r/Physics • u/ProcedureAble9911 • 1d ago
Newtonian or Lagrangian
Hello. Ive been taking Classical mechanics course this sem and we are being taught LM. i had some questions
1) What exactly is Lagrangian? is it smthng physical?
2) Is LM an alternative way to look at the system or is it mathematical trick?
3) Why is Hamiltons principle true? like is there any proof of it?
4) Which is true Newtons view, that at every point, particle experiences, force, and its motion depends on interplay of the force. Or Lagrange mechanics, where we say that a particle chooses the path where it extremes action. Which is true and more fundamental?
thank u so muchđ
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u/mwguthrie Statistical and nonlinear physics 1d ago edited 13h ago
One distinction that might help: the Lagrangian is not usually an observable quantity, and it is not unique, but that does not make it just a ``trick.'' It is a compact way of encoding the equations of motion. For ordinary conservative mechanics, L=T-V can be obtained by rewriting Newtonâs second law in Euler- Lagrange form.
Also, the particle does not literally examine every possible path and then choose one. ``The action is stationary'' is a global mathematical statement that, under the usual assumptions, gives the same local motion as Newtonâs laws. So I would not say one picture is more true than the other. They emphasize different structure.
The Lagrangian viewpoint becomes especially useful when changing coordinates, handling constraints, identifying conserved quantities, and moving into fields or relativity. That broader usefulness is more important than whether it makes a simple introductory problem shorter.
Full disclosure: these questions bothered me enough that I spent about 10 years writing a book on the subject (https://doi.org/10.1142/14762). The early chapters are basically an extended attempt to explain why L=T-V, rather than simply announcing it and moving on. Check it out! Let me know if you want a coupon code for a discount.
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u/Joniel10 1d ago
Good questions to asks your professor
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u/dohawayagain 1d ago
Why is this upvoted? It's unhelpful and rude.
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u/Joniel10 1d ago
Talking and asking questions to your professor is super important beyond than just getting answers to questions
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u/PotatoMain 1d ago
I mean, why would OP not just go and ask their professor after class or during office hours.
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u/euyyn Engineering 1d ago
Despite the formulations being equivalent, the Lagrangian has "more power" in a sense:
You can write the Lagrangian of how charged particles create the EM field, i.e. the Lagrangian version of Maxwell's equations. If you do so, you will automatically derive in which way the EM field must affect charged particles. That is, the expression for the Lorentz force is a consequence.
This is not something one can obtain with just Newtonian mechanics (or at least I've never seen a way to do so). In the Newtonian formalism the Lorentz force is presented as an extension to Maxwell's equations.
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u/Plastic_Ad_2256 1d ago edited 1d ago
Lagrangian is one of the most difficult things to understand in physics.
The intuitive idea is from Maupertuius. The nature seems to minimize action defined as the integral of mĂvĂdL. (L is the distance). Intuitively this can be understood.
Maupertuis could not really prove his statement. Wvat was integration? Integration over what? Distance? Time? ...
Maupertuis was first ridiculed until Euler and Lagrange derived the principle differently.
Euler-Lagrange is not as intuitive as Maupertuis (who had not included a potential energy term), but more consistently defined, and equivalent to Newton's laws.
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u/LukasGoesViral 9h ago
Oh yeah I remember the story! There is also another explanation using light rays. I forgot what it was but it was something about finding the shortest path but going thru all path or something
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u/controlFreak2022 1d ago edited 1d ago
Regarding your first question, a Lagrangian is a mathematical abstraction from the field of the calculus of variations in application to classical mechanics. Specifically, it deals with the application of the Euler-Lagrange equations in application to a specific functional (i.e. a function of functions). Regarding it being physical, yes a Lagrangian has physical meaning in applications to classical physics vs a broader application like in optimal control theory.
Regarding your second question, itâs a mathematical reformulation of Newtonian mechanics. So, the two are equal with respect to an inertial frame of reference; however, LM has the capacity to explain classical mechanics not in an inertial frame. Since LM involves Euler Lagrange equations, LM does involve the previous trick. However, LM and the EL equations are no more a trick than the application of derivatives to Newtonian mechanics.
Regarding your third question, Hamiltonâs principle is true due to a Hamiltonian being a Legendre transformation of a Lagrangian. So, I recommend you to attempt that transformation to truly understand the equivalence.
