Deeper Meaning to the Nature of Lagrangian

Question

Is there a more fundamental reason for the Classical Lagrangian to be $T-V$ and Electromagnetic Lagrangian to be $T-V+ qA.v$ or is it simply because we can derive Newton's Second Law and Lorentz Force Law with these Lagrangians respectively?

Answer 1

By defining the lagrangian as $$L = T-V$$ you're restricting yourself. This definition works only in classical mechanics.

A lagrangian is a vastly more general concept which I'll summatrize: $L(q(t),\eta(t),t)$ is an arbitrary differentiable map $$L:A\times\mathbb{R}\to\mathbb{R}\qquad A\subseteq\mathbb{R}^n\times\mathbb{R}^n$$ where $q(t)$ are generalized coordinates and $\eta(t)$ are generalized velocities. With this we define the action functional $$S[q,\eta] = \int_{t_0}^tL(q(t),\eta(t),t)dt$$

Then, basically all physics can be derived from Hamilton principle of stationary action which says that

The natural path of a system $q(t)\in Q$, where $$Q = \{q:\mathbb{R}\to\mathbb{R}^n | q\in C^2([t_0,t]), \, q(t_0) = q_0, q(t) = q\}$$ is the set of all possible varied paths with fixed boundaries, is the one which makes the action stationary.

By evaluating the variation of the action $\delta S$ and by imposing the stationary condition one finds that a path $q(t)$ makes the action stationary if is a solution to the equation

$$ \frac{d}{dt}\frac{\partial L}{\partial \eta_i}(q(t),\dot{q}(t))-\frac{\partial L}{\partial q_i}(q(t),\eta(t)) = 0\\ \det\left(\frac{\partial^2L}{\partial\eta_i\partial\eta_j}\right)\neq 0 $$

One can then use the change of variables $\dot{q}(t) = \eta(t)$. Note that in all of this we never talked about a specific form of the lagrangian such as $T-V$.

This general formulation of a lagrangian is what is used everywhere in physics and can be extended even to quantum theories like quantum field theory. Even though in QFT we mostly speak of lagrangian densities, which does not change much.

So in the end, what is really fundamental is Hamilton's principle of stationary action.

Answer 2

It depends on what you take as 'fundamental reason'. From a mathematical point of view, the Lagrangian $L= T - V$, with $q A\cdot v$ included in $V$ in the electromagnetic case, can be derived from the equations of Motion. These are in principle the Newton's law you mentioned. However, the Lagrangian $L=T-V$ is by far not unique. For example, if you add a total time derivative, you get a boundary contribution in the action integral $S=\int L$ by partial integration, which is vanishing in the variation $\delta S$. Thus, you can say the form $L=T-V$ is not fundamental, since it's not unique.

But note, however, that we are comparing two formalims here. Writing $L=T-V$ assumes that the kinetic term $T$ as well as the potential term $V$ are specified from the Newton formalism. On the other hand, it's both just models, and asking which one is more fundamental is somehow like asking whether spacetime is really curved in nature, or whether it's just a model helping to visulize things.

To come to an end, I think it's a bit of a subjective question which one is more fundamental. Me for my part would take the Lagrangian as more fundamental since, from a mathematical point of view, it is a single scalar function containing all the information about the equations of motions given by Newton's law.

Cheers!

Answer 3

Not only can we derive Newton's Laws and Classical Electromagnetism from the Lagrangians, but, in the first instance, we actually find these Lagrangians from Newton's Laws and Maxwell's equations together with the Lorentz force law. Then, from the point of view of mathematics, the assumptions are equivalent and any decision which to take as more fundamental is a matter of subjective choice.

From the point of view of science, I would always take empirical assumptions as more fundamental, and for this reason I regard the Lagrangian formulations as a mathematical curiosity, rather than as physically fundamental.

At a deeper level, in quantum electrodynamics, I do not use a Lagrangian. Instead I find Schrodinger's equation from the probability interpretation, by way of establishing unitarity and using Stone's theorem, and imposing relativistic constraints. This leads to the Dirac equation when interactions are introduced it is possible to derive conservation of momentum in particle interactions (equivalent to Newton's laws) and it is possible to derive Maxwell's equations and the Lorentz force law A Construction of Full QED Using Finite Dimensional Hilbert Space. I have also given a full treatment in The Mathematics of Gravity and Quanta

Answer 4

Many empirical facts about the nature can be reduced to set of symmetries. For example conservation of momentum is equivalent to homogenity of space. Lagrangian is then great method to seek available theories from symmetry.

For example in the case of classical mechanics, you can start by free particle and general lagrangian $L=L(q,\dot{q},t),$ where $q$ is position of the particle. Now you can assume symmetries of space and time - that space is homogeneous and isotropic and time is homogeneous in certain frame (called inertial). From this you get that the lagrangian of free particle in an inertial frame should be of the form $L=L(\dot{q}^2).$ Then you can assume galileo relativity principle - that is invariance of physics under the transformation $q'=vt+q$ and $t'=t.$ For infinitesimal $v$ this will lead to lagrangian of the form $L=k\dot{q}^2,$ where $k$ is some constant to be called $m/2$ and you have have $L=T$ for free particle. Now many particles that do not interact together should have their motion decoupled, thus lagrangian of N particles should be sum of free particle lagrangians. To add interactions between them, you can modify the lagrangian so that it best suits the nature of the interaction - for example if interactions seems spherically symmetrical, you can modify lagrangian to depend also on radius vector between the particles.

In classical mechanics, the interactions are supposed to be infinitely propagated (like gravity). This is suggested by galileo relativity principle, because if the interactions would not be infinitely propagated, then switching to another inertial frame of reference would mean there is different speed of propagation and thus the two inertial frames are not equivalent. (This can be of course circumvented by tricks similar to including eather in electromagnetic theory, as it was done before Einstein modified relativity principle). This infinite speed of interaction propagation leads to lagrangian that takes the form $L=T(\dot{q_i})-V(q_i).$

So to answer the question - yes there is more fundamental reason. In case of classical mechanics, it is symmetry of space and time plus galileo relativity principle.

You can read more about this approach in Mechanics by Landau and Lifshitz. The point is that the difference between seeking the force and seeking the lagrangian is in the assumptions you are starting with. If you measured acceleration, then seeking the force might be the most straightforward approach. If you have instead only some general assumptions about the physics in question, writing down the lagrangian obeying these general assumptions is probably more straightforward.