intuition of probability density . It would be well , to comment on the electron analysis : ‘ the density of probability composes non linearly ‘ ‘ intrinsic probability [ entity ] composes linearly ‘ Indeed identifying such a non linear ‘ cosine delta ‘ term is the Logical way of recovering the original entities . ….. There are many restrictions here , and this may not be an electron analysis , … so it’s interesting , to which type of experimental prediction this algebra is speaking of.

[@monkerud2108]

So i give you a Hamiltonian for an expanding spacetime, it conserves energy sort of, but i also give you a system where stuff is slowing down and getting smaller in a non expanding universe. One has dissipative forces one does not, intrinsically they can be identical.

[@ozachar]

Would it be correct to say the divergence less of the Hamiltonian is a statement of the conservation of energy - there are no sources.

[@gcarcassi]

It's an indirect relationship. It is not the Hamiltonian that is divergenceless (it is a scalar field), it is the flow of states that is divergenceless. There are no "source or sinks" of states, as states evolve. That allows to create a potential for the state flow, and the Hamiltonian is that potential.

[@Rupadarshi-Ray]

(1) HM-1D is true only in 2 real dimensions of phase space, so dim=2 case for (2) (2) HM-G is true in any (even real) dimensions and is the definition of Hamiltonian vector field of H on symplectic manifolds (2) DR-JAC = DR-VOL = DR-DEN = DI-SYMP = DI-POI = DI-CURL (and also the last four, I think) are only locally Hamiltonian vector fields in 2 dimensions, as counterexamples: - on the punchered plane, we can have a outward flow which is volume preserving, locally Hamiltonian vector field of the angle but not given by any Hamiltonian globally - in dim 4, we can have linear vector fields which preserve volume but do not preserve the symplectic form. Yes I am a math major, so I'm into the edge cases, haha.

[@gcarcassi]

I say it specifically that I am looking at the single degree of freedom case (i.e. dim 2). Preparing another video for the general case.

[@Comomevoyapasarlavida]

Could you explain it with python?

[@gcarcassi]

Like this? def isDeterministic(classicalSystem): return isHamiltonian(classicalSystem) def isVolumePreserving(classicalSystem): return isHamiltonian(classicalSystem) def isInformationPreserving(classicalSystem): return isHamiltonian(classicalSystem) ...

[@eliavrad2845]

Maybe a nicer one: import numpy as np from typing import Callable def assert_evolver_is_hamiltonian(evolver: Callable, states: np.array, dt: int): new_states = evolve(states, dt) assert len(new_states)==len(states) # volume preserving assert states == evolve(new_states, dt=-dt) # reversible assert new_states == evolve(states, dt=dt) # deterministic assert ... # example def hamiltonian_simple(state, dt): return state assert_evolver_is_hamiltonian(hamiltonian_simple, np.array([1,2,3,4,5]), dt=1) # no AssertionError def hamiltonian_less_simple(state, dt, v=1): return state + v*dt # for example, states are x positions with constant speed=1 assert_evolver_is_hamiltonian(hamiltonian_less_simple, np.array([1,2,3,4,5]), dt=1) # no AssertionError def non_hamiltonian_cause_not_reversible(states, dt): return states + 1 assert_evolver_is_hamiltonian(non_hamiltonian_cause_not_reversible, np.array([1,2,3,4,5]), dt=1) # AssertionError from assert states == evolve(new_states, dt=-dt) : [1, 2, 3, 4, 5] != [3, 4, 5, 6, 7] def non_hamiltonian_cause_not_preserving(states, dt, v=1, wall=6): new_states = states + v*dt # states are positions x; move all positions good_states = state < wall # did not hit the wall at x=6 return new_states[good_states] # filter all states that hit the walls assert_evolver_is_hamiltonian( non_hamiltonian_cause_not_preserving, np.array([1,2,3,4,5]), dt=1) # AssertionError from assert len(new_states)==len(states): 5 != 4 def non_hamiltonian_cause_not_deterministic(states, dt): return states + dt * np.random.rand(states.shape) assert_evolver_is_hamiltonian(non_hamiltonian_cause_not_deterministice, np.array([1,2,3,4,5]), dt=1) # AssertionError will already come from reversible, but even if we deletde that, we would get assertion from "assert new_states == evolve(states, dt=dt)"

[@monkerud2108]

So you have to be a bit careful about what you think these things mean, because dissipation is not a binary thing, a dissipative theory can be dual to a theory that conserves energy and momentum.

[@biolinux2307]

Like

[@m_c_8656]

ew, snap!

[@EstParum]

What if we just lived in mud huts instead.