[HN Gopher] Structure and Interpretation of Classical Mechanics ...
___________________________________________________________________
Structure and Interpretation of Classical Mechanics (2014)
Author : the-mitr
Score : 129 points
Date : 2025-10-27 04:27 UTC (18 hours ago)
(HTM) web link (tgvaughan.github.io)
(TXT) w3m dump (tgvaughan.github.io)
| noosphr wrote:
| I've often dreamed of a "Structure and interpretation" series of
| books.
|
| Scheme is pretty close to a universal computation substrate that
| provides enough ergonomics to be human understandable and writing
| anything out in it provides genuine illumination to what's going
| on under the hood.
|
| The "little" books are a tease of what that series could be.
| fouronnes3 wrote:
| I want to write Structure and Interpretation of Geometric
| Optics. I have an outline already in my notes and I'm convinced
| that the computing-first approach would benefit the field
| immensely. I've been learning optics for a while and writing a
| python library [0]. With a background in software it's very
| obvious that there is strong SICP vibes in lenses, refraction,
| etc. I just need someone to trust me and write me a check for 1
| or 2 years salary so I can go full bunker mode and write it =)
|
| [0] https://victorpoughon.github.io/torchlensmaker/
| noosphr wrote:
| Sicp is not computation first. Sicp is understanding first.
|
| Doing the calculations automatically is a happy side effect
| of finding the right abstractions for describing what's
| happening physically and those abstractions being expressed
| in scheme already.
|
| E.g. Exercise 3.73 in SCIP asks how to implement an
| electrical circuit using a stream data structure. Because of
| all the work done beforehand you end up with an expression
| which describes the time behaviour of the circuit using the
| same expressions that describe its layout.
| throwaway81523 wrote:
| I didn't get anywhere trying to read this book. Then I watched a
| youtube video about calculus of variations and suddenly
| Lagrangian dynamics made total sense to me. I should probably try
| reading the book again.
| arunix wrote:
| Do you remember which video that was?
| griffzhowl wrote:
| I don't know which it was but Dr. Jorge Diaz has an excellent
| video on Lagrangian mechanics as part of a series on quantum
| mechanics (this video just pertains to the formalism
| applicable classically)
|
| https://www.youtube.com/watch?v=QbnkIdw0HJQ
| Schiphol wrote:
| Does anybody know of a way to run the code in this book? I've
| tried a couple of times but never quite succeeded.
| nemoniac wrote:
| You can run it in Racket with the SICP language.
|
| https://docs.racket-lang.org/sicp-manual/SICP_Language.html
| Schiphol wrote:
| Ah, nice, I'll try that. SICM in particular relies on
| numerical routines and things for scientific computing that
| this perhaps doesn't cover. We'll see. Thanks!
| noosphr wrote:
| https://github.com/perceptual-ai/sicm-scheme-exercises
| Schiphol wrote:
| This is what I tried, unsuccessfully if I remember correctly.
| I'll give it another try, thanks!
| gsf_emergency_4 wrote:
| https://groups.csail.mit.edu/mac/users/gjs/6946/installation...
|
| https://stackoverflow.com/questions/62518079/scmutils-for-si...
|
| Easter egg: if you dig deeper into the code you will find the
| "amazing bug" (footnote on page 5)
|
| https://arxiv.org/abs/1211.4892
| gugagore wrote:
| The bug being "perturbation confusion"?
| gsf_emergency_4 wrote:
| The specific bug in sicm is discussed on pages 19-21
|
| (Sorry, it's been a while, but iirc the code comments call
| it the amazing bug, with credit to Radul)
| kkylin wrote:
| MIT Scheme (and ScmUtils) are unfortunately not getting enough
| maintainence, but they still work with a little effort.
| Probably better on Linux than any other environment. If you
| have a Mac you may try this:
|
| https://github.com/kkylin/mit-scheme-intel-mac-patch?tab=rea...
|
| Works well on Intel Macs and (with effort) _mostly_ works on
| Apple Silicon.
