Not the person you were responding to, but: the Lagrangian formulation describes physics in terms of Least Action (so minimizing (or maximizing) S = ∫ L dt) on a manifold in (x,v) coordinates, and its equations of motion are like L_x - d_t L_v = 0. The second term tends to be second-order in time.
The Hamiltonian formulation performs a Legendre transformation[1] on L giving H = v L_v - L, which is essentially a convenient trick: it reparameterizes L in (q,p) coordinates, where p = L_v, and writes S as ∫ (pv - H) dt. This changes the E.o.M. to (qdot, pdot) = (H_p, -H_q), which is (a) first-order in time and therefore easier to deal with and (b) geometrically elegant because it is a rotation in (q,p) space, which is easy to think about.
At least those are the reasons everyone gives why it's important. I think the real reason is that QM is formulated in terms of H so you need to know it, and also that this (q,p) thing makes statistical mechanics easier because it has good geometric properties: it amounts to saying that time evolution conserves area in (q,p) space, which means that you can treat the evolution of many-particle systems as being in a whole block of states at once, treated as a geometric object that flows over time.
I've never been able to understand if there is something "truly fundamental" about H compared to L, or if H is more of a mathematical convenience for making the equations first-order.
In classical mechanics the Lagrangian and Hamiltonian formulations are mostly equivalent (though, confusingly the Lagrangian formulation is due to Hamilton, who has shown that the Lagrange equations can be derived from a variational principle, while the complete Hamiltonian formulation is due to some later phycisists; for what has become the Hamiltonian formulation, Hamilton has shown only how to obtain the system of first order equations from the system of second order equations, but that had already been done before by Cauchy in 1831 in a journal that few were reading, so it was ignored).
Still, in classical mechanics some consider the Hamiltonian formulation to be more fundamental, because the first order equations can be applicable to some problems that have discontinuities incompatible with second-order equations, though such problems are artificial (real systems are continuous enough, strong discontinuities appear only through approximations).
However this changes completely in relativistic mechanics, where the Hamiltonian is not invariant, while the Lagrangian is a relativistic invariant quantity.
This makes the Lagrangian formulation a far better choice in relativistic mechanics and it is a strong argument to consider the Lagrangian formulation as the fundamental one and the Hamiltonian formulation as only an approximation that can be used at small velocities or only as a mathematical trick for numeric solutions.
When the Lagrangian formulation is used, after a coordinate system is chosen, it is always possible to use the Legendre transformation to obtain a Hamiltonian system of first order equations. However, in the relativistic case the system depends on the coordinate system. Therefore, if the coordinate system is changed, the Hamiltonian equations must be derived again from the invariant Lagrangian formulation.
The reason why the Lagrangian is a relativistic invariant is that this scalar value is the projection of the energy-momentum 4-vector on the trajectory curve in space-time. The Hamiltonian is just the temporal component of the 4-vector, which is changed by any coordinate transformation. Therefore L is more fundamental than H, in the same sense that the magnitude of a vector is more fundamental than any of the components that the vector happens to have in some particular coordinate system.
The traditional formulation of the quantum mechanics using H is a serious inconvenience for extending it to the relativistic case. Coherent formulations of the relativistic quantum mechanics must also use L instead of H.
Thank you! The Lagrangian as projection of energy-momentum actually makes sense, unlike the "let's just subtract potential from kinetic. No reason, it just works" story. I'd been idly wondering that for a while (and this is as someone with a physics degree, though in astro, which is a good bit more applied).
Hamilton has introduced what he has called the "Principal Function S", which is used in his variational principle on which the Lagrangian formulation is based.
Nowadays this function is frequently called "Hamilton's action", though this is not a good idea because it causes confusions with what Hamilton, like all his predecessors, called "action", which is the integral of the kinetic energy.
The "Principal Function S", which is a scalar value, i.e. a relativistic invariant quantity, is the line integral of the Lagrangian over the trajectory in space-time, i.e. it is the line integral of the energy-momentum 4-vector over the trajectory in space-time.
Like any line integral of a vector, the line integral of the energy-momentum 4-vector is equal to the line integral over the trajectory of its projection on that trajectory.
This is why the Lagrangian is the projection of the energy-momentum 4-vector. Hamilton has found the correct form of this line integral in relativistic theory, even if that was about 3 quarters of century before the concept of 4-vectors became understood.
The "Principal Function S", i.e. the integral of the energy-momentum, can be considered as a more fundamental quantity than the Lagrangian, which is its derivative (the energy-momentum vector is its gradient). In quantum mechanics the "Principal Function S" is the phase of the wave function, so it is even more obvious that it must be an invariant quantity.