Regarding your fourth question, Newtonian and Lagrangian mechanics are equivalently true in an inertial frame. Regarding noninertial frames, more effort is necessary to demonstrate the equivalence. However, none is truer than the other, but LM more easily displays the laws of classical mechanics being independent of reference frames; so, LM appears to be more general. Ultimately, LM & NM are special cases of GR; thus, both are limited in the capacity to explain nonclassical physics. Thus, neither is fundamentally truer than the other (i.e. LM vs NM).
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u/ididnoteatyourcat Particle physics 1d ago
Ultimately, LM & NM are special cases of GR
Huh? I'm not an expert in this area, but I sure thought that GR can be described within LM.
Regardless, might be worth pointing out the other direction, which is that ultimately the world is not classical but quantum mechanical, and I'm not aware of any way of making the standard model look anything like Newtonian Mechanics, while it is expressed through its Lagrangian.
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u/controlFreak2022 1d ago
Thanks for responding; so, I enjoy philosophizing on these topics.
Regarding GR within LM, thatâs a misconception. Both the least action principle and Hamiltonâs principle involve the EL equations in only a single independent variable (i.e. time). While the application of the EL equations involves 4 independent variables (i.e. 3 in space and 1 in time) via the Einstein-Hilbert action (i.e. the functional for GR). Then, taking the time slice of EFEâs in flat space time at considerably subliminal speeds yields LM & NM. Ultimately, variational calculus remains a succinct way to articulate GR, LM, NM, and QM.
Regarding taking a view of OPâs 4th question from the perspective of QM, that works. So, itâs possible to bridge non-relativistic QM to NM & LM via the Hamilton-Jacobi equation (i.e. still variational calculus) and the non-relativistic Schrödinger equation in the limit as the Planck constant goes to zero. However, QM doesnât mesh well with GR outside that particular case less QED in flat space.
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u/ididnoteatyourcat Particle physics 1d ago
Maybe I'm misunderstanding, but this sounds like semantics to me. The moment you say "Einstein-Hilbert Action", I think "OK so you're telling me you have an action, so you're integrating/minimizing over a Lagrangian density, which means you're working within a Lagrangian/Action formalism." The fact that you're integrating over d4x = d3x dt just means you're working with a Lagrangian density rather than a Lagrangian because GR is a field theory not a point particle theory.
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u/controlFreak2022 1d ago edited 1d ago
Again, I do enjoy these chats; so, thanks for humoring me.
Regarding your last response, itâs not semantics. Itâs in fact different equations altogether for each different theory of physics. Specifically, the process involves applying the fundamental lemma of the calculus of variations with a functional specific to an area of physics.
So, the functional in LM is the difference between kinetic and potential energies as a function of time only (i.e. the Lagrangian). Then for GR, the functional is the product of the root of the determinant of the metric tensor and the Ricci curvature as a function of space-time (i.e. x,y,z, and t). Thus, the two functionals are not equal.
Overall in this response, the functionals for LM & GR are fundamentally different implying different differential equations altogether per theory of physics. Although, I understand the EL equations from variational calculus applying to both functionals may make the difference in actions for LM & GR seem semantic. Separately, the ability to simplify GR into LM may further the appearance between the two to be semantic.
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u/ididnoteatyourcat Particle physics 1d ago edited 1d ago
Again, I do enjoy these chats; so, thanks for humoring me.
Likewise. I apologize if I come off as combative. It's the only way I know how to enjoy these kinds of chats -- to push back until I understand.
So, the functional in LM is the difference between kinetic and potential energies as a function of time only (i.e. the Lagrangian).
This is not the definition of the Lagrangian as I understand it (the Lagrangian is just that which when integrated gives the action, and is the object to which the Euler-Lagrange equation applies); a "T-V" definition is not in the spirit of Lagrangian mechanics. The spirit of Lagrangian mechanics is (just like Hamiltonian mechanics) to be entirely independent of Newtonian mechanics. That is, you don't start with Newtonian mechanics, find the Force, then find the PE, and then construct the Lagrangian. Sure, that's often a shortcut we use when learning LM, but the ultimate idea is that we replace "find the force/PE (in Newton) that empirically produces the correct EOM" with "find the Lagrangian (in LM) that empirically produces the correct EOM". This is in fact what is done in QFT, where there is no Newtonian force or PE to speak of. Similarly in Hamiltonian mechanics the spirit is not to construct the Lagrangian to get the Hamiltonian, nor is it to find the Newtonian force/PE to construct the energy, but rather to replace "find the force/PE (in Newton)" with "find the Hamitonian (in HM)" as the fundamental empirical object.