| HexDecOctBin wrote:
| I found this, will try tomorrow:
| https://hub.docker.com/r/sritchie/mechanics/
| zkmon wrote:
| Funny that we call it classical. Newton wouldn't have called it
| so. Maybe we should categorize sciences based on the spatial
| scale at which they operate.A specific scale might define a world
| that has it's logic system, purpose, reasoning etc. For example,
| quantum scale, human scale and cosmic scales have their own
| physics, logic and causality.
| kergonath wrote:
| > Newton wouldn't have called it so.
|
| Of course. To him that would be modern mechanics. Or just
| mathematical natural philosophy, or whatever.
|
| > Maybe we should categorize sciences based on the spatial
| scale at which they operate.
|
| That would not be very useful, because there is no boundary.
| Nothing in general relativity says "below this everything is
| Newtonian". As a matter of fact we need to consider
| relativistic effects in quantum chemistry calculations that
| involve some heavy elements, at length scales smaller than 0.1
| nm. Similarly, they just gave a Nobel prize for work on
| "Quantum properties on a human scale".
|
| > For example, quantum scale, human scale and cosmic scales
| have their own physics, logic and causality.
|
| That is not at all how these frameworks are built, and that is
| not the dominant epistemological approach. The mainstream view
| is that there is a theory of everything that exists but is
| unknown to us, and that our various theories are approximations
| of that theory under different assumptions. They look
| categorically different because we don't understand the
| overarching framework, not because nature is fundamentally
| different depending on scale.
|
| Also, I don't see how the logic is fundamentally different
| between e.g. quantum mechanics and general relativity. Both
| rely heavily on things like Hamiltonian mechanics or
| symmetries. Some behaviours are different (like photons
| following geodesics and not straight lines, or superpositions
| of quantum states), but these are not a fundamental problem: a
| straight line is a limit case of a geodesic in a flat space,
| and a unique state is a limit case of superposition.
|
| I am not saying that everything is fine and we know everything,
| just that there is no clear boundary between the situations in
| which different theories are required and we cannot neatly
| decompose the universe into different realms where different
| theories apply.
| zkmon wrote:
| From my little knowledge, logic at Quantum scale appears
| quite different:
|
| * Things don't have their own location or identity
|
| * Spatial and temporal extents don't exist
|
| * Something may be true and false at the same time, or
| concept of true and false may not be defined
|
| * cause and effect goes for a toss, as behavior of time is
| different
|
| * Existence and non-existence co-exist, or come into
| existence together
|
| Similar effects at relatively-infinite scale (maybe purely
| mathematical)
|
| * Comparisons (big/small/equal) breakdown
|
| * Regular arithmetic and logic breaks down
| simiones wrote:
| Most of these things are misunderstandings of quantum
| mechanics, as we know it today.
|
| The main thing that is at the root of all of them is the
| word "things". In QM, the ground truth of the world is the
| wavefunction of the system. The wavefunction assigns some
| amplitude (potentially 0) to any possible state of the
| system that it describes. It then evolves purely
| deterministically from past to future, according to
| Schrodinger's equation (or Dirac's equation, if you want to
| discuss speeds close to that of light). The only kink is
| interaction with a measurement device (what constitutes a
| measurement device is one of the big mysteries that we
| don't yet have an answer for). After a measurement, the
| wavefunction collapses non-deterministically to one of the
| states that the measurement device was set up to detect,
| with a probability that is proportional to the amplitude of
| the wave function of that state.
|
| Now, this is the "ground truth" of QM. Everything else,
| such as particles and space-time and so on are just stories
| we tell to make sense of the wavefunction and its behavior.
| Sometimes those descriptions break down, and they start
| assigning weird fanciful ideas, such as retrocausality etc
| - but these just prove that the stories are wrong, that
| they are misinterpreting the math of the wavefunction.
|
| I'd also note that the main "time is weird" factoid you
| encounter related to QM experiments, the delayed-choice
| quantum eraser, is mostly a misunderstanding /
| sensationalization of the actual physics and the
| experiment. It again only proves that certain
| interpretations of what the wavefunction and its collapse
| represent are not temporally consistent, but the direct
| conclusion from this should be that the interpretations are
| wrong, not that "cause and effect goes for a toss, as
| behavior of time is different".