In an earlier answer I gave more information about that resource. To find that earlier answer: go up to the entire thread, and search on the page for my nick: Cleonis
This is the exact point that confused me a lot (and still confuses me) when I tried to read the "The Theoretical Minimum: What You Need to Know to Start Doing Physics" : "Hey, let's just fix/define the lagrangian as T - V and you'll see that after some magical math stuff in the following chapter, we'll find back newtonian equations. Trust me for now".
If anyone has a reference/book/paper that allows you to learn this concept more intuitively, I'd be grateful.
I have created a resource for the purpose of making Hamilton's stationary action transparent.
It is possible to go in all forward steps from F=ma to Hamilton's stationary action; that is what I present.
The path from F=ma to Hamilton's stationary action consists of two stages:
(1) Derivation of the work-energy theorem from F=ma
(2) Demonstration: when the conditions are such that the work-energy theorem holds good then Hamilton's stationary action will hold good also.
These presentations are illustrated with interactive diagrams. Each diagram has one or more sliders for manipulation of the contents of the diagram. That way a single diagram can offer a range of cases/possibilities.
About my approach:
I think of Hamilton's stationary action as an engine with moving parts. To show how an engine works: construct a model out of translucent plastic, so that the student can see all the way inside, and see how all of the moving parts interconnect. My presentation is in that spirit.
Unfortunately there are tons of books about the history of physics that get many details wrong, because apparently the authors have not actually read many of the primary sources, especially when they had not been written in English, but in German, French, Latin or other languages.
Even the original works written in English, like those of Hamilton, pose serious problems when you are not careful, because many words used in physics have changed in meaning during the time, some of them multiple times (e.g. "energy", "action" or "force"). Those who are not aware of this frequently reach wrong conclusions about who has said what.
A few authors have actually read the primary sources, but those typically do not understand physics so well as to be able to distinguish the important concepts from those of little importance, so they are not able to trace the evolution of the important concepts :-(
The only foolproof method to understand the history of physics is to read very carefully the original works (carefully, because the words and the notations used may be very different from those used now). Fortunately, that has become much easier now than before, because there are many online repositories with digitized scientific works from the previous centuries.
For example, "Sir William Rowan Hamilton (1805-1865): Mathematical Papers":
The most important for this thread are (especially the 2nd, which introduces the modern Lagrangian formulation):
1834-04: “On a General Method in Dynamics”, and
1834-10 (but published in 1835): “Second Essay on a General Method in Dynamics”.
The first who has shown how to rewrite the second order system of Lagrange equations into the first order system of equations now called Hamilton's equations (i.e. by using the equivalent of the Legendre transformation) was Poisson in 1809, but the last time when I have searched for that work online I could not find it.
After Poisson, Cauchy has presented in 1831 a method equivalent with that published by Hamilton in 1835. I also could not find online the 1831 work, but an extract of it has been republished in 1837 and this can be found online in many places, for instance at:
Note sur la variation des constantes arbitraires dans les problèmes de Mécanique,
Cauchy, Augustin, pp. 406-412.
You can read some books about the history of physics for a general acquaintance with the authors and the published works from the previous centuries, but you must remain skeptical about any opinions presented there until you read yourself the primary works to verify if they really contain what is claimed about them, or they contain something else.
I have found the reading of many old scientific papers, especially from the 19th century, surprisingly useful for a better understanding of the modern theories that I use now.
It's worth mentioning that Feynman's dissertation was on a Lagrangian formulation of quantum mechanics. However, just because Feynman thought it was interesting doesn't mean it's a good idea for the rest of us (although it's a relatively short 69 pages and an interesting read).
> I've never been able to understand if there is something "truly fundamental" about H compared to L, or if H is more of a mathematical convenience for making the equations first-order.
Actually, Hamiltonian formulation, being equivalent, offers more room for finding solutions. Lagrangian formulation of the Least Action principle allows you to search for a solution employing arbitrary smooth re-parameterizations of the configuration variables `q`. The Hamiltonian formulation, on the other hand, allows you to re-parametrize the entire phase space (q,p) and find solutions that are much harder to get in Lagrangian formulation.
That sounds interesting. Do you know of a particular example where that helps?
I guess maybe it's 'all of them'. The pedagogy on Hamiltonian mechanics had been strangely hard for me to learn beyond the elementary level, like it doesn't make enough sense for my brain to organize it in a memorable way.
