[HN Gopher] A canonical Hamiltonian formulation of the Navier-St...
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A canonical Hamiltonian formulation of the Navier-Stokes problem
Author : Anon84
Score : 151 points
Date : 2024-04-07 00:46 UTC (22 hours ago)
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| jackhalford wrote:
| 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.
| Horffupolde wrote:
| That's a correct usage of the factorial.
| fransje26 wrote:
| 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..
| om2 wrote:
| 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?
| bdjsiqoocwk wrote:
| Since you sound like you know what you're talking about, wanna
| tell us what is the difference between the hamiltonian and
| lagrangian formulations?
| ajkjk wrote:
| 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.
|
| [1]: https://blog.jessriedel.com/2017/06/28/legendre-
| transform/ is a good exposition, if still pretty tough to
| understand. Legendre transforms are hard to grok.
| adrian_b wrote:
| 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.
| oddthink wrote:
| 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).
| adrian_b wrote:
| 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.
| ajkjk wrote:
| Do you know of a good source on the history of this
| stuff, such as the relationship with cauchy? Or is it
| just something you pick up?
| kurthr wrote:
| 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).
|
| http://files.untiredwithloving.org/thesis.pdf
|
| The path integral method of QED does make the Lagrangian
| for field theories easier.
|
| https://en.wikipedia.org/wiki/Path_integral_formulation
| lr1970 wrote:
| > 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.
| ajkjk wrote:
| 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.)
| om2 wrote:
| > 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?
| techas wrote:
| 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.
| tekla wrote:
| Haha, great paper. I read the title and the abstract and went
| WTF?!?
|
| > Given the title of this paper, it is incumbent on the authors
| to assure the reader that we do not claim to have done the
| impossible
|
| Awesome. Though I have no clue what the Hamiltonian formulation
| is.
| comment_ran wrote:
| It descripts a system using the energy concept. The total
| energy of the system, which is the sum of kinetic energy and
| potential energy. Its formula often looks like H = T + V, where
| T represents kinetic energy and V represents potential energy.
|
| Both the quantum mechanics and molecular dynamics have shared a
| similar concept.
|
| In structural mechanics, we use the virtual method to calculate
| the hyperstatic structure to determine displacements in a
| structure, given forces acting on the structure. Another kind
| of Hamiltonian.
| tekla wrote:
| I'm classically educated on fluids so asking me to deviate
| from Newtonian mechanics for viscous fluids is ... difficult.
|
| Nothing against this tho, I just don't have the foundation
| for this.
| comment_ran wrote:
| I still remember in high school, we only need two or three
| equations to solve a free-fall problem. In another book,
| basically the same question, but someone uses Hamiltonian
| framework to solve really complex PDEs and couple pages
| with those crazy equations to basically solve the same
| thing, and eventually got the same results.
|
| I still remember that was mind-blowing. High school physics
| is so simple, whereas the Hamilton is so complex. I later
| on notice that Hamilton is kind of a more standard way to
| solve the problem. Never mind, I'm not an expert on it, but
| I'm just kind of amazed by the Hamiltonian mechanics.
| jfengel wrote:
| The Hamiltonian is a lot more flexible with respect to
| frame. The Newtonian formulation works great for simple
| cases but as it gets more complex it's harder and harder
| to pick a reference frame that's easy to compute.
|
| Effectively, by working with energy rather than force,
| you can avoid working with vectors. That ends up being
| simpler as the components add up.
| spenczar5 wrote:
| I am curious about what you mean by "classically educated."
| In my undergrad physics education, computing Hamiltonians
| was pretty much an entire semester of classical mechanics
| in my junior year.
|
| We didn't really touch fluids though. Does "classical" mean
| something different there?
| kyykky wrote:
| I assume that the classical exposition to fluid mechanics
| uses newtonian concepts, mainly forces.
| abnry wrote:
| I have no clue about this paper. Only comment is that this was
| published April 1st.
| mikewarot wrote:
| Does this result mean that aerodynamic simulation can be made far
| less compute intensive, like I think it does?
|
| This could be as useful as Feynman diagrams are to physics
| calculations.
| 2four2 wrote:
| Okay I gotta be honest, and maybe I'm a little ignorant, but
| when did we start saying compute instead of computation? I feel
| like I went away on vacation and missed out on an in-joke.
| Should I be saying compute instead?
| javajosh wrote:
| 'Compute' connotes hardware, 'computation' does not.
| 2four2 wrote:
| Thank you! I always just assumed the difference was
| verb/noun.
| cvoss wrote:
| Rather, 'compute' is a verb, 'computation' is a noun, and
| 'computationally' is an adverb (which is the most
| appropriate choice in the context of GGP's statement). All
| of these words refer to the same concept and may involve
| hardware computers (a more modern use of these words) or
| people-as-computers (the older, but still valid, use of the
| words).
| 2four2 wrote:
| Digging more into other internet discussions, a lot of
| people disagree about the usage. Many believe it's fine
| to use compute as a noun since it's a truncation of
| compute resources or compute nodes. Seems like an
| industry shibboleth like how data/stats people say "data"
| as a countable plural.
| ajross wrote:
| "Compute" as used upthread is in fact a noun. That's the
| point, it's a new word. And sort of a dumb one, but new
| nonetheless.
| hasmanean wrote:
| I think Compute is a gerund (a noun derived from a verb,
| like "swimming" as in the activity.)
