The Closest We’ve Come to a Theory of Everything

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Strong rec Connor Colestock · 2025-03 · Physics Stack Exchange · on “Is linear momentum times distance any meaningful quantity? $\vec{r} \cdot \vec{p}$ or $pr = mvr$ comparing to $\vec{r} \times \vec{p}$”

EDIT: After some research, your intuition is correct. Momentum times distance (also some dot product or projection of it into a given vector) represents the concept of "action," which is deeply related to Newtonian and Lagrangian mechanics. It often shows up in formulations of the "Principle of Least Action": $\delta\int mvds=0$ Another form of this is the ubiquitous Euler Lagrange Equations: $\frac{d}{dt}\frac{\partial L}{\partial \dot q_k}-\frac{\partial L}{\partial q_k}=0$ From what I've learned about the history of it all, many 17-18th century contemporaries were debating about which properties truly described motion. At the time, it was yet to be decided. Was it momentum, energy (i.e. V

Recommends Perttu Hilla · 2024-11 · Physics Stack Exchange · on “Lagrangian - does it rely on Newtonian mechanics?”

Energy, $T + V$ , which is a conserved quantity resulting from the time-translation invariance of physical reality, is in my view the fundamental quantity. One does not need force to arrive at energy, but you can equate work with energy in the Newtonian sense. It is also true that all physical theories that we know arise from the principle of least action. It just so happens that the thing one has to integrate to arrive at the correct equations of motion is the Lagrangian, $T - V$ . There is a very good recent video [this video] on Veritasium about the principle of least action, check it out.

Recommends VinalV · 2025-01 · Physics Stack Exchange · on “Intuition for $L = T - V$ in classical mechanics without Newton's 2nd law?”

There is a nice video by Veritasium that goes into some of the history of the Lagrangian which may help understand where it comes from. Link: [this video]

Recommends Qmechanic · 2025-04 · Physics Stack Exchange · on “Intuition for Maupertuis' action and the principle of least action”

On one hand, in the principle of stationary action/Hamilton's principle [LINK] $$ \delta \int_{t_i}^{t_f}\! Ldt~=~0, \tag{1}$$ the initial and final times $t_i$ and $t_f$ are kept fixed, while the total energy $E$ is free. On the other hand, in the abbreviated action/Maupertuis' principle [LINK] $$ \delta \int_{t_i}^{t_f}\! p_k dq^k~~=~0, \tag{2}$$ the total energy $E$ of all paths is kept fixed and the same; while the initial and final times $t_i$ and $t_f$ are free. TL;DR: The variational principle [LINK] (1) is more important in physics and its applications as compared to (2), which is why (2) is typically not taught in lower level physics. For a more rigorous derivation and generalizatio

Recommends Cort Ammon · 2025-06 · Physics Stack Exchange · on “What are the conceptual foundations of Lagrangian formulation?”

There are a lot of things in physics that we use because they work. Lagrangian mechanics gets used a lot because it works a lot. It's just a tool, like addition or subtraction or Fourier transforms. I recommend Vertasium's video, The Closest We've Gotten to a Theory of Everything [this video] . He did a great job of showing how these concepts evolved and got selected over time. It's a good background to start from. In my own research into the history of Lagrangian mechanics, I note that Lagrangian mechanics did not start out as an attempt to solve everything. Lagrange used it to solve a few problems like the brachistochrone curve [LINK] . His approach was more elegant than past attempts. In doing

Cites Connor Colestock · 2025-04 · Physics Stack Exchange · on “Intuition for Maupertuis' action and the principle of least action”

I'm familiar with Lagrangian mechanics and Newtonian mechanics, though I'm currently building an intuition for the concept of "action" or "least action" (i.e what unites the two theories). The principle of least action, as formulated by the Euler-Lagrange theory is generally what comes to mind for me. $$\frac{d}{dt}\frac{\partial L}{\partial \dot q_k}-\frac{\partial L}{\partial q}=0.$$ However, I've recently learned how this theory came to be through a Veritasium video [this video] . If "action", or $$S_0=\sum mvs$$ is such a useful concept, as put by Maupertuis, then why is it not taught in lower level physics? The intuitive gap b/w energy or force and action would seem to be the reason. If one