Interpretations of Lagrangian vs. Hamiltonian mechanics

Question

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.

Answer 1

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.