| gyrovagueGeist wrote:
| It's been commonly used as a noun in the HPC / scientific
| computing space for a long time (relevant to this thread)
| typically talking about "compute bound" and "memory bound"
| algorithms. It's spilled over everywhere now that number
| crunching is hip for machine learning.
| hansvm wrote:
| People favor shorter words for common concepts. We've started
| needing something equivalent to the 4-syllable "computation"
| frequently lately, so we nouned the verb "compute."
|
| In the short-term, I think both words are fine. If it
| matters, use whichever one will best help you achieve your
| goals (e.g., if you're talking to a person who frequently
| uses n>>1 devices to perform computationally intensive
| workloads, you'll probably sound very mildly out-of-place
| nowadays if you don't choose "compute," which may or may not
| be the image you want to portray of yourself).
|
| Slightly longer term, I expect this will be one of those
| cases where dictionaries and pedants try to settle the
| matter, nobody complies, and we have a "gray" vs "grey"
| scenario indefinitely. Beyond the next year or two I'm
| hesitant to place bets on the multitude of possible outcomes.
|
| Way off-topic, patterns of speech like that might be a fun
| way to fingerprint the training data used for an LLM.
| smallnamespace wrote:
| Compute as a noun may be a newer usage, but a noun sharing
| the same form as the verb is common in English, so it at
| least conforms to the patterns of the language.
|
| For example dispute/disputation or repute/reputation ("a
| person of ill repute").
| meindnoch wrote:
| "compute" is used by the same people who call a collection of
| things a "stack".
| ajkjk wrote:
| I think of "compute" as referring to the resources for
| computations, rather than the computations themselves. For
| instance "compute-intensive" for an algorithm means that it
| is expensive in terms of computation resources, i.e. CPU
| cycles, as opposed to say I/O or network usage or something.
| Sometimes it refers to the actual physical hardware,
| sometimes to the theoretical resource of CPU time.
| semi-extrinsic wrote:
| This exact topic came up in another thread a.few months ago,
| and I did some digging:
|
| I believe the term "compute node" was first used in the
| context of the Intel ASCI Red supercomputer that was
| installed at Sandia in 1997. This later led to the Cray XT3,
| XT4, XT5 families of machines that used the same terminology.
|
| But I believe the term was not in generic usage outside of
| those specific supercomputers until around 2005-2010.
|
| And it is more recently that the term has been extended
| beyond referring to hardware.
| SilasX wrote:
| It looks like the general trend of truncating nouns down to
| the verb they're based on instead of requiring the noun-ified
| form. You may have seen these:
|
| 1) an invitation to the party -> an invite to the party
|
| 2) I like that quotation. -> I like that quote.
|
| 3) I gave him a consultation. -> I gave him a consult.
|
| You could even count:
|
| 4) What's the request? -> What's the ask?
|
| Also, broken English seen on receptacle in China:
|
| 5) "The environment needs your conserve." (instead of
| "conservation")
| kyykky wrote:
| What makes you think so? I quickly skimmed through and found
| nothing of the sort.
|
| Edit: My take home message was that there is an equivalent
| higher order formulation which allows more structure and might
| be theoretically interesting. Usually higher order formulations
| are numerically more challenging.
| somat wrote:
| Do Feynman diagrams reduce computation? my understanding is
| that Feynman diagrams are a clever simple way to document
| particle interaction. That is, more for human consumption than
| machine. My gut tells me it is the same as saying "we were able
| to reduce the computational complexity of our simulation
| because we had a good flowchart", by which I mean, It may help,
| but the flowchart is not the code(yes, uml, I am looking at
| you)
| splittingTimes wrote:
| I worked in that field doing numerical many-body simulations of
| electron dynamics interacting with their environment in solid
| state devices like quantum dots, graphene, photonic waveguides
| & cavities , etc.
|
| You would start from a Lagrangian formulation of the classical
| interaction, let's say Light-Matter, that would yield for
| example the Schrodinger and Maxwell equations. Following a
| Legendre transformation (there a post on the HN front page the
| other day on that) you end up with a so-called Hamilton
| operator from which you can derive a (huge) set of coupled
| differential equations which you then solve.
|
| Here, if you wanted to increase temporal accuracy, it typically
| leads simply to longer calculation times.
|
| We also tried a different approach using Feynman's path
| integrals and boy did that explode numerically. We optimized
| our programs to the point where everything was reduced to work
| on bits, but to no avail it was numerically unstable and the
| memory consumption when through the roof the longer or more
| accurate you wanted to make the simulation.
|
| So, I would argue that NO, Feynman does not make it easier per
| se.
|
| However, other groups made it work somehow.
|
| As a starting point you can check that paper and it's
| references from the introduction section.
|
| https://doi.org/10.1002/pssb.201000842
| nextworddev wrote:
| For a second I thought they solved it
| neonate wrote:
| https://www.linkedin.com/posts/johnwsanders_a-canonical-hami...
| cherryteastain wrote:
| So, the gist of the paper seems to be:
|
| 1. Put all terms of the momentum and mass conservation equations
| of Navier Stokes on the right hand side. These should be normally
| identically 0.
|
| 2. Define an error term ('residual') R for each equation, this is
| basically the value of the RHS from step 1. The error term is 0
| when pressure and velocities satisfy N-S.
|
| 3. Define the Lagrangian as the sum of residuals squared
|
| 4. Apply the Legendre transformation to the Lagrangian to get the
| Hamiltonian and the conjugate momenta
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