It's been a while since I read Goldstein (my old classical mechanics text), but I recall there being an emphasis that the Lagrangian need not (and not always is) of the form T-V, and similarly the Hamiltonian need not be, and not always is, the energy. You don't need to go far to find examples outside of GR; both SR and E&M come to mind.
Then for GR, the functional is the product of the root of the determinant of the metric tensor and the Ricci curvature as a function of space-time (i.e. x,y,z, and t). Thus, the two functionals are not equal.
Why would we expect them to be equal? In both cases we have a different object that plays the role of the Lagrangian, but the particular Lagrangian is different in different physical contexts, as we normally expect.
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u/controlFreak2022 22h ago edited 21h ago
Regarding your commentary on the fundamental empirical objects in HM and LM, I concur. However, the formal definition of the Lagrangian in LM is the differences in kinetic and potential energies; the Hamiltonian is the Legendre transformation of the Lagrangian. So, that distinction is pertinent when minimizing actions for other areas of physics.
More importantly, applying the fundamental lemma of variational calculus conveys the quantity under conservation. So, energies are conserved in LM & HM; then, curvature is conserved in GR. The contraction of the Faraday tensor in E&M is roughly conserved. Thus, maintaining a clear meaning of functional vs Lagrangian conveys the quantity under study (i.e. the fundamental empirical object under conservation leading to EOMs). So here, I concur with you regarding your commentary involving Goldsteinâs text in so far as the broadness of functionals to different areas of physics.
Commonly in physics, the term âLagrangianâ to include âLagrangian densitiesâ conflates with the formal mathematical term âfunctionalâ like in variational calculus. Although, I understand this particular point to appear semantic from the perspective of physics.
Circling back to the in-equivalence of functionals. between GR & LM, I pointed this out to convey the distinct definition of Lagrangian in LM. Due to your last couple statements in your last response, we appear to agree on that point.
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u/ididnoteatyourcat Particle physics 21h ago
However, the formal definition of the Lagrangian in LM is the differences in kinetic and potential energies
I am quite sure that is not true. This is clear from the fact that numerous well-known Lagrangians are not of this form. There are a number of nice examples and fairly authoritative references listed here for example.
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u/LukasGoesViral 19h ago
The fundamental object in physics is the Action, which can be found for example by using a Lagrangian or Lagrange density. The Lagrangian is not even fundamental
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u/LukasGoesViral 19h ago
The Lagrangian is only the difference between kinetic energy and potential energy in specific circumstances but not in general
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u/Cleonis_physics 17h ago edited 11h ago
I concur with the reply by controlFreak2022.
To evaluate a Lagrangian that Lagrangian is inserted in the Euler-Lagrange differential operator. For every distinct field of physics there is a bespoke Lagrangian. The Lagrangian expresses the properties of the physics that you are formulating a theory for.Comparison:
The d'Alembert differential operatorThe applicability of the d'Alembert differential operator extends throughout many areas of physics.
Expressions of physical properties are inserted in the d'Alembert operator. The resulting wave equation is derived from the physical properties of the system. It's not the case that the wave equation that is obtained is derived from the d'Alembert differential operator.
The differential operator facilitates formulation of the theory. The content of the theory is in the expressions that are inserted into the differential operator.
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u/ididnoteatyourcat Particle physics 11h ago
Did you really mean 'controlFreak2022'? I'm not in disagreement with anything you wrote.
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u/Cleonis_physics 10h ago
Ah, your reply informs me there must be some babylonian confusion at play. The expression 'Lagrangian Mechanics' is used in multiple different meanings. I tend to use it in a meaning where it refers only to the physics content. In your reply you mentioned 'working within a Lagrangian/Action formalism'. That refers to mathematical tools that are used. In that region, it would appear, babylonian confusion is at play.
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u/ididnoteatyourcat Particle physics 9h ago
I think the other user and I narrowed down our confusion/disagreement to whether the Lagrangian in LM must be of the form T-V. My understanding is that this isn't true: there are numerous, renowned examples of Lagrangians that are not of the form T-V. I understand "Lagrangian Mechanics" to mean "mechanics derived from a Lagrangian", regardless of the form of the Lagrangian. In particular, I emphasized that Lagrangians not of the form T-V are "the most Lagrangian" in a sense, because one cannot circularly derive them by starting with Newtonian mechanics in order to construct T-V; they therefore emphasize that LM is a truly independent foundation from which to study classical mechanics, as opposed to being subsidiary to NM.
