Good questions to asks your professor
Why is this upvoted? It's unhelpful and rude.
Hello. Ive been taking Classical mechanics course this sem and we are being taught LM. i had some questions
thank u so much😊
Good questions to asks your professor
Why is this upvoted? It's unhelpful and rude.
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.
This is a common misunderstanding of what a lagrangian is. Like, okay, here is an example of the core idea: Take the equation F=ma. In english, we'd pronounce this as "Force quals mass times acceleration". This is Newton's second law of motion. If we divide both sides by m, we get F/m=a, which can be read as "Force divided by mass equals acceleration". And I think it's pretty clear that if one is true, the other also is true. I think most people would go as far as to say that they aren't even different statements about the universe. There would, for example, be no reason to ask which one is true. It's 12 of one half a dozen of another - just two ways of stating the same core idea. F=ma and F/m=a are not philosophically different ideas. Lagrangians are, essentially a much more abstract version of this. The statements "Force equals mass times acceleration" and "The action is defined as the integral of Kinetic Energy minus Potential energy over time and the action is stationary" are as identical as "Force equals mass times acceleration" and "Force divided by mass equals acceleration". If one is true, all of the statements are provably true. The math of getting to the Lagrangian from the Newtonian is just more complicated. This isn't to say that there aren't legitimate mathematical insights to the Lagrangian, but it's not a new law of physics. By understanding the fact that F=ma necessarily means that there are rules about Kinetic Energy minus Potential Energy, you can better understand F=ma. Additionally, it is an important insight that just as neither F=ma nor F/m=a have more of a claim to truth than the other, neither has more of a claim to truth than the Lagrangian version. Neither CAN be more fundamental. Lagrangians are ultimately a way that plenty of physical laws can be 'reworded'. Electromagnetism, in the context of special relativity, can be summed up into a singular equation rather than 5 if you allow for it to be reframed in terms of a Lagrangian.
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.
This isn’t correct. You asked about this exact explanation on r/AskPhysics 19 days ago, and multiple people explained why this explanation doesn’t 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 doesn’t 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. There’s also the important distinction that when you divide by m on both sides of Newton’s second law, you aren’t making a different formulation of classical mechanics. You are starting with the same axioms and arriving at a (trivially) different form of Newton’s laws. Lagrangian mechanics starts with entirely different axioms, and reproduces Newton’s second law only in a subset of physical systems (since you cannot use the Lagrangian for every problem).
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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Well it is indeed something physical, is a relation between the kinetic energy of a system and the forces that acts upon the system.
It’s pretty common to read that Newtonian, Lagrangian, and Hamiltonian formulations are equivalent; I don’t think that’s actually true. For a subset of systems, they provide the same answers. But it isn’t 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. Newton’s 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 won’t 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 Newton’s laws; for Newton you need no knowledge of Hamilton’s principle, etc. The as for what the Lagrangian is, it’s a quantity whose time integral is the action. Here are a few key things worth noting—-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 doesn’t 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 it’s 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 Hamilton’s 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 doesn’t get cancelled out in the classical limit is the path that extremizes the action. The natural follow-up is “why does this also work in GR?” Unfortunately, I don’t 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 it’s worth noting that Newton’s laws don’t hold in GR, and in quantum mechanics, the idea of force is ill-defined (although Ehrenfest’s theorem does give you something like Newton’s laws). You can use a Lagrangian approach in both GR and QM, however.
Why is newton law not valid in gr? You can write the geodesic equation in a form that you have newtons law (christoffel symbol cobtracted witb velociy on the right side)
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.
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.
That these concepts are central in how we formulate QM is true, but I think totally beside the point here, and doesn’t address any of OP’s 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 Noether’s theorem). If they weren’t so useful, they wouldn’t have been studied so carefully for 150 years before QM!
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 Newton’s 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 Newton’s 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 the better part of the last decade writing a book (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.
Learning mechanics from Reddit ur cooked
Regarding your first question, a Lagrangian is a mathematical abstraction from the field of the calculus of variation 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, it’s 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, Hamilton’s 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 a 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).
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.
„The Lagrangian“ 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.
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.
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.
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 m×v×dL. (L is the distance). Maupertuis was first ridiculed until Euler and Lagrange derived the priciple differently.