| zkmon wrote:
| If we can't even define a "thing", by identifying inside
| and outside of it, what it is and what it is not, where
| it is and where it is not, that itself a big contrast
| with Newtonian (human scale) mechanics. Everything one
| can talk about Quantum mechanics is coompletly alien to
| the human perceived world. That should justify the
| distinction by scale.
| nyrikki wrote:
| Most of these happen at most scales, but is more to do with
| the classic _laws_ of classic logic that we accept _A
| priori_ because they are useful.
|
| * PNC: at most one is true; both can be false
|
| * PEM, at least one is true; both can be true
|
| * PNC + PEM, exactly one is true, exactly one is false
|
| If you stick to the familiar computational complexity
| classes P=co-P, but NP != co-NP, and those both relate to
| the accessibility of T/F, where both P and NP are by
| definition decision problems, specifically the ones that
| can be verified in poly time.
|
| If you ignore the Heisenberg uncertainty principle to avoid
| that complexity, the standard model of QM is just mapping
| continuous functions to discrete space, and will have
| problems with the above.
|
| This happens in math too, where we use the rationals over
| the reals or Cauchy sequences to construct the reals,
| because almost all reals are normal and non-computable, and
| even equality between to "real" real numbers is
| undecidable.
|
| This is also related to why after Weierstrass show that
| almost all continuous functions are nowhere smooth, we
| moved to the Epsilon-Delta definition of a limits etc...
|
| We have the lier's paradox, which is easier to understand
| than Berry's paradox, which relates to Chaitin proof that
| there is an upper limit to what any algorithm can prove.
|
| Things at Quantum scale do act very different than our
| typical intuition, but lots of maps from a continuous space
| to discrete categories can exhibit the same behavior even
| at macro scale, we just can often use a model that lets us
| ignore that to accomplish useful work.
|
| Often we can even use repeated approximation or other
| methods to reduce those problems to something that is
| practical, but that is still the map and not the territory.
|
| Superposition is just that: f(a + b) =
| f(a) + f(b)
|
| And: f(sa) = s*f(a)
|
| If you have two vectors, one at i (0,1) and one at 1 (1,0),
| but a map that maps only to choice(1,i) and think of that
| as up and down, how you divide that continuous arc from
| that segment of the unit circle to UP or DOWN, almost all
| of quantum mechanics still works, and it will be the
| contradictions that conflict with our intuitions that will
| become the barrier. (ignoring Heisenberg)
|
| If you think about Heisenberg uncertainty as indeterminacy
| instead, Independence in math (Like ZFC + CH) which are
| neither provable as false or true, Chatlin's complexity
| limits, the breakdown of Laplacian determinism, or even
| modal logic all apply at multiple levels.
|
| The amazing thing in my mind is that we found useful models
| despite these limits and using methods which are effective.
|
| But IMHO it is best to think that in the quantum world, we
| aren't so lucky rather than those limits aren't still
| lurking under the surface at the macro scale, which they
| very much are.
| IAmBroom wrote:
| > > Maybe we should categorize sciences based on the spatial
| scale at which they operate.
|
| > That would not be very useful, because there is no
| boundary. Nothing in general relativity says "below this
| everything is Newtonian". As a matter of fact we need to
| consider relativistic effects in quantum chemistry
| calculations that involve some heavy elements, at length
| scales smaller than 0.1 nm. Similarly, they just gave a Nobel
| prize for work on "Quantum properties on a human scale".
|
| You are just saying "well ackshually". I dare you to build a
| cabinet using the Hamiltonian. I double-dog-dare you.
|
| > > For example, quantum scale, human scale and cosmic scales
| have their own physics, logic and causality.
|
| > That is not at all how these frameworks are built, and that
| is not the dominant epistemological approach.
|
| Again, 99.999% of all functional mechanics don't involve
| epistemology.
|
| > The mainstream view is that there is a theory of everything
| that exists but is unknown to us, and that our various
| theories are approximations of that theory under different
| assumptions.