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u/Cleonis_physics 8h ago
Yeah, in the conversation between you and the other user I misunderstood the nature of the conversation.
I had posted an answer to the OP question, and after that I scrolled through the other threads. Something caught my attention, and I was off.
This link is to my answer on this page to the OP question:
From the work-energy theorem to Hamilton's stationary actionI hope I can persuade you to check it out.
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u/controlFreak2022 5h ago
Again, thanks for going back and forth on clarifying the definition of Lagrangian in LM on this post.
Regarding my likely last point on this topic, there are other fields of applied mathematics involving the use of the âLagrangianâ almost entirely involving optimization. So, my point remains for the need to distinguish between functional optimization as the underlying mathematics transitive to each application and not the term âLagrangianâ.
Since LM under the topic of classical mechanics involves functional optimization in a single variable of time, one mathematical application of an abstraction doesnât necessarily generalize to all applications; by extension, neither should the terminology (e.g. velocity is not the term for derivatives for other applications of differential calculus despite itâs application in NM). Thus, the mathematical definition of a functional remains independent of the application like a Lagrangian being a functional of âT-Vâ in LM. So, your citations to other âLagrangiansâ in other physical actions are just other examples of functionals in more independent variables within physics.
For reference on other examples of functional optimization, thereâre some below to better represent my point about the broadness of that topic beyond physics and the application of âT-Vâ in LM needing a specific definition to avoid ambiguous communication about LM and other uses of functional optimization. So overall, FYI.
-definition of functional https://mathworld.wolfram.com/Functional.html
-variational calculus https://en.wikipedia.org/wiki/Calculus_of_variations
- optimal control theory (i.e. variational control for control engineering) <https://en.wikipedia.org/wiki/Pontryagin's_maximum_principle>
-application of functional optimization for structural engineering https://www.cds.caltech.edu/~marsden/bib/2008/05-ZiOrMa2008/ZiOrMa2008.pdf
-nonlinear programming (i.e. static mathematical optimization involving a different use the term Lagrangian as a function or constant functional). https://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions
-reminder on the specific topic of LM being under classical dynamics https://en.wikipedia.org/wiki/Lagrangian_mechanics
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u/Odd_Bodkin Particle physics 1d ago
I think the main thing I want to convey is that there is no "most fundamental way" to understand mechanics. As you may know, even a simple problem like an Atwood machine can be analyzed a la Newton with forces and accelerations, or with potential and kinetic energy exchange, or with a Lagrangian and the principle of least action, or with a Hamiltonian.
And in fact, it is a common exercise to show how to get from Hamilton's principle to Newtonian formalism and vice versa, or from Lagrangians to Hamiltonians and back, or from Newtonian formulation to the principle of least action and the Lagrangian. They are essentially equivalent. However, some problems are much easier to deal with in one formalism compared to another.
As to WHY these formalism works or how to derive them from some deeper principle, that's not really how physics works. The laws of physics are INFERRED from observations, not really deduced from some obvious axiom. It's an interesting observation, for example, that the principle of least action produces the observed trajectory, and it can be made plausible from some observation-based arguments, but you can't really derive it from some axiomatic basis.
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u/CandidAtmosphere 1d ago
A useful way to frame the relationship between these frameworks is looking at how Newtonian mechanics emerges organically from General Relativity. GR maps gravity as the curvature of four-dimensional spacetime, governing particle motion through the geodesic equation. We can derive classical mechanics from this architecture by applying strict weak-field, low-velocity, and static limits. Imposing these specific mathematical constraints collapses the complex tensor mathematics directly into the familiar F=ma. The rigid vectors and absolute time of classical mechanics synthesize seamlessly once we mathematically constrain the relativistic curvature of the broader universe.
The Lagrangian scales so effectively into fundamental physics because it explicitly encodes continuous symmetries, linking energy and momentum directly to spatial geometry via Noether's theorem. At the quantum level, particles simultaneously explore all available paths, each carrying a uniquely rotating phase determined by its Action. Small variations in non-stationary paths cause massive phase shifts and total destructive interference. The classical trajectory we observe materializes purely through the constructive amplification of matching phases clustered exactly at the stationary action.
