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This might seem like a duplicate question; however, rest assured, it is not. My question is pointed and particular:

Some background:

Given a system we describe its Lagrangian $L$ as $T-V$, where each $T=\int_{\alpha_0}^{\alpha_1} F_i \cdot dx_i$ and $V$ is some context dependent potential. So we have, $L=T-V$, and the famous E-L equation: $\frac{\partial L }{\partial x_j }-\frac{d}{dt}(\frac{\partial L}{\partial \dot x_j})$. Taking the Lagrangian as fundamental we can define the Hamiltonian $H$, knowing that the generalized momentum is $\frac{\partial L}{\partial \dot x_j}$, let's call it $\bar m$, as $H=\bar m_j \dot x_j-L$. From this, it's easy to derive the usual kinematic equations taking partials with respect to each $x_k, k \in \{i,j,k\}$ and $\bar m_k, k \in \{i,j,k\}$.

Interestingly enough, this shows that the two formalisms: $H,T$ are mathematically equivalent since they are mutually derivable. (My equations could be wrong since its been a while that I have studied them)

Finally, here is my question, due to my limited knowledge of physics beyond that of classical mechanics, I cannot see, whether, on the level of physical description, the two formalisms diverge. In particular, I am wondering whether there are any physical parameters such that the two formalisms diverge in their description of it.

For example, the parameter could be such that in the Lagrangian its description is that of a constant. Whereas, the same parameter, in the Hamiltonian is a variable. How could this be possible, one might wonder, this is not an issue because in the end they are quantitatively equal, except qualitatively (i.e. description-wise) they are different.

This is, of course, just an example. It could be any sort of divergence: like, for instance, you could say the Hamiltonian describes the world (i.e. some physical system (or one of its particular component)) like this...(Place holder), but the Lagrangian implies the world (i.e. the same physical system(or its particular component)) is like this...(place holder). One such example would do. If such a divergence is impossible please describe why it is so.

In particular, consider Classical Quantum Mechanics, and Quantum Field Theory. Although the two are, to the extent that calculations are possible, quantitatively equivalent. However, former takes particles (with their spin and charges) as fundamental while the latter posits a permeating field which gives rise to particles when perturbed. They are clearly different descriptions. Do we get something like this with Hamiltonian/Lagrangian formalism or are the differences too low-level to give any meaningful descriptive variations.

Bertrand Wittgenstein's Ghost
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The Lagrangian and Hamiltonian formalisms are exactly equivalent, so any physical observable that can be computed in one formalism, can be computed in the other, and the results must match.

You gave the argument for this statement yourself in your question. You can derive the Hamiltonian formalism from the Lagrangian formalism, and vice versa, so they are equivalent.

Based on some clarifying discussion in the comments, I will add that I think what this question is getting at is whether there are different interpretations of classical mechanics, similar to different interpretations of quantum mechanics (including both what is normally meant by interpretations of quantum mechanics, but also other issues such as the fact that you can look at quantum field theory as a theory of quantized fields or a theory of an indefinite number of identical particles). I would tend to say that classical mechanics is much better understood than quantum field theory, both because it is a more mature subject that has been around longer, and because it is intrinsically less conceptually complicated, and therefore there are not major differences in interpretation of the meaning of classical physics. At the end of the day, a classical theory computes a particle trajectory, or the evolution of a field, as a solution to a differential equations. The different formalisms give different mathematical representations, but at the end of the day typically it is clear what we mean by a particle's trajectory.