|
| Oh! You're so close to seeing the point... There are multiple
| levels of approximation (at least two), and the one we all
| experience is Newtonian. Perhaps more accurately, our senses
| mostly believe pre-Newtonion approximations, which is why it
| took until Newton to realize how inaccurate they were.
|
| > Also, I don't see how the logic is fundamentally different
| between e.g. quantum mechanics and general relativity.
|
| You're pretty radically moving the goalposts here. GP was
| talking about Newtonian mechanics, not Hamiltonian.
| simiones wrote:
| > You are just saying "well ackshually". I dare you to
| build a cabinet using the Hamiltonian. I double-dog-dare
| you.
|
| The Hamiltoninan (and Lagrangian) are much more amenable to
| actual physical calculations, at least on a computer, than
| the Newtonian formulations of classical mechanics - but
| otherwise they are perfectly equivalent mathematically. I'm
| not sure where you'd need any kind of dynamical laws in the
| building of a cabinet, on the other hand. Are you trying to
| arrange for a system of inclined planes and pullies to slot
| the pieces into place?
|
| > Perhaps more accurately, our senses mostly believe pre-
| Newtonion approximations, which is why it took until Newton
| to realize how inaccurate they were.
|
| This is a bit of misnomer. Our senses and intuitions are in
| fact remarkably accurate for a certain range of values, and
| quite equivalent to what Newton's laws of motion say about
| these. To some extent, Newton "only" found a simple
| formalism to represent our existing intuitions. Our
| intuitions of course break down in other places, such as at
| very high speeds, or very high altitudes , where
| relativistic corrections start to become significant.
|
| QM however is a paradigm shift in how the world is
| described, and it is completely non-intuitive, even in
| regimes where its predictions are fully aligned with our
| intuitions and senses. You can use QM to compute the
| collision of two ideal balls on an ideal plane, and the
| results will exactly match your intuitions. But the
| computation and even the representation of the system will
| not, in any way.
| kergonath wrote:
| > You are just saying "well ackshually". I dare you to
| build a cabinet using the Hamiltonian. I double-dog-dare
| you.
|
| We do that every day using things like finite elements.
| It's just a different Hamiltonian that accounts for the
| fact that we simplify bunches of atoms into a continuum.
|
| > Again, 99.999% of all functional mechanics don't involve
| epistemology.
|
| Discussing the nature of scientific theories _is_
| epistemology. The parent's point is epistemological in
| nature.
|
| > ! You're so close to seeing the point... There are
| multiple levels of approximation (at least two), and the
| one we all experience is Newtonian.
|
| You are off base. There are many, many approximations that
| may or may not overlap. It's not just onion layers.
|
| > You're pretty radically moving the goalposts here. GP was
| talking about Newtonian mechanics, not Hamiltonian.
|
| GP was talking about different theories applying at
| different scales. Sorry, it may not have been clear in
| context but Hamiltonian here is not just the generalisation
| of Newtonian mechanics we learn about in 2nd year Physics.
| these theories (quantum mechanics, Newtonian mechanics, and
| relativity) can be written using the Hamiltonian formalism.
| BigTTYGothGF wrote:
| > the one we all experience is Newtonian
|
| The one we all experience is Aristotelian.
| IAmBroom wrote:
| We call music from Newton's age "classical".
|
| As the past recedes, "the golden age" advances in time.
| "Hollaback Girl" is now a classic oldie.
| hyperjeff wrote:
| These days "classical" just distinguishes all things quantum
| from non-quantum. General relativity is considered a classical
| theory, for example.
| michaelsbradley wrote:
| There's also _Functional Differential Geometry_ by the same
| Sussman and Wisdom:
|
| https://mitp-content-server.mit.edu/books/content/sectbyfn/b...
| in_a_hole wrote:
| Does anyone know a text which justifies why the Lagrangian
| approach works? This text and many others I have encountered just
| start with the Principle of Least Action taken as given and go
| from there but I'm left wondering why we define the Action as
| this object and why we should expect it to be minimised for the
| physical trajectory in the first place.