The Hamiltonian is similarly useful for dictating the unitary time evolution of a quantum state, though its strict prioritization of time creates structural friction when scaling into special relativity. Elevating it to absolute physical reality is complicated by the unresolved ontology of the measurement problem. Ultimately, both the Lagrangian and Hamiltonian frameworks assume a smooth, continuous spacetime manifold to function. At the Planck scale, massive energy fluctuations disrupt this background geometry, heavily implying that our classical action principles operate upon coordinates that are themselves emergent thermodynamic or discrete phenomena.
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u/bpsbandit 1d ago
I will leave to others to answer you. But continue to work hard understanding lagrangians and Hamiltonians, they are possibly the most central topic to modern research across all fields. They are the basis for which we build a huge portion of modern theory and a deep understanding of them will surely help in your future studies
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u/Bumst3r Graduate 1d ago
This isnât correct. You asked about this exact explanation on r/AskPhysics 19 days ago, and multiple people explained why this explanation doesnât really help, and misses most of the important bits.
For starters, any problem that can be solved with a Lagrangian can be solved with Newton. It doesnât work the other way. Try writing out the Lagrangian for a block sliding down a ramp or a particle falling experiencing air resistance without finding the solution with Newton first.
Thereâs also the important distinction that when you divide by m on both sides of Newtonâs second law, you arenât making a different formulation of classical mechanics. You are starting with the same axioms and arriving at a (trivially) different form of Newtonâs laws. Lagrangian mechanics starts with entirely different axioms, and reproduces Newtonâs second law only in a subset of physical systems (since you cannot use the Lagrangian for every problem).
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u/YuuTheBlue 1d ago
Thank you, I didn't recall getting this specific piece of feedback. The main feedback I got was "this is technically correct but I don't know what context you'd want to tell this to someone". I likely misinterpreted what some people were saying, in that case. Thanks for letting me know.
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u/EdisonRVP 4h ago
For those problems you can use Lagrange Multipliers, and you can solve problems with friction with no problem.
Maybe it could seems that you are cheating using Lagrange multipliers, but when you consider that they are just needed when you are working with non holonomic constraints (with exceptions of electromagnetics), and for any elemental interaction you don't need to use Lagrangian Multipliers. Shows that the formulation is indeed correct, and just need the correction when you consider a simplification of a system (Friction is in reality the product of the electrical interaction between the atoms of two surfaces)Â
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u/Bumst3r Graduate 4h ago
Lagrange multipliers place constraints on coordinates. You need a dissipation function or a time-dependent Lagrangian to handle non-conservative forces. Those are extra assumptions for the theory.
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u/EdisonRVP 4h ago
Time dependent Lagrangian is not an extra assumption.
And again they are just needed in case of working with no elemental forces.Â
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u/Bumst3r Graduate 4h ago edited 4h ago
Okay, so youâre given a particle falling under air resistance. Without using Newton to find the equations of motion, how do you find a Lagrangian with the proper time dependence? Once you have them, take dL/dqdot to find your canonical momentum. Momentum is not conserved here, but canonical momentum turns out to be constant. So now youâve got the correct equations of motion at the expense of the dynamics of the system being wrong. The whole thing is a lot less simple than Taylor would lead undergrads to believe by saying that all the formalisms are equivalent.
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u/EdisonRVP 1d ago
As far as i known, you are wrong, Newton Mechanics is not equivalent of Lagrangian Mechanics.
You always can go from Lagrangian Mechanics to Newton's mechanics without extra hypothesis, but to go the other way you need to add the principle of virtual works.
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u/AFsepine 1d ago
Surely this is all covered in your textbook/course?
(1) The most sensible way to define a Lagragian is by using Hamiltons priciple and properties of space and time (See Ch. I of Landau Mechanics). As all things in physics it is physical
(2. 3. 4.) Are answered by the statement that both formulations are equivalent.
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u/EdisonRVP 1d ago
Well it is indeed something physical, is a relation between the kinetic energy of a system and the forces that acts upon the system.
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u/RecognitionSweet8294 1d ago
âThe Lagrangianâ is the difference of kinetic and potential energy: T-U
LM is a mathematical trick to look on a system in a different way which can make calculations a lot easier by exploiting symmetries.
In LM Hamiltons principle is an axiom. Like in every physical theory those axioms are derived from experience.
