# Interpretations of Lagrangian vs. Hamiltonian mechanics

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.

• Related: Equivalence between Hamiltonian and Lagrangian Mechanics and links therein. Commented Apr 25, 2022 at 7:29
• – J.G.
Commented Apr 25, 2022 at 14:00
• @Qmechanic definitely related but not duplicate . I know the two are equivalent, but that not the issue. Can you point me to a scenario such that we have one physical interpretation given one, and a different physical interpretation given the other? Commented Apr 25, 2022 at 15:44
• In case anyone else wants to give an answer, based on some discussion with the OP, I think another way to phrase this question is: are there multiple interpretations of classical mechanics, similar to how QFT can be interpreted either as a theory of quantized fields, or a theory of an indefinite number of identical quantum particles? My answer is, essentially, no, but others may have a different perspective. Commented Apr 25, 2022 at 16:08

• @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. Commented Apr 25, 2022 at 5:07