|
| Failing a full derivation from the ground up, a proof of the
| equivalence to Newtonian mechanics would be interesting.
| wbpaelias wrote:
| I believe Veritasium had a series where they derive the
| equivalence to Newton's laws
| ano-ther wrote:
| It's been a while but I seem to remember that the first book of
| Landau-Lifschitz' Theroretical Mechanics starts with a 20 page
| discussion that does this and culminates in the Lagrangian.
| in_a_hole wrote:
| I recently got hold of a copy of that. I started Hand & Finch
| - Analytical Mechanics but their woolly discussion of virtual
| work and virtual displacement was very frustrating and
| unenlightening. Perhaps I'll have a better time with L&L.
| nchagnet wrote:
| Regarding the "why the action is this object" part of the
| question, I find that the easiest way to think about it is from
| the Hamiltonian perspective. There you can think of it as
| minimising energy along a trajectory. From that point, a
| Lagrangian is just a mathematical trick to express the
| symplectic structure differently.
|
| But if your question was more about "why minimizing something
| yields trajectories", I personally would argue this is beyond
| physics. As an empirical science, physicists have seen this
| kind of behaviour broadly (optics, classical mechanics, quantum
| mechanics) and just unified it as an overarching principle.
|
| Finally regarding the proof to newtonian mechanics, I don't
| have anything handy from the pure Newtonian perspective beyond
| the usual "minimises the lagrangian and your equations of
| motions look the same". However, you might be interested in
| proofs which show newtonian gravity as low energy approximation
| of general relativity. And since general relativity has a nice
| action formulation, it all gets nicely tied in.
|
| Hope this helps!
| in_a_hole wrote:
| But simply getting to the Lagrangian picture from the
| Hamiltonian picture would just leave me wondering why the
| Hamiltonian picture works!
|
| My motivation for getting to the bottom of all this is to
| fill the gaps in my physics understanding at least up to
| quantum mechanics. I have a grasp of QM but I would like to
| have some insight into the conceptual leaps that brought us
| there from classical mechanics. QM works in the Hamiltonian
| picture and I recall from my undergrad days that you get
| there from a Legendre transformation on the Lagrangian (or
| something to that effect) so I'm trying to understand the
| justification of that approach before moving up the
| conceptual ladder.
|
| Ideally I would like to be able to trace my way from simple
| postulates based on observation of the physical world all the
| way to QM, then maybe to QFT after that.
| aeonik wrote:
| You know about the Ultraviolet catastrophe?
|
| Physics Explained Ultraviolet Catastrophe:
| https://youtu.be/rCfPQLVzus4
|
| This Veritassium Video goes back further in time and talks
| about Action, and the Genesis of this idea.
| https://youtu.be/qJZ1Ez28C-A
|
| Chemistorian, The history of Atomic Theory.
| https://youtu.be/SqYPrA7upiE
| nchagnet wrote:
| "QM works in the Hamiltonian picture and I recall from my
| undergrad days that you get there from a Legendre
| transformation on the Lagrangian (or something to that
| effect) so I'm trying to understand the justification of
| that approach before moving up the conceptual ladder."
|
| That is only one approach to QM. As you rightly point out,
| Hamiltonian and Lagrangian approaches are always two sides
| of the same coin: one is only the Legendre transformation
| of the other and so they describe the same physics.
|
| So to that end there is a neat QM Lagrangian
| representation: the path integral formulation. You can
| apply it to basic QM as well as to QFT or even QFT in
| curved spacetimes and string theory.
|
| So if your goal is to "trace your way from simple
| postulates", that is a good way: assume your system can be
| described by an action, and go from there. It works in
| pretty much every scenario. In most research I've been
| involved, you always end up constructing generic actions
| whose coefficients eventually determine the behaviour of
| the theory.
|
| And to get back to what I assume is your real conundrum
| (why do we extremize something to begin with), I just don't
| think there's any true answer as to why nature behaves this
| way.