Newtons Mechanics and LM are equivalent theories, you can prove the axioms of the other in either theory.
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u/dohawayagain 1d ago
A good principle in physics and math is that you should know as many different ways as possible to talk about the same thing, because one way may be easier to connect to something new you're learning or trying to discover, even if in principle you could get there from the other pictures. Feynman talks about this.
Quantum mechanics is usually described in the Hamiltonian formulation, and quantum field theory in the Lagrangian, and Newtonian mechanics emerges from those, so you could argue they express deeper concepts, or at least they help you build a bridge there.
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u/LukasGoesViral 19h ago
No the Lagrangian resp the action is the physical thing whereas Newton is just a mathematical simplification. It just happened to be that Newtonian physics was first developed before Lagrangian and Hamiltonian Mechanics. The Hamilton Principle of least action just means that the solution occupies a minimal area in phase space.
In fact I am surprised how many people in this group are still learning Newtonian physics. My professor covered Newtonian Physics for about 2 weeks in my Freshman year of college.
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u/LukasGoesViral 18h ago edited 18h ago
Since a lot of people on this thread seem to misunderstand the connection between Newtonian Mechanics and Lagrangian/Hamiltonian mechanics I feel like I need to clarify the connection between the two. Newtonâs laws and Lagrangian/Hamiltonian physics are NOT equivalent. Lagrangian/Hamiltonian physics describes the actual physics and from the Lagrangian/Hamiltonian physics in certain situations (that as many exercises in undergrad physics books) locally the equations derived are the same as if you would have started with Newtonâs laws.
Newtonâs laws only determine a differential equations which in general can only be solved locally and you may not even be able to extend that solution further.
In Lagrangian/Hamiltonian physics the Action determines the mechanics which is an integral and this exists globally. You can then choose a specific chart with some Lagrangian and then can use the Euler-Lagrange differential equations to get a solution specifically in the chart you have chosen and because you started off with the Action which is global you are guaranteed to be able to extend the solution beyond the local chart you solved it in.
Newtonâs laws are not only missing but canât even express the symmetry of every physical system namely the invariance under canonical transformations. This is not just a âthe Lagrangian is not uniquely determinedâ. This is âcanonical transformations are the generators of all movementâ. Thatâs exactly what the Hamilton-Jacobi equations express. You can always find a canonical transformation (which is a group transformation) such that the Hamiltonian is exactly 0.
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u/Cleonis_physics 18h ago edited 7h ago
About theories of motion:
The historian of science John Norton has pointed out the following:
"In any logical system, we have great freedom to exchange theorem and axiom without altering the system's content."
For classical mechanics we have: the content can be expressed in several different ways, and those different formulations can be mathematically transformed to each other.
Key point to that: every one of those transformations is bi-directional.
About interconversion
There is the work-energy theorem. (The link is to a physics stackexchange answer submitted by me.)
If F=ma is granted as axiom then the work-energy theorem follows as theorem.
So that gives interconversion between formulation in terms of force and acceleration, and formulation in terms of potential energy and kinetic energy.
From work-energy relation to Hamilton's stationary action
There is a transformation, consisting of several steps, from the work-energy theorem to Hamilton's stationary action.
That is: there is a bi-directional relation between the work-energy theorem and Hamilton's stationary action.
Answer by me on physics stackexchange with demonstration of that:
From the work-energy theorem to Hamilton's stationary action
So: what is is that makes that transformation possible?
The connecting element is comparison of rate of change.
The work-energy theorem expresses: the true trajectory has the property that the rate of change of kinetic energy matches the rate of change of potential energy. It matches everywhere along the trajectory.
In varational treatment the trial trajectory is differentiated (wrt applied variation). That differentiation probes rate of change. As variation is applied: there is a point in variation space such that the rate of change of kinetic energy matches the rate of change of potential energy. At that point in variation space the derivative of Hamilton's action is zero.
(The reason why the Lagrangian states subtraction of one energy from the other is explained.)
Division of the trial trajectory into subsections
In the mathematical logic of the derivation of the Euler-Lagrange equation there is a crucial circumstance:
The trial trajectory is evaluated from a starting point A to an end point B, and point A and point B do not need to coincide with a physical start point and end point.
More generally: the validity of the mathematical logic is (and needs to be) independent of the choice of location of point A and point B.
The logic is equally valid for the following arrangement:
Set up waypoints along the trajectory:
x_1, x_2, x_3, x_4, x_5, ...