Andrew
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  • Hi, thank you for the input. I totally understand the two are quantitatively equivalent, but it's the descriptive (qua qualitative) equivalence that I am looking for. That is, are they, unfortunately, descriptively equivalent as well? – Bertrand Wittgenstein's Ghost Apr 25 '22 at 04:58
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    @BertrandWittgenstein'sGhost I'm sorry, I don't understand what that means. – Andrew Apr 25 '22 at 04:58
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    The concepts in the two formalisms are not identical. In Hamiltonian mechanics you have a phase space, position and momenta, conjugate transformations, Poisson brackets, Liouville's theorem... In Lagrangian mechanics, you have the principle of least action, generalized coordinates, manifest Lorentz invariance (for relativistic theories), Noether's theorem... So the two formalisms are useful for different things and certain concepts are easier to express in one vs another. But they are fundamentally equivalent, so the exactly agree on any physical prediction. – Andrew Apr 25 '22 at 05:03
  • I don't know how else to put it: but suppose, for examples sake, that the Hamiltonian formalism describes a component $x$ of some arbitrary system $S$ in a particular way, let's call it $P(x)$ and the Lagrangian for the same system, and component, gives us another description $P_L(x)$. – Bertrand Wittgenstein's Ghost Apr 25 '22 at 05:03
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    @BertrandWittgenstein'sGhost (1) A trivial example might be that the variables used in Lagrangian mechanics are $q, \dot{q}$ (the position and velocity), whereas in Hamiltonian mechanics they are $q, p$ (position and momentum). This feeds into things like the energy being $E=\frac{1}{2} m \dot{q}^2$ in Lagrangian mechanics and $E=p^2/2m$ in Hamiltonian mechanics. (2) Even in one framework, you can have multiple representations of one physical system. For example, you can use Cartesian or spherical coordinates to represent the position of a particle, in either formalism. – Andrew Apr 25 '22 at 05:07
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    @BertrandWittgenstein'sGhost I think what you're getting at is that you can represent a system in multiple ways. This is true. Even within Lagrangian mechanics, there is not a unique Lagrangian that represents a system. Same with Hamiltonian mechanics. One representation might be more useful for solving a given problem, or making a certain feature of a physical system. more clear. For example, it's much easier to prove Liouville's theorem in Hamiltonian mechanics, or solve the Kepler problem in appropriate coordinates. But it's important to keep in mind that all representations are equivalent. – Andrew Apr 25 '22 at 05:12
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    It's hard to give more detail than that though without more information about what you are interested in. Certainly you can think about a physical system in multiple ways that are psychologically different and may make a given feature easier or harder to see, or may be easier or harder to generalize to a deeper theory. But a very important thing a physicist must learn is what aspects of a formalism are just mathematical niceties and what aspects are invariant physical statements that are true in any formalism. – Andrew Apr 25 '22 at 05:16
  • I really appreciate the answer, and the subsequent clarification given as comments. I really do. I understand that each H/T has its own merits, and they agree on physical predictions (due to the fact that they are equivalent). In a nutshell this is what I was looking for: consider QFT and QM they are, to a very large extent, equivalent however the former posits we have a permeating field giving rise to particles. Whereas, in terms of QM we fundamentally have particle. Although they both agree on numerical predictions their description of the physical world is vastly different. – Bertrand Wittgenstein's Ghost Apr 25 '22 at 05:23
  • In other words, in terms of QFT fields are fundamental, but in terms of classical QM it's the particles that are fundamental. So, we get vastly differing descriptions of reality, but quantitatively exact overlap. Do we have something like this in terms of Hamiltonian vs Lagrangian? – Bertrand Wittgenstein's Ghost Apr 25 '22 at 05:26
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    @BertrandWittgenstein'sGhost Besides the trivial fact that the Lagrangian and Hamiltonian formalisms are different if you take them literally, I don't think the interpretation of classical mechanics is as interesting as the interpretation of quantum mechanics or quantum field theory. At the end of the day, you have a differential equation that you solve to get the trajectory of a particle or evolution of a field. You can express these equations in different variables or languages, but there are typically no issues interpreting what a trajectory means. – Andrew Apr 25 '22 at 05:39
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    Also as an aside, I don't think it's fair to say that QFT and QM conflict. In the regime where interacting, relativistic QFT is relevant, you cannot use ordinary QM, because you necessarily have processes that change the particle number, but ordinary QM only works for a fixed particle number. On the other hand, there are different approaches to QFT: for example, the "straightforward" approach where fields are fundamental and quantized, and the "Weinberg" approach where you derive fields as a way to encode local, Lorentz invariant, and unitary interactions between relativistic particles. – Andrew Apr 25 '22 at 05:43
  • That was my gut feeling too! I was thinking that, unfortunately, the difference b/w H/L is a little too low-level to give any meaningful description-let alone varying description. That's sad. – Bertrand Wittgenstein's Ghost Apr 25 '22 at 05:57