|
| What we can answer is: what is the action, what does it
| represent, what does it mean to extremize it? The short
| answer to this (provided by the path integral formulation I
| mentioned earlier) is that the action is essentially
| controlling a probability distribution of paths in a given
| geometry. In quantum mechanics, we interpret this
| distribution with that of actual particles. When you
| extremize the distribution, you essentially find the most
| likely trajectory, and if your distribution is peaked
| enough around that trajectory, then you can take this path
| as representative of your system when fluctuations are
| ignored.
|
| So in the classical limit of QM, that trajectory is all
| that's left (and that would be the classical mechanics
| trajectory).
|
| Interestingly a similar interpretation exists in
| statistical physics. If you "complexify" your time
| dimension, your action is again on a Euclidean (instead of
| lorentzian) spacetime and the time direction behaves like a
| circle whose radius sets a scale akin to a temperature.
| This might sound a bit complex but where I'm going with
| this is that once again, you can think of this euclidean
| path integral as a distribution of paths over fluctuations
| (this time thermal), and the extremum is this time the
| system's behaviour when at equilibrium.
| omnicognate wrote:
| It's an old book and I can't vouch for it as I only just
| discovered it myself, but it appears to be very highly
| regarded, it focuses on precisely the questions you (and I)
| have, and just from the preface I like the author already [1]:
| The Variational Principles of Mechanics, by Cornelius Lanczos.
|
| There's a PDF here:
| https://pages.jh.edu/rrynasi1/PhysicalPrinciples/literature/...
|
| [1] An appetising quote:
|
| > The author is well aware that he could have shortened his
| exposition considerably, had he started directly with the
| Lagrangian equations of motion and then proceeded to Hamilton's
| theory. This procedure would have been justified had the
| purpose of this book been primarily to familiarize the student
| with a certain formalism and technique in writing down the
| differential equations which govern a given dynamical problem,
| together with certain "recipes" which he might apply in order
| to solve them. But this is exactly what the author did not want
| to do. There is a tremendous treasure of philosophical meaning
| behind the great theories of Euler and Lagrange, and of
| Hamilton and Jacobi, which is completely smothered in a purely
| formalistic treatment, although it cannot fail to be a source
| of the greatest intellectual enjoyment to every mathematically-
| minded person. To give the student a chance to discover for
| himself the hidden beauty of these theories was one of the
| foremost intentions of the author.
| in_a_hole wrote:
| I had heard about this book and that quote makes me want to
| read it. Thank you.
| DarmokJalad1701 wrote:
| The author of this book is the same Lanczos that the "Lanczos
| Resampling" algorithm (used in image resizing for example),
| is named after:
|
| https://en.wikipedia.org/wiki/Lanczos_resampling
| whatshisface wrote:
| The least action principle conceptually emerged from the least
| time principle for light. Light refracts along the path that
| gets it from the starting to the ending point the quickest, and
| the index of refraction is what regulates its speed. The
| question went like this: we know that potential and kinetic
| energy work together to regulate the speed of moving objects.
| Is there a way to combine the two quantities into something
| like an index of refraction? The analogy between potential
| fields and optics isn't just conceptual - beams of charged
| particles are focused using electromagnetic "lenses," made out
| of fields.
| in_a_hole wrote:
| Do you know any references that discuss this in detail? I'm
| interested in the history of these developments. Who noticed
| this? Who asked this question?
| lucozade wrote:
| > why we define the Action as this object and why we should
| expect it to be minimised for the physical trajectory in the
| first place.
|
| The most coherent explanation I've heard was from Feynnman [0].
| As far as I understand it (and I may well not have understood
| it at all well), at the quantum level, all paths are taken by a
| particle but the contributions of the paths away from the
| stationary point tend to cancel each other. So, at a
| macroscopic level, the net effect appears to be be that the
| particle is following the path of least action.
|
| > a proof of the equivalence to Newtonian mechanics
|
| The Lagrangian method isn't really equivalent to Newton's
| method. Again, Feynman talks about this in [0]. It's that for a
| certain class of action, the Euler-Lagrange equations are
| equivalent to Newton's laws.
|
| It's perfectly plausible to come up with actions that recover
| systems that represent Einsteinian relativity or quantum
| mechanics. This is the main reason (as I understand it) why
| it's considered a more powerful formalism.