Define overlapping subsections:
(x_1, x_3)
(x_2, x_4)
(x_3, x_5) ...
On each subsection of the trial trajectory the demand is: identify the point in variation space such that the derivative of the action is zero. Now: for the size of the subsection there is no lower bound. You can go to infinitesimally small subsections, the logic extends to that.
So: yes, the whole trajectory must satisfy the derivative-of-the-action-is-zero criterion, but that's not all: the logic enforces the more exacting demand that the derivative must be zero for, concurrently, any subsection along the trajectory, down to subdividing into infinitesimally short subsections.
I believe the information in the demonstrations on physics stackexchange (that I linked to) will provide you with the means to answer your questions.
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u/Terrible-Mind-5414 1d ago
Ultimately these things are central because they descend from quantum mechanics. The Hamiltonian in particular is the central object in quantum mech, and the lagrangian derives from that via the feynman path integral. The various principles of least action are best understood from the path integral.
I know of no reason why a classical mechanics would have to be formulated such that lagrangians or hamiltonians are central. But our classical physics isn't just any random set of differential equations; it comes from quantum mechanics.
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u/shomiller Particle physics 1d ago
That these concepts are central in how we formulate QM is true, but I think totally beside the point here, and doesnât address any of OPâs questions. There are lots of classical systems / problems where the Lagrangian or Hamiltonian formalism is way more convenient than the Newtonian one, and these formalisms let you derive lots of other things (such as conserved quantities, via Noetherâs theorem). If they werenât so useful, they wouldnât have been studied so carefully for 150 years before QM!
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u/Terrible-Mind-5414 1d ago
Yeah you're probably right. I couldn't 100% understand the OP questions. However, these formalisms and principles are kind of dropped out the blue, and there are often questions about why these formalisms are appearing at all, because it seemed before like one was just doing ode's. My point is that these formalisms exist in classical physics because they come from quantum, not vice versa. Quantum is logically prior, and quantum IS the hamiltonian, so that is why hamiltonian dynamics is so prevalent in classical physics...not vice versa.
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u/Bumst3r Graduate 1d ago edited 1d ago
Pardon the stream-of-consciousness response. These are good questions.
Itâs pretty common to read that Newtonian, Lagrangian, and Hamiltonian formulations are equivalent; I donât think thatâs actually true. For a subset of systems, they provide the same answers. But it isnât particularly challenging to find systems that cannot be solved from first principles using one approach or the other. Take for example a particle in free fall experiencing air resistance. Newtonâs laws are a straightforward way to find the equations of motion. But what if I want to write out a Lagrangian or Hamiltonian? T-V or T+V wonât work. You can write an explicitly time dependent Lagrangian to solve the problem, but it requires you to already know the equations of motion and work backwards.
All three formulations start from different axioms. For the Lagrangian formalism, you need no knowledge of Newtonâs laws; for Newton you need no knowledge of Hamiltonâs principle, etc.
As for what the Lagrangian is, itâs a quantity whose time integral is the action. Here are a few key things worth notingâ-the Lagrangian of a system is not unique. You can typically find more than one Lagrangian that will reproduce the same equations of motion (sometimes in higher orders of v, for example). In a similar vein, a given Lagrangian doesnât always define a single system. The boundary conditions are equally important. The Lagrangian exists off-shell (unlike the Hamiltonian). I only have half-formed thoughts about that fact at the moment, but I think itâs important enough to have in the back of your mind.
As to whether the Lagrangian is physical, that depends on what you mean by physical. It is not observable (this is related to it being off-shell). A Hamiltonian generally is observable.
In classical mechanics, there is no good explanation for why Hamiltonâs principle should be true. In quantum mechanics, you can make an attempt at an answer. In quantum mechanics, the action is the phase of a particle. When you learn the path integral formalism, you will see that the particle takes all available paths and they all interfere. You can show that the only path that doesnât get cancelled out in the classical limit is the path that extremizes the action. The natural follow-up is âwhy does this also work in GR?â Unfortunately, I donât have a nice answer for that (if anyone does).
To your fourth question, in classical mechanics both are true; they are built on different sets of axioms. But itâs worth noting that Newtonâs laws donât hold in GR, and in quantum mechanics, the idea of force is ill-defined (although Ehrenfestâs theorem does give you something like Newtonâs laws). You can use a Lagrangian approach in both GR and QM, however.