|
| [0] https://www.feynmanlectures.caltech.edu/II_19.html
| abdullahkhalids wrote:
| There is no explanation for this, same as there is no real
| explanation for why energy is conserved or why closed systems
| have non-decreasing entropy. As others have pointed out, you
| can show correspondence to Newtonian mechanics under some
| assumptions, but the Lagrangian approach is applicable to a
| wide variety of areas in physics - classical mechanics, optics,
| quantum mechanics, quantum field theory, etc.
|
| The universe has these weird laws, and for now, all we can do
| is accept them as is. But hopefully, in the future, someone
| will figure out deeper and simpler principles.
| nh23423fefe wrote:
| Seems false. Energy conservation is explained via time
| translation symmetry and noether's theorem. 2nd law is
| explained by boltzmann's H-theorem.
| aap_ wrote:
| Unfortunately I can't help with the classical picture, but in
| quantum physics it all comes out very nicely: You can interpret
| the Lagrangian as giving all possibilities to build a
| trajectory through spacetime. In the path integral formulation
| we then follow one such trajectory from one configuration to
| another configuration and find its amplitude. And then we
| integrate over _all_ possible trajectories that we could have
| picked. For incoherent trajectories there will always be
| another one that cancels out the amplitude. Where the
| amplitudes add up constructively you will find stationary
| action and the classical behavior in the limit. So this is a
| depth-first approach: first follow one trajectory completely,
| then add up all possible trajectories.
|
| The Hamiltonian approach in contrast is breadth-first: you
| single out a time axis, start with some initial state, and
| consider all possibilities that a particle (or field in QFT)
| could evolve forwards in time just a tiny bit (this is what the
| Hamiltonian operator does). Then you add up all these
| possibilities to find the next state, and so you move forwards
| through time by keeping track of all possible evolutions all at
| once. This massive superposition of everything that is possible
| (with corresponding amplitudes) is what you call a state (or
| wavefunction) and the space that it lives in is the Hilbert (or
| Fock) space.
|
| So Lagrangian/path-integral: follow full trajectories, then add
| up all possible choices. depth-first
|
| Hamiltonian/time-evolution: add up all choices for a tiny step
| in time, then simply do more steps: breadth-first
|
| I imagine it a bit like a scanline algorithm calculating an
| image as it moves down the screen (Hamiltonian) vs something
| like a stochastic raytracer that can start with an empty image
| and refine it pixel by pixel by shooting more rays (Lagrangian)
|
| This is my layman explanation anyways...hopefully it helps,
| even though i can't say much about their relationship in
| classical physics.
| mindaslab wrote:
| What software they used to create this wonderful book?
| dang wrote:
| Related. Others?
|
| _Structure and Interpretation of Classical Mechanics (2015)_ -
| https://news.ycombinator.com/item?id=40805136 - June 2024 (12
| comments)
|
| _Structure and Interpretation of Classical Mechanics_ -
| https://news.ycombinator.com/item?id=31568387 - May 2022 (1
| comment)
|
| _Structure and Interpretation of Classical Mechanics_ -
| https://news.ycombinator.com/item?id=23153778 - May 2020 (40
| comments)
|
| _Structure and Interpretation of Classical Mechanics (2015)_ -
| https://news.ycombinator.com/item?id=19765019 - April 2019 (87
| comments)
|
| _Structure and Interpretation of Classical Mechanics_ -
| https://news.ycombinator.com/item?id=9560567 - May 2015 (20
| comments)
|
| _Structure and Interpretation of Classical Mechanics_ -
| https://news.ycombinator.com/item?id=6947257 - Dec 2013 (37
| comments)
|
| _Structure and Interpretation of Classical Mechanics_ -
| https://news.ycombinator.com/item?id=1581696 - Aug 2010 (20
| comments)
| Jtsummers wrote:
| By the same authors so related but not SICM itself:
|
| _Functional Differential Geometry (2012) [pdf]_ -
| https://news.ycombinator.com/item?id=7884551 - June 2014 (38
